GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA John Rognes February 8th 2005 Abstract. We introduce the notion of a Galois extension of commutative S-* *algebras (E1 ring spectra), often localized with respect to a fixed homology theor* *y. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of* * commu- tative rings, real and complex topological K-theory, Lubin-Tate spectra a* *nd cochain S-algebras. We establish the main theorem of Galois theory in this genera* *lity. Its proof involves the notions of separable (and 'etale) extensions of commut* *ative S- algebras, and the Goerss-Hopkins-Miller theory for E1 mapping spaces. We* * show that the global sphere spectrum S is separably closed (using Minkowski's * *discrimi- nant theorem), and we estimate the separable closure of its localization * *with respect to each of the Morava K-theories. We also define Hopf-Galois extensions * *of com- mutative S-algebras, and study the complex cobordism spectrum MU as a com* *mon integral model for all of the local Lubin-Tate Galois extensions. Contents 1. Introduction 2. Galois extensions in algebra 2.1. Galois extensions of fields 2.2. Regular covering spaces 2.3. Galois extensions of commutative rings 3. Closed categories of structured module spectra 3.1. Structured spectra 3.2. Localized categories 3.3. Dualizable spectra 3.4. Stably dualizable groups 3.5. The dualizing spectrum 3.6. The norm map 4. Galois extensions in topology 4.1. Galois extensions of E-local commutative S-algebras 4.2. The Eilenberg-Mac Lane embedding 4.3. Faithful extensions 5. Examples of Galois extensions 5.1. Trivial extensions ______________ 1991 Mathematics Subject Classification. 13B05, 13B40, 55N15, 55N22, 55P43, * *55P60. Key words and phrases. Galois theory, commutative S-algebra. Typeset by AM S-T* *EX 1 2 JOHN ROGNES 5.2. Eilenberg-Mac Lane spectra 5.3. Real and complex topological K-theory 5.4. The Morava change-of-rings theorem 5.5. The K(1)-local case 5.6. Cochain S-algebras 6. Dualizability and alternate characterizations 6.1. Extended equivalences 6.2. Dualizability 6.3. Alternate characterizations 6.4. The trace map and self-duality 6.5. Smash invertible modules 7. Galois theory I 7.1. Base change for Galois extensions 7.2. Fixed S-algebras 8. Pro-Galois extensions and the Amitsur complex 8.1. Pro-Galois extensions 8.2. The Amitsur complex 9. Separable and 'etale extensions 9.1. Separable extensions 9.2. Symmetrically 'etale extensions 9.3. Smashing maps 9.4. 'Etale extensions 9.5. Henselian maps 9.6. I-adic towers 10. Mapping spaces of commutative S-algebras 10.1.Obstruction theory 10.2.Idempotents and connected S-algebras 10.3.Separable closure 11. Galois theory II 11.1.Recovering the Galois group 11.2.The brave new Galois correspondence 12. Hopf-Galois extensions in topology 12.1.Hopf-Galois extensions of commutative S-algebras 12.2.Complex cobordism References 1. Introduction The present paper is motivated by (1) the "brave new rings" paradigm coined by Friedhelm Waldhausen, that structured ring spectra are an unavoidable gener- alization of discrete rings, with arithmetic properties captured by their algeb* *raic K-theory, (2) the presumption that algebraic K-theory will satisfy an extended form of the 'etale- and Galois descent foreseen by Dan Quillen, and (3) the alg* *ebro- geometric perspective promulgated by Jack Morava, on how the height-stratified moduli space of formal group laws influences stable homotopy theory, by way of complex cobordism theory. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 3 We here develop the arithmetic notion of a Galois extension of structured ri* *ng spectra, viewed geometrically as an algebraic form of a regular covering space,* * by always working intrinsically in a category of spectra, rather than at the na"iv* *e level of coefficient groups. The result is a framework that well accommodates much recent work in stable homotopy theory. We hope that this study will eventually lead to a conceptual understanding of objects like the algebraic K-theory of the sphere spectrum, which by Waldhausen's stable parametrized h-cobordism theorem bears on such seemingly unrelated geometric objects as the diffeomorphism groups of manifolds, in much the same way that we now understand the algebraic K-theory spectrum of the ring of integers. Let E be any spectrum and G a finite group. We say that a map A ! B of E-local commutative S-algebras is an E-local G-Galois extension if G acts on B through commutative A-algebra maps in such a way that the two canonical maps i: A ! BhG and Y h: B ^A B ! B G induce isomorphisms in E*-homology (Definition 4.1.3). When E = S this means that the maps i and h are weak equivalences, and we may talk of a global G-Galo* *is extension. In more detail, the map i is the standard inclusion into the homotopy fixed points for the G-action on B and h is given in symbols by h(b1 ^ b2) = {g* * 7! b1 . g(b2)}. To make the definition homotopy invariant we also assume that A is* * a cofibrant commutative S-algebra and that B is a cofibrant commutative A-algebra. There are many interesting examples of such "brave new" Galois extensions. Examples 1.1. (a) The Eilenberg-Mac Lane functor R 7! HR takes each G-Galois extension R ! T of commutative rings to a global G-Galois extension HR ! HT of commu- tative S-algebras (Proposition 4.2.1). (b) The complexification map KO ! KU from real to complex topological K- theory is a global Z=2-Galois extension (Proposition 5.3.1). (c) For each rational prime p and natural number n the profinite extended Morava stabilizer group Gn = Sn o Gal acts on the even periodic Lubin-Tate spectrum En, with ss0(En) = W(Fpn)[[u1, . .,.un-1 ]], so that LK(n)S ! En is a K(n)-local pro-Gn-Galois extension (Theorem 5.4.4(d)). (d) For most regular covering spaces Y ! X the map of cochain HFp-algebras F (X+ , HFp) ! F (Y+ , HFp) is a Galois extension (Proposition 5.6.3(a)). A map A ! B of commutative S-algebras will be said to be faithful if for each A-module N with N ^A B ' * we have N ' * (Definition 4.3.1). The map A ! B is separable if the multiplication map ~: B ^A B ! B admits a bimodule section up to homotopy (Definition 9.1.1). A commutative S-algebra B is connected (in t* *he sense of algebraic geometry) if its space of idempotents E(B) is weakly equival* *ent to the two-point space {0, 1} (Definition 10.2.1). There are analogous definit* *ions in each E-local context. 4 JOHN ROGNES In commutative ring theory each Galois extension is faithful, but it remains* * an open problem to decide whether each Galois extension of commutative S-algebras is faithful (Question 4.3.6). Rather conveniently, a commutative S-algebra B is connected if and only if the ring ss0(B) is connected (Proposition 10.2.2). Here is our version of the Main Theorem of Galois theory for commutative S- algebras. The first two parts (a) and (b) of the theorem are obtained by specia* *lizing Theorem 7.2.2 and Proposition 9.1.4 to the case of a finite, discrete Galois gr* *oup G. The recovery in (c) of the Galois group is Theorem 11.1.1. The converse part (d) is the less general part of Theorem 11.2.2. Theorem 1.2. Let A ! B be a faithful E-local G-Galois extension. (a) For each subgroup K G the map C = BhK ! B is a faithful E-local K-Galois extension, with A ! C separable. (b) For each normal subgroup K G the map A ! C = BhK is a faithful E-local G=K-Galois extension. If furthermore B is connected, then: (c) The Galois group G is weakly equivalent to the mapping space CA (B, B) of commutative A-algebra self-maps of B. (d) For each factorization A ! C ! B of the G-Galois extension, with A ! C separable and C ! B faithful, there is a subgroup K G such that C ' BhK as an A-algebra over B. In other words, for a faithful E-local G-Galois extension A ! B with B con- nected there is a bijective contravariant Galois correspondence K $ C = BhK be- tween the subgroups of G and the weak equivalence classes of separable A-algebr* *as mapping faithfully to B. The inverse correspondence takes C to K = ss0CC (B, B). The main theorem fully describes the intermediate extensions in a G-Galois extension A ! B, but what about the further extensions of B? We say that a connected E-local commutative S-algebra A is separably closed if there are no connected E-local G-Galois extensions A ! B for non-trivial groups G (Defini- tion 10.3.1). The following fundamental example is a consequence of Minkowski's discriminant theorem in number theory, and is proved as Theorem 10.3.3. Theorem 1.3. The (global) sphere spectrum S is separably closed. The absence of localization is crucial for this result. At the other extrem* *e the K(n)-local category is maximally localized, for each p and n. Here the Lubin-Ta* *te spectrum En admits a K(n)-local pro-n^Z-Galois extension En ! Enrn, with ss0(Enrn) = W(~Fp)[[u1, . .,.un-1 ]] given by adjoining all roots of unity of order prime to p (x5.4.6). We expect that each further G-Galois extension Enrn! B of such a Landweber exact even periodic spectrum must again be Landweber exact and even periodic, and such that ss0(Enrn) ! ss0(B) will be a G-Galois extension of commutative rings. But W(~Fp)[[u1, . .,.un-1 ]] is separably closed as a commutative ring, so such a s* *s0(B) cannot be connected, and B the cannot be connected for non-trivial groups G. Therefore we expect: GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 5 Conjecture 1.4. The extension Enrn of the Lubin-Tate spectrum En is K(n)- locally separably closed. In particular, the Galois group Gnrn= Sn o ^Zof LK(n)* *S ! Enrn is the K(n)-local absolute Galois group of the K(n)-local sphere spectrum LK(n)S. Partial results supporting this conjecture have been obtained by Andy Baker * *and Birgit Richter [BR:r], for global Galois extensions that are furthermore assume* *d to be faithful and abelian. The substantial supply of pro-Galois extensions in the K(n)-local category, * *like LK(n)S ! En, is not available in the E(n)-local category (see x5.5.4). This dra* *ws extra attention to the non-smashing Bousfield localizations, and thus to the di* *stinc- tion between the whole category of modules over LK(n)S and its full subcategory of K(n)-local modules. A study of the sphere spectrum as an algebro-geometric scheme- or stack-like object, that only involves smashing localizations or only* * treats the whole module categories over the various Bousfield localizations, does thus* * not capture these very interesting examples of regular geometric covering spaces. Similarly, the THH- or TAQ-based replacements for K"ahler differentials, in * *the context of associative or commutative S-algebras (see Sections 9.2 and 9.4), ne* *ed not be K(n)-local when applied to K(n)-local S-algebras (see Example 9.2.3). Therefore the notions of (formally) 'etale extensions of associative or commuta* *tive S-algebras will again give a richer theory when considered within the K(n)-local subcategory, rather than in the whole module category over LK(n)S. Thus also a study of the geometry of the sphere spectrum with respect to the 'etale topolog* *y will become more substantial by taking these Bousfield local subcategories into acco* *unt. This phenomenon differs from that which is familiar in discrete algebraic geome* *try, since there all localizations are, indeed, smashing. The (mono-)chromatic localizations LK(n)S of the sphere are of course even m* *ore drastic than the p-localizations S(p), so that many of the principal examples s* *tudied in this paper are of an even more local nature than e.g. local number fields. * *But the arithmetic properties of a global number field can usefully be studied by a* *d`elic means, in terms of the system of local number fields that can be obtained from it by the various completions that are available. We are therefore also intere* *sted in finding global models for the system of naturally occurring K(n)-local Galois extensions of LK(n)S, for varying p and n. The obvious candidate, given Quillen's discovery of the relation of formal g* *roup law theory to complex cobordism, is the unit map S ! MU to the complex cobor- dism spectrum. The following statement is proved in Corollary 9.6.6, Proposi- tion 12.2.1 and the discussion surrounding diagram (12.2.6). In the second par* *t, S[BU] is the commutative S-algebra 1 BU+ . In summary, MU is very close to such a global model, up to formal thickenings by Henselian maps. This makes the author inclined to think of S ! MU as a kind of (large) ramified global Galois extension, with S[BU] playing the part of the functional dual of its imaginary * *Ga- lois group. To make good sense of this, we introduce the notion of a Hopf-Galois extension of commutative S-algebras in Section 12.1. Theorem 1.5. For each prime p and integer n 0 the K(n)-local pro-Gn-Galois extension LK(n)S ! En factors as the composite of the following maps of commu- 6 JOHN ROGNES tative S-algebras LK(n)S ! LMUK(n)MU -q!E[(n) ! En . Here the first map admits the global model S ! MU, by Bousfield K(n)-localizati* *on in the categories of S-modules and MU-modules, respectively. The second map q is a formal thickening, or more precisely, symmetrically (and possibly commutative* *ly) Henselian. The third map is a finite Galois extension (and can be avoided by pa* *ssing to the even periodic version MUP of MU and adjoining some roots of unity). Furthermore, the global model S ! MU is an S[BU]-Hopf-Galois extension of commutative S-algebras, with coaction fi :MU ! MU ^ S[BU] given by the Thom diagonal. For each element g 2 Gn its Galois action on En can be directly recov* *ered from this S[BU]-coaction, up to the adjunction of some roots of unity. Here are some more detailed references into the body of the paper. Chapter 2 contains a review of the basic Galois theory for fields and for co* *mmu- tative rings, together with some algebraic facts that we will need for our exam* *ples. We also make a comparison with the theory of regular covering spaces, for the benefit of the topologically minded reader. As hinted at above, we sometimes consider more general Galois groups G than finite (and profinite) groups. For the initial theory, all that is required is* * that the unreduced suspension spectrum S[G] = LE 1 G+ admits a good Spanier- Whitehead dual in the E-local stable homotopy category, i.e., that G is stably dualizable (Definition 3.4.1). We review the basic properties of stably dualiz* *able groups and their actions on spectra in Chapter 3, referring to the author's pa- per [Ro:s] for most proofs. This chapter also contains a discussion of the var* *ious categories of E-local S-modules and (commutative) S-algebras in which we work. The precise Definition 4.1.3 of a Galois extension of commutative S-algebras* * is given in Chapter 4, followed by a discussion showing that the Eilenberg-Mac Lane embedding from commutative rings preserves and detects Galois extensions (Propo- sition 4.2.1). We also consider the elementary properties of faithful modules * *over structured ring spectra, flatness being implicit in our homotopy invariant work* *. We shall often make use of how various algebro-geometric properties of S-algebras * *are preserved by base change, or are detected by suitable forms of faithful base ch* *ange. Chapter 5 is devoted to the many examples of Galois extensions mentioned above, including all the intermediate K(n)-local Galois extensions between LK(n* *)S and the maximal unramified extension Enrnof En. We also go through the K(1)- local case of the Lubin-Tate extensions in much detail, making explicit the clo* *se analogy with the classification of abelian extensions of the p-adic and rationa* *l fields Qp and Q. Finally we extend the example of cochain algebras of regular covering spaces to cochain algebras of principal G-bundles P ! X, for stably dualizable groups G. Chapter 6 develops the formal consequences of the Galois conditions on A ! B, including the basic fact that B is a dualizable A-module (Proposition 6.2.1), t* *wo useful alternate characterizations of (faithful) Galois extensions (Proposition* *s 6.3.1 and 6.3.2), and two further characterizations of faithfulness (Proposition 6.3.* *3 and Lemma 6.5.3). These let us prove in Chapter 7 that faithful Galois extensions a* *re preserved by arbitrary base change (Lemma 7.1.1) and are detected by faithful a* *nd GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 7 dualizable base change (Lemma 7.1.4(b)). From these results, in turn, the "forw* *ard" part of the Galois correspondence (Theorem 7.2.2) follows rather formally, sayi* *ng that for a faithful G-Galois extension A ! B the homotopy fixed point spectra C = BhK give rise to K-Galois extensions C ! B for subgroups K G, and to G=K-Galois extensions A ! C when K is normal. When this much of the Galois correspondence has been established, we can make sense of the notion of a pro-Galois extension, which we do somewhat informally * *in Section 8.1. The "converse" part of the Galois correspondence (Theorem 11.2.2) relies on * *the possibility of recovering the Galois group G in a G-Galois extension A ! B from the space CA (B, B) of commutative A-algebra self-maps B ! B, or more generally, to recover the subgroup K from the mapping space CC (B, B), when C = BhK is a fixed S-algebra of B (Proposition 11.2.1). This is achieved in Chapter 11, b* *ut relies on three preceding developments. First of all, we use the commutative form of the Hopkins-Miller theory, as d* *evel- oped by Paul Goerss and Mike Hopkins [GH04], to study such mapping spaces. We use an extension of their work, from dealing with spaces of E1 ring spectrum m* *aps, or commutative S-algebra maps, to spaces of commutative A-algebra maps. This is discussed in Section 10.1, where we also touch on the consequences for this the* *ory of working E-locally. The main computational tool is the Goerss-Hopkins spec- tral sequence (10.1.4), whose E2-term involves suitable Andr'e-Quillen cohomolo* *gy groups, which fortunately vanish in all relevant cases for the Galois extension* *s we consider. Second, the recovery of the Galois group G from CA (B, B) only has a chance, judging from the discrete algebraic case, when B is connected in the geometric sense that it has no non-trivial idempotents. For a commutative S-algebra B the* *re is a space E(B) of idempotents, which in turn is a commutative B-algebra mapping space of the sort that can be studied by the Goerss-Hopkins spectral sequence. So in Section 10.2 we treat connectivity in this geometric sense for commutative S-algebras, reaching a convenient algebraic criterion in Proposition 10.2.2. T* *his also lets us define separably closed commutative S-algebras in Section 10.3. Thirdly, not all commutative A-algebras C mapping faithfully to B occur in t* *he Galois correspondence as fixed S-algebras C = BhK . As in the discrete algebraic case, the characteristic property is that C is separable over A, and in Section* * 9.1 we develop the basic theory of separable extensions of S-algebras. As further g* *en- eralizations of separable maps we have the 'etale maps, which we discuss in thr* *ee related contexts in Sections 9.2 through 9.4, leading to the notions of symmetr* *ically (=thh-)'etale, smashing and (commutatively) 'etale maps of S-algebras, respecti* *vely. Topological Hochschild homology THH controls the K"ahler differentials in the associative setting, while topological Andr'e-Quillen homology TAQ takes on the same r^ole in the purely commutative setting. Our discussion here relies heavil* *y on the work of Maria Basterra [Ba99] and Andrej Lazarev [La01]. There is much con- ceptual overlap between the triviality of the topological Andr'e-Quillen homolo* *gy spectrum TAQ(B=A) for (formally) 'etale maps A ! B, and the vanishing of the Goerss-Hopkins Andr'e-Quillen cohomology groups DsB*T(BA*(B), tB) for finite Galois extensions A ! B, but the direct connection is not as well understood as 8 JOHN ROGNES might be desired. The remainder of the paper is concerned with the interpretation of S ! MU as a Hopf-Galois extension that provides an integral model, up to Henselian map* *s, for all of the Lubin-Tate extensions LK(n)S ! En. Thus we consider square-zero extensions, singular extensions and Henselian maps as various forms of infinite* *simal and formal thickenings in Section 9.5. We then obtain a good supply of relevant examples in Section 9.6, using work of Baker and Lazarev on I-adic towers. We have already cited Corollary 9.6.6 as relevant for part of Theorem 1.5. The idea of Hopf-Galois extensions is to replace the action by the Galois gr* *oup G on a commutative A-algebra B by a coaction by the functional dual DG+ = F (G+ , S) of the Galois group, which is a commutative Hopf S-algebra. In the algebraic situation such coactions have been useful, e.g. to classify inseparab* *le Ga- lois extensions of fields [Ch71]. In the absence of an actual Galois group, the condition that i: A ! BhG is a weak equivalence must be rewritten, by using a cosimplicial resolution for the coaction (the Hopf cobar complex), in place of * *the homotopy fixed points. This rewriting can naturally go through a second cosimpl* *i- cial resolution associated to A ! B, which we know as the Amitsur complex. We discuss the Amitsur complex in Section 8.2, so as to have the accompanying noti* *on of completion of A along B available in Chapter 10, and give the definitions of* * the Hopf cobar complex and of Hopf-Galois extensions in Section 12.1. To conclude the paper, in Section 12.2 we go through some of the details of * *how the inseparable extension S ! MU is an S[BU]-Hopf-Galois extension, and how the Hopkins-Miller theory and the Lubin-Tate deformation theory work together to show that the global S[BU]-coaction on MU captures the Morava stabilizer group action on En, at all primes p and chromatic heights n. Acknowledgments. The study of idempotents in Chapter 10 first got going during an Oberwolfach hike with Neil Strickland, and the proper use of separability in Chapter 11 was* * at last found in a discussion with Birgit Richter. I am very grateful for their in* *terest. Most of this work was done in the year 2000 and announced at various confer- ences. I apologize for the long delay in publication, which for much of the tim* *e was due to the unresolved Question 4.3.6, on the faithfulness of Galois extensions. 2. Galois extensions in algebra 2.1. Galois extensions of fields. We first recall the basics about Galois extensions of fields. Let G be a fi* *nite group acting effectively (only the unit element acts as the identity) from the * *left by automorphisms on a field E, and let F = EG be the fixed subfield. Let j :E ! Hom F (E, E) be the canonical associative ring homomorphism taking e1g to the homomorphism e2 7! e1 . g(e2), from the twisted group ring of G over E to the F -module endo* *mor- phisms of E. Then j is an isomorphism, for by Dedekind's lemma j is injective, and dim F(E) equals the order of G, so j is also surjective by a dimension coun* *t. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 9 See [Dr95, App.] for elementary proofs. Let Y h: E F E ! E G be the canonical commutative ring homomorphism taking e1 e2 to the sequence {g 7! e1 . g(e2)}, from the tensor product of two copies of E over F to the pro* *d- uct of G copies of E. Then also h is an isomorphism, for it is the E-module dual ofQj, by way of the identifications Hom E(E F E, E) ~= Hom F(E, E) and Hom E( G E, E) ~=E (using that G is finite). 2.2. Regular covering spaces. There is a parallel geometric theory of regular (= normal) covering spaces [* *Sp66, 2.6.7], [Ha02, 1.39]. Let G be a finite discrete group acting from the right o* *n a compact Hausdorff space Y . Let X = Y=G be the orbit space, and let ss :Y ! X be the orbit projection. There is a canonical map , :Y x G ! Y xX Y (to the fiber product of ss with itself), taking (y, g) to (y, y . g). This map* * is always surjective, by the definition of X as an orbit space, and it is injective if an* *d only if G acts freely on Y . So , is a homeomorphism if and only if Y ! X is a regular co* *vering space, with G as its group of deck transformations, acting freely and transitiv* *ely on each fiber. In general, the possible failure of , to be injective measures the * *extent to which G does not act freely on Y , which in turn can be interpreted as a mea* *sure of to what extent Y is ramified as a cover of X. The theory of Riemann surfaces provides numerous examples of the latter phenomenon. Dually, let R = C(X) and T = C(Y ) be the rings of continuous (real or compl* *ex) functions on X and Y , respectively. As usual the points of X can be recovered * *as the maximal ideals in R, and similarly for Y . The group G acts from the left o* *n T , by the formula g(t) = g * t: y 7! t(y . g), and the natural map R ! T dual to ss identifies R with the invariant ring T G, by the isomorphism C(Y )G ~=C(Y=G). The map , above is dual to the canonical homomorphism Y h: T R T ! T G takingQt1 t2 to the function g 7! t1 . g(t2), considered as an element in the* * product G T . Then , is a homeomorphism if and only if h is an isomorphism, by the categorical anti-equivalence between compact Hausdorff spaces and their function rings. The surjectivity of , ensures that h is always injective, and in genera* *l the possible failure of h to be surjective measures the extent of ramification in Y* * ! X. For a moment, let us also consider the more general case of a principal G-bu* *ndle ss :P ! X for a compact Hausdorff topological group G. The map , :P x G ! P xX P is a homeomorphism, now with respect to the given topology on G. Let R = C(X), T = C(P ) and H = C(G). Then H is a commutative Hopf algebra with coproduct _ :H ! H H, if the map H ! C(G x G) dual to the group multiplication G x G ! G factors through the canonical map H H ! C(G x G). 10 JOHN ROGNES Likewise H coacts on T from the right by fi :T ! T H, if the map T ! C(P xG) induced by the group action P xG ! P factors through T H ! C(P xG). These factorizations can always be achieved by using suitably completed tensor produc* *ts, but we wish to refer to the algebraic tensor products here. Then the freeness o* *f the group action on P is expressed by saying that the composite map h: T R T -1-fi!T R T H -~-1! T H is an isomorphism. We shall return to this dualized context in Chapter 12 on Hopf-Galois extensions. 2.3. Galois extensions of commutative rings. Generalizing the two examples above, for finite Galois groups, Auslander and Goldman [AG60, App.] gave a definition of Galois extensions of commutative rings as part of their study of separable algebras over such rings. Chase, Harrison * *and Rosenberg [CHR65, x1] found several other equivalent definitions, and developed the Galois theory for commutative rings to also encompass the fundamental Galois correspondence. We now recall their basic results. Let R ! T be a homomorphism of commutative rings, making T a commutative R-algebra, and let G be a finite group acting on T from the left through R-alge* *bra homomorphisms. Let i: R ! T G be the inclusion into the fixed ring, let Y h: T R T ! T G be the commutative ring homomorphism that takes t1 t2 to the sequence {g 7! t1 . g(t2)}, and let j :T ! Hom R (T, T ) be the associative ring homomorphismQthat takes t1g to the R-module homomor- phism t2 7! t1 . g(t2). We give G T the pointwise product (tg)g . (t0g)g = (t* *gt0g)g and T the twisted product t1g1 . t2g2 = t1g1(t2)g1g2, using the left G-acti* *on on T . Definition 2.3.1. Let G act on T over R, as above. We say that R ! T is aQG- Galois extension of commutative rings if both i: R ! T G and h: T R T ! G T are isomorphisms. Here we are following Greither [Gr92, 0.1.5]. Auslander and Goldman [AG60, p. 396] instead took the condition below on i, j and T to be the defining prope* *rty, but Chase, Harrison and Rosenberg [CHR65, 1.3] proved that the two definitions are equivalent. Proposition 2.3.2. Let G act on T over R, as above. Then R ! T is a G- Galois extension if and only if both i :R ! T G and j :T ! Hom R(T, T ) are isomorphisms and T is a finitely generated projective R-module. The condition that i is an isomorphism means that we can speak of R as the f* *ixed ring of T . The homomorphism h measures to what extent the extension R ! T is ramified, and Galois extensions are required to be unramified. The injectivity * *of j is a form of Dedekind's lemma, and ensures that the action by G is effective. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 11 Example 2.3.3. If K ! L is a G-Galois extension of number fields, then the corresponding extension R = OK ! OL = T of rings of integers is a G-Galois extension of commutative rings if and only if K ! L is unramified as an extensi* *on of number fields [AB59]. More generally, if is a set of prime ideals in OK ,* * and 0 the set of primes in OL above those in , then the extension OK, ! OL, 0 of rings of -integers is G-Galois if and only if contains all the primes that r* *amify in L=K [Gr92, 0.4.1]. Thus OK ! OL becomes a G-Galois extension precisely upon localizing away from (= inverting) the ramified primes. To see this, note that if T = R{t1, . .,.tn} is a free R-module of rank n, t* *hen T R T is a free T -module on the generators 1 t1, . .,.1 tn, and h is repr* *esented as a T -module homomorphism by the square matrix A = (g(ti))g,iof rank n, with g 2 G and i = 1, . .,.n. The discriminant of T=R is d = det(A)2, by definition,* * and the prime ideals in OK that ramify in L=K are precisely the prime ideals divid* *ing the discriminant. So h is an isomorphism if and only if det(A) and d are units in R, or equivalently, if there are no ramified primes. A local version of the * *same argument works when T is not free over R. Here are some further basic properties of Galois extensions of commutative r* *ings, which will be relevant to our discussion. Proposition 2.3.4. Let R ! T be a G-Galois extension. Then: (a) T is faithfully flat as an R-module, i.e., the functor (-) R T preserve* *s and detects (=reflects) exact sequences. P (b) The trace map tr :T ! R (taking t 2 T to g2G g(t) 2 T G = R) is a split surjective R-module homomorphism. (c) T is invertible as an R[G]-module, i.e., a finitely generated projective* * R[G]- module of constant rank 1. For proofs, see e.g. [Gr92, 0.1.9], [Gr92, 0.1.10] and [Gr92, 0.6.1]. Bewar* *e that part (b) does not extend well to the topological setting, as Example 6.4.4 demo* *n- strates. 3. Closed categories of structured module spectra 3.1. Structured spectra. We now adapt these ideas to the context of "brave new rings," i.e., of commu- tative S-algebras. These can be interpreted as the commutative monoids in either one of the popular symmetric monoidal categories of structured spectra, such as the S-modules of Elmendorf, Kriz, Mandell and May [EKMM97], the symmetric spectra of Hovey, Shipley and Smith [HSS00] or the simplicial functors of Segal* * and Lydakis [Ly98], according to the reader's needs or preferences. But to be concr* *ete, and to have a convenient source for the more technical references, we shall work with the S-modules of Peter May et al. Let S be the sphere spectrum, and let MS be the category of S-modules. Among other things, it is a topological category with all limits and colimits and all* * topo- logical tensors and cotensors. A map f :X ! Y of S-modules is called a weak equivalence if the induced homomorphism ss*(f): ss*(X) ! ss*(Y ) of stable homo- topy groups is an isomorphism. The category DS obtained from MS by inverting 12 JOHN ROGNES the weak equivalences is called the stable homotopy category, and is equivalent* * to the homotopy category of spectra constructed by Boardman [Vo70]. The smash product X ^ Y and function object F (X, Y ) make MS a closed symmetric monoidal category, with S as the unit object. For each topological space T the topological tensor X ^ T+ equals the smash product X ^ S[T ], and the topological cotensor Y T = F (T+ , Y ) equals the function spectrum F (S[T * *], Y ), where S[T ] = 1 T+ denotes the unreduced suspension S-module on T . An (associative) S-algebra A is a monoid in MS , i.e., an S-module A equipped with a unit map j :S ! A and a unital and associative multiplication ~: A^A ! A. A commutative S-algebra A is a commutative monoid in MS , i.e., one such that the multiplication ~ is also commutative. We write AS and CS for the categories of S-algebras and commutative S-algebras, respectively. More generally, for a c* *om- mutative S-algebra A we write MA , AA and CA for the categories of A-modules, associative A-algebras and commutative A-algebras, respectively [EKMM97, VII.1]. 3.2. Localized categories. Our first examples of Galois extensions of structured ring spectra will be m* *aps A ! B of commutative S-algebras, with a finite group G acting on B through A- algebraQmaps, such that there are weak equivalences i: A ' BhG and h: B ^A B ' G B. The formal definition appears in Section 4.1 below. However, there are interesting examples that only appear as Galois extensions to the eyes of weaker invariants than the stable homotopy groups ss*(-). More precisely, for a fixed homology theory E*(-) we shall allow ourselves to work in the E-local stable homotopy category, where have arranged that each map f :X ! Y such that E*(f): E*(X) ! E*(Y ) is an isomorphism, is in fact a weak equivalence. In particular, we will encounter situations where we only have that E*(i) and E*(h) are isomorphisms, in which case we shall interpret A ! B as an E-local G-Galois extension. Note the close analogy between the E-local theory and the case (Example 2.3.* *3) of rings of integers localized away from some set of primes. Doug Ravenel's inf* *luen- tial treatise on the chromatic filtration of stable homotopy theory [Ra84, x5],* * brings emphasis to the tower of cases when E = E(n), the n-th Johnson-Wilson spectrum. To us, the most interesting case is when E = K(n) is the n-th Morava K-theory spectrum. The K(n)-local stable homotopy category is studied in detail in [HSt9* *9, xx7-8], and captures the n-th layer, or stratum, in the chromatic filtration. Definition 3.2.1. Let E be a fixed S-module, with associated homology theory X 7! E*(X) = ss*(E ^ X). By definition, an S-module Z is said to be E-acyclic if E ^ Z ' * (so E*(Z) = 0), and an S-module Y is said to be E-local if F (Z, Y ) * *' * for each E-acyclic S-module Z (so [Z, Y ]* = 0). Let MS,E MS be the full subcategory of E-local S-modules. A map f :X ! Y of E-local S-modules is a weak equivalence if and only if it is an E*-equivalence, i.e., if E*(f) is an isomor* *phism. There is a Bousfield localization functor LE :MS ! MS,E MS [Bo79], [EKMM97, VIII.1.6], and an accompanying natural E*-equivalence X ! LE X for each S-module X. We may assume that this E*-equivalence is the identity when X is already E-local, so that the localization functor LE is idempotent. T* *he homotopy category DS,E of MS,E is the E-local stable homotopy category. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 13 More generally, for a commutative S-algebra A we let MA,E MA be the full subcategory of E-local A-modules, with homotopy category DA,E . To be precise, there is an A-module FA E of the homotopy type of A^E, and a localization funct* *or LAFA E:MA ! MA,E , with respect to FA E in the category of A-modules, which amounts to E-localization at the level of the underlying S-modules [EKMM97, VIII.1.7]. We shall allow ourselves to simply denote this functor by LE . Notation 3.2.2. We write LnX = LE(n)X for the Bousfield localization of X with respect to the Johnson-Wilson spectrum E(n) [JW73], with ss*E(n) = Z(p)[v1, . .,.vn-1 , vn1 ], for each non-negative i* *nte- ger n, and LK(n)X for the Bousfield localization of X with respect to the Morava K-theory spectrum K(n) [JW75], with ss*K(n) = Fp[vn1 ], for each natural number n. The smash product X ^ Y of two E-local S-modules will in general not be E- local, although this is the case when LE is a so-called smashing localization,* * i.e., one that commutes with direct limits [Ra84, 1.28]. The Johnson-Wilson spectra E = E(n) provide interesting examples of smashing localizations Ln = LE(n) [Ra9* *2, 7.5.6], while localization LK(n) with respect to the Morava K-theories E = K(n) is not smashing [HSt99, 8.1]. Likewise, the unit S for the smash product is rar* *ely E-local. So in order to work with S-algebras and related constructions interna* *lly within MS,E, we first perform each construction as usual in MS , and then apply the Bousfield localization functor LE . Definition 3.2.3. We implicitly give MS,E all colimits, topological tensors, s* *mash products and a unit object by applying Bousfield localization to the constructi* *ons in MS . So colimi2IXi means LE (colimi2I Xi), X ^ Y means LE (X ^ Y ), S means LE S and S[T ] means LE 1 T+ . All limits, topological cotensors and function objects formed from E-local S-modules are already E-local, so no Bousfield loca* *l- ization is required in these cases. With these conventions, MS,E is a topologi* *cal closed symmetric monoidal category with all limits and colimits. The same consi* *d- erations apply for MA,E . There is a natural map LE X ^ LE Y ! LE (X ^ Y ), making LE a lax monoidal functor, so that LE S is always a commutative S-algebra. When E is smashing, the category MS,E of E-local S-modules is equivalent (at the level of homotopy categories) to the category MLE S of LE S-modules, so the study of E-local S- modules is a special case of the study of modules over a general commutative S- algebra A = LE S. However, when E is not smashing, as is the case for E = K(n), the two homotopy categories are not equivalent, and we shall need to consider t* *he more general notion. When E = S, every S-module is E-local and MS,E = MS , etc., so the E-local context specializes to the "global", unlocalized situation. For brevity, we sha* *ll often simply refer to the E-local S-modules as S-modules, or even as spectra, but exc* *ept where we explicitly assume that E = S, the discussion is intended to encompass also the general E-local case. 14 JOHN ROGNES Remark 3.2.4. By analogy with algebraic geometry, we may (heuristically) wish to view A-modules M as suitable sheaves M~ over some geometric "structure space" Spec A. This structure space would come with a Zariski topology, with (open?) subspaces UA,E Spec A corresponding to the various localization functors LE on the category of A-modules, in such a way that the restriction of the sheaf M~ over Spec A to the subspace UA,E would be the sheaf (LE M)~ corresponding to the E-local A-module LE M. For smashing E this would precisely amount to an LE A-module, so that UA,E could be identified with the structure space Spec LE * *A. However, for non-smashing E the condition of being an E-local A-module is strictly stronger than being an LE A-module. Therefore, the geometric structure on Spec A is not simply that of an "S-algebra'ed space" carrying the (commutati* *ve) S-algebra LE A over UA,E , by analogy with the ringed spaces of algebraic geome* *try. If we wish to allow non-smashing localizations E to correspond to Zariski opens, then the geometric structure must also capture the additional restriction it is* * for an LE A-module to be an E-local A-module. This exhibits a difference compared to the situation in commutative algebra, where localization at an ideal commutes w* *ith direct limits, and behaves as a smashing localization, while completions behave more like non-smashing localizations. It does not seem to be so common to do commutative algebra in such implicitly completed situations, however. A continuation of this analogy would be to consider other Grothendieck-type topologies on Spec A, with coverings built from E-local Galois extensions LE A * *! B (Definition 4.1.3) or more general 'etale extensions (Definition 9.4.1), subjec* *t to a combined faithfulness condition (Definition 4.3.1). In the unlocalized cases, s* *uch a (big) 'etale site on the opposite category of CS , and associated small 'etale * *sites on the opposite category of each CA , have been developed by Bertrand To"en and Gabrie* *le Vezzosi [TV:h, x5.2]. However, the rich source of K(n)-local Galois extensions* * of LK(n)S discussed in Section 5.4 provides, by Lemma 9.4.4, an equally rich supply of K(n)-local 'etale maps from LK(n)S. It appears, by extension from the case n = 1 discussed in Section 5.5, that these are not globally 'etale maps, in whi* *ch case the 'etale topology proposed in [TV:h] will be too coarse to encompass the* *se examples. The author therefore thinks that a finer 'etale site, taking non-smas* *hing localizations like LK(n) into account, would lead to a stronger and more intere* *sting theory. 3.3. Dualizable spectra. In each closed symmetric monoidal category there is a canonical natural map :F (X, Y ) ^ Z ! F (X, Y ^ Z) . It is right adjoint to a map ffl ^ 1: X ^ F (X, Y ) ^ Z ! Y ^ Z, where the adju* *nction counit ffl: X ^ F (X, Y ) ! Y is left adjoint to the identity map on F (X, Y ). Dold and Puppe [DP80] say that an object X is strongly dualizable if the can* *on- ical map :F (X, Y ) ^ Z ! F (X, Y ^ Z) is an isomorphism for all Y and Z. Lew* *is, May and Steinberger [LMS86, III.1.1] say that a spectrum X is finite if it is s* *trongly dualizable in the stable homotopy category, i.e., if the map is a weak equiva* *lence. We shall instead follow Hovey and Strickland [HSt99, 1.5(d)] and briefly call s* *uch spectra dualizable. By [LMS86, III.1.3(ii)] it suffices to verify this conditio* *n in the GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 15 special case when Y = S and Z = X, so we take this simpler criterion as our definition. Definition 3.3.1. Let DX = F (X, S) be the functional dual of X. We say that X is dualizable if the canonical map :DX ^ X ! F (X, X) is a weak equiva- lence. More generally, for an (implicitly E-local) module M over a commutative S-algebra A, let DA M = FA (M, A) be the functional dual, and say that M is a dualizable A-module if the canonical map :DA M ^A M ! FA (M, M) is a weak equivalence. Lemma 3.3.2. (a) If X or Z is dualizable, then the canonical map :F (X, Y ) ^ Z ! F (X, Y ^ Z) is a weak equivalence. (b) If X is dualizable, then DX is also dualizable and the canonical map ae * *:X ! DDX is a weak equivalence. (c) The dualizable spectra generate a thick subcategory, i.e., they are clos* *ed under passage to weakly equivalent objects, retracts, mapping cones and (de-)suspensi* *ons. Here ae: X ! DDX = F (F (X, S), S) is right adjoint to F (X, S)^X ! S, which is obtained by twisting the adjunction counit ffl: X ^ F (X, S) ! S. For proofs* *, see [LMS86, III.1.2 and III.1.3]. We sometimes also use to label the conjugate map Y ^F (X, Z) ! F (X, Y ^Z). The corresponding results hold for E-local A-modules, by the same formal proofs. One justification for the term "finite" is the following converse to Lemma 3* *.3.2(c), in the unlocalized setting E = S. Proposition 3.3.3. Let A be commutative S-algebra. A global A-module M is dualizable in MA = MA,S if and only if it is weakly equivalent to a retract of* * a finite cell A-module. The proof [EKMM97, III.7.9] uses in an essential way that stable homotopy X 7! ss*(X) = [A, X]A*commutes with coproducts, which amounts to A being small in the homotopy category DA of A-modules. This fails in some E-local contexts. For example, the K(n)-local sphere spectrum LK(n)S is not small in the K(n)-loc* *al category [HSt99, 8.1], and consequently ss*(X) is not a homology theory on this category. So in general there will be more dualizable E-local A-modules than the semi-finite ones, i.e., the retracts of the finite cell LE A-modules. In this p* *aper we shall prefer to focus on the notion of dualizability, rather than on being semi* *-finite, principally because of Proposition 6.2.1 and (counter-)Example 6.2.2 below. 3.4. Stably dualizable groups. For our basic theory of G-Galois extensions of commutative S-algebras the gr* *oup action by G appears through the module action by its suspension spectrum S[G] = LE 1 G+ , and the finiteness condition on G only enters through the property t* *hat S[G] is a dualizable spectrum. We then say that G is an E-locally stably dualiz* *able group. Only when we turn to properties related to separability will it be relev* *ant that G is discrete, and then usually finite. So we shall develop the basic theo* *ry in the greater generality of stably dualizable topological groups G. Definition 3.4.1. A topological group G is E-locally stably dualizable if its * *sus- pension spectrum S[G] = LE 1 G+ is dualizable in MS,E. Writing DG+ = 16 JOHN ROGNES F (G+ , LE S) for its functional dual, the condition is that the canonical map :DG+ ^ S[G] ! F (S[G], S[G]) is a weak equivalence in the E-local category. Examples 3.4.2. (a) Each compact Lie group G admits the structure of a finite CW complex, so S[G] is a finite cell spectrum and G is stably dualizable, for e* *ach E. (b) The Eilenberg-Mac Lane spaces G = K(Z=p, q) are loop spaces and thus admit models as topological groups. They have infinite mod p homology for each q 1, so S[G] is never dualizable in MS by Proposition 3.3.3. However, the Mor* *ava K-homology K(n)*K(Z=p, q) is finitely generated over K(n)* by a calculation of Ravenel and Wilson [RW80, 9.2], so G = K(Z=p, q) is in fact K(n)-locally stably dualizable by [HSt99, 8.6]. We are curious to see if these and similar topolog* *ical Galois groups play any significant r^ole in the K(n)-local Galois theory. 3.5. The dualizing spectrum. W Q The weak equivalence S[G] = G S ! GS = DG+ for a finite group G generalizes to an E-local self-duality of the suspension spectrum S[G], when G * *is an E-locally stably dualizable group. The self-duality holds up to a twist by a so* *-called dualizing spectrum SadG . When G is a compact Lie group this is the suspension spectrum on the one-point compactification of the adjoint representation adG of* * G, thus the notation, and so SadG = S for G finite. John Klein [Kl01, x1] introduc* *ed dualizing spectra SadG for arbitrary topological groups, and Tilman Bauer [Ba0* *4, 4.1] established the twisted self-duality of S[G] in the p-complete category, w* *hen G is a p-compact group in the sense of Bill Dwyer and Clarence Wilkerson [DW94]. In [Ro:s] we have extended these results to all E-locally stably dualizable gro* *ups, as we now review. Definition 3.5.1. Let G be an E-locally stably dualizable group. The group multiplication provides the suspension spectrum S[G] = LE 1 G+ with mutually commuting left and right G-actions. We define the dualizing spectrum SadG to be the G-homotopy fixed point spectrum SadG = S[G]hG = F (EG+ , S[G])G of S[G], formed with respect to the right G-action [Ro:s, 2.4.1]. Here EG = B(*, G, G) is the standard free, contractible right G-space. The remaining left action on S[G] induces a left G-action on SadG . When G is finite, there is a natural weak equivalence SadG = S[G]hG ' DGhG+' S . Here the last equivalence involves the collapsing homotopy equivalence c: EG ! * **, which is a G-equivariant map, but not a G-equivariant homotopy equivalence. For general stably dualizable groups G, the dualizing spectrum is indeed dualizable and smash invertible [Ro:s, 3.2.3 and 3.3.4], so smashing with SadG induces an equivalence of derived categories. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 17 The left G-action on S[G] functorially dualizes to a right G-action on DG+ , with module action map ff :DG+ ^ S[G] ! DG+ . The group multiplication on G induces a coproduct _ :S[G] ! S[G] ^ S[G], using [EKMM97, II.1.2]. These combine to a shear map sh: DG+ ^ S[G] -1^_-!DG+ ^ S[G] ^ S[G] -ff^1-!DG+ ^ S[G] , which is equivariant with respect to each of three mutually commuting G-actions [Ro:s, 3.1.2] and is a weak equivalence [Ro:s, 3.1.3]. Taking homotopy fixed po* *ints with respect to the right action of G on S[G] in the source and the diagonal ri* *ght action on DG+ and S[G] in the target induces a natural Poincar'e duality equiva* *lence [Ro:s, 3.1.4] (3.5.2) DG+ ^ SadG -'! S[G] . This identification uses the stable dualizability of G, and expresses the twist* *ed self- duality of S[G]. The weak equivalence is equivariant with respect to both a lef* *t and a right G-action. The left G-action is by the inverse of the right action on DG* *+ , the standard left action on SadG and the standard left action on S[G]. The rig* *ht G-action is by the inverse of the left action on DG+ , the trivial action on Sa* *dG and the standard right action on S[G]. 3.6. The norm map. Let X be any E-local S-module with left G-action, and equip it with the triv* *ial right G-action. The smash product X ^ S[G] then has a diagonal left G-action, a* *nd a right G-action that only affects S[G]. Consider forming homotopy orbits (-)hG with respect to the left action and forming homotopy fixed points (-)hG with respect to the right action, in either order. There is then a canonical colimit* */limit exchange map ~: ((X ^ S[G])hG )hG ! ((X ^ S[G])hG )hG . The source of ~ receives a weak equivalence from (X ^SadG )hG (this uses the st* *able dualizability of G; see the proof of Lemma 6.4.2), and the target of ~ maps by a weak equivalence to XhG (this is easy). The composite of these three maps is t* *he (homotopy) norm map [Ro:s, 5.2.1] (3.6.1) N :(X ^ SadG )hG ! XhG . If X = W ^G+ = W ^S[G] for some spectrum W with left G-action, with G acting in the standard way on S[G], then the norm map for X is a weak equivalence [Ro:* *s, 5.2.4]. That reference only discusses the case when G acts trivially on W , bu* *t in general there is an equivariant equivalence i :w ^ g 7! g(w) ^ g from W ^ S[G] * *with G acting only on S[G] to W ^ S[G] with the diagonal G-action. We can define the G-Tate construction XtG to be the cofiber of the norm map (X ^ SadG )hG -N! XhG -! XtG . Then XtG ' * if and only if N is a weak equivalence, which in turn holds if and only if the exchange map ~ is a weak equivalence. From this point of view, XtG * *is the obstruction to the commutation of the G-homotopy orbit and the G-homotopy fixed point constructions, when applied to X ^ S[G]. 18 JOHN ROGNES 4. Galois extensions in topology 4.1. Galois extensions of E-local commutative S-algebras. Fix an S-module E, and consider the categories MS,E and CS,E of E-local S-modules and E-local commutative S-algebras, respectively. These are the fi- brant objects in suitable topological (closed) model category structures on MS and CS , respectively, with E*-equivalences as the weak equivalences, as explai* *ned in [EKMM97, VII.4 and VIII.1]. The cofibrations in these model structures are t* *he same as in the unlocalized cases. (The reader may, if preferred, alternatively * *work with the "convenient" S-model structures of Jeff Smith and Brooke Shipley [Sh04* *], but this will not be necessary.) Let A ! B be a map of E-local commutative S-algebras, making B a commu- tative A-algebra, and let G be an E-locally stably dualizable group acting cont* *in- uously on B from the left through commutative A-algebra maps. For example, G can be a finite discrete group. Suppose that A is cofibrant as a commutative S-algebra, and that B is cofi- brant as a commutative A-algebra. The commutative A-algebra B tends not to be cofibrant as an A-module, but the smash product functor B ^A (-) is still homo- topically meaningful when applied to (other) cofibrant commutative A-algebras, * *as explained in [EKMM97, VII.6]. Let (4.1.1) i: A ! BhG be the map to the homotopy fixed point S-algebra BhG = F (EG+ , B)G that is right adjoint to the composite G-equivariant map A ^ EG+ ! A ! B, collapsing the contractible free G-space EG to a point. Let (4.1.2) h: B ^A B ! F (G+ , B) be the canonical map to the product (cotensor) S-algebra F (G+ , B) that is rig* *ht adjoint to the composite map B ^A B ^ G+ ! B ^A B ! B, induced by the action B ^ G+ ~= G+ ^ B ! B of G on B, followed by the A-algebra multiplication B ^A B ! B in B. We consider B ^A B and F (G+ , B) as B-modules by the multiplication in the first (left hand) copy of B in B ^A B, and in the target of F (G+ , B). Then h * *is a map of B-modules. The group G acts from the left on the second (right hand) copy of B in B ^A B, and by right multiplication in the source of F (G+ , B). Then h* * is also a G-equivariant map. These B- and G-actions clearly commute, and combine to a left module action by B[G]. Here is our key definition, which assumes that E, A, B and G are as above, and uses the maps i and h just introduced. We introduce the related map j in Section 6.1. Definition 4.1.3. We say that A ! B is an E-local G-Galois extension of com- mutative S-algebras if the two canonical maps i: A ! BhG = F (EG+ , B)G and h: B ^A B ! F (G+ , B), formed in the category of E-local S-modules, are both weak equivalences. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 19 The assumption that A and B are E-local ensures that BhG and F (G+ , B) are E-local, without any implicit localization. But B ^A B formed in S-modules needs not be E-local, unless E is smashing. The condition that h is a weak equivalenc* *e in MS,E amounts to asking that the corresponding map B ^A B ! F (G+ , B) formed in MS is an E*-equivalence, i.e., that E*(h) is an isomorphism. Lemma 4.1.4. Subject to the cofibrancy conditions, the notion of an E-local G- Galois extension A ! B is invariant under changes up to weak equivalence in A, B and the stabilized group S[G] = LE 1 G+ . Proof. By [EKMM97, VII.6.7] the cofibrancy conditions ensure that the construc- tions A, BhG , B ^A B and F (G+ , B) preserve weak equivalences in A and B, whether implicitly E-localized or not. The natural E*-equivalences 1 G+ ! S[G] and 1 EG+ ! S[EG] induce a (not implicitly localized) map FS[G](S[EG], B) ! F 1 G+ ( 1 EG+ , B) ~=F (EG+ , B)G , which is a weak equivalence when B is E-local. Thus the construction BhG also preserves weak equivalences in S[G]. Thus the E-local Galois conditions, that G is stably dualizable and the maps* * i and h are weak equivalences, are invariant under changes in A, B or G that amou* *nt to E-local weak equivalences of A, B and S[G]. When E = S, so there is no implicit E-localization, we may simply say that A ! B is a G-Galois extension, or for emphasis, that A ! B is a global G- Galois extension. However, most of the time we are implicitly working E-locall* *y, for a general spectrum E, but omit to mention this at every turn. Hopefully no confusion will arise. Q When G is discrete, we often prefer to write the target F (G+ , B) of h as * * G B. When G is finite and discrete, we say that A ! B is a finite Galois extension. 4.2. The Eilenberg-Mac Lane embedding. The Eilenberg-Mac Lane functor H, which to a commutative ring R associates a commutative S-algebra HR with ss*HR = R concentrated in degree 0, embeds the category of commutative rings into the category of commutative S-algebras. The two notions of Galois extension are compatible under this embedding. For this to make sense, we must assume that G is finite and that E = S. Proposition 4.2.1. Let R ! T be a homomorphism of commutative rings, and let G be a finite group acting on T through R-algebra homomorphisms. Then R ! T is a G-Galois extension of commutative rings if and only if the induced map HR ! HT is a (global) G-Galois extension of commutative S-algebras. Proof. Suppose first that R ! T is G-Galois. Then T is a finitely generated projective R-module by Proposition 2.3.2, hence flat, so TorRs(T, T ) = 0 for s* * 6= 0. Furthermore, T is finitely generated projective (of constant rank 1) as an R[G* *]- module, by Proposition 2.3.4(c). There is an isomorphism of left R[G]-modules R[G] ~= Hom R (R[G], R), since G is finite, so Ext sR[G](R, R[G]) ~= ExtsR(R, R* *) = 0 20 JOHN ROGNES for s 6= 0. Therefore Ext sR[G](R, T ) = 0 for s 6= 0, by the finite additivity* * of Ext in its second argument. It follows that the homotopy fixed point spectral sequence E2s,t= H-s (G; sstHT ) = Ext-s,-tR[G](R, T ) =) sss+t(HT hG) derived from [EKMM97, IV.4.3], and the K"unneth spectral sequence E2s,t= TorRs,t(T, T ) =) sss+t(HT ^HR HT ) of [EKMM97, IV.4.2], both collapse to the originQs = tQ= 0. So (HT )hG ' H(T G) = HR and HT ^HR HT ' H(T R T ) ~= H( G T ) ' G HT are both weak equivalences. Thus HR ! HT is a G-Galois extension of commutative S- algebras. Conversely, suppose that HR ! HT is G-Galois. Then by the same spectral sequencesQT G ~=Qss0(HT hG) ~= ss0HR = R and T R T ~=ss0(HT ^HR HT ) ~= ss0( G HT ) ~= G T , so R ! T is a G-Galois extension of commutative rings. 4.3. Faithful extensions. Galois extensions of commutative rings are always faithfully flat, and it wi* *ll be convenient to consider the corresponding property for structured ring spectra. * * It remains an open problem whether Galois extensions of commutative S-algebras are in fact always faithful, but we shall verify that this is the case in most * *of our examples, with the possible exception of some cases in Section 5.6. Definition 4.3.1. Let A be a commutative S-algebra. An A-module M is faithful if for each A-module N with N ^A M ' * we have N ' *. An A-algebra B, or G-Galois extension A ! B, is said to be faithful if B is faithful as an A-modul* *e. A set of A-algebras {A ! Bi}i is a faithful cover of A if for each A-module N with N ^A Bi ' * for each i we have N ' *. In particular, a single faithful A-algebra B covers A in this sense. By the following lemma, this corresponds well to the algebraic notion of a f* *aith- fully flat module [Gr92, 0.1.7]. Flatness (cofibrancy) is implicit in our homo* *topy invariant work, so we only refer to the faithfulness in our terminology. Lemma 4.3.2. Let M be a faithful A-module. (a) A map f :X ! Y of A-modules is a weak equivalence if and only if f ^ 1 :X ^A M ! Y ^A M is a weak equivalence. (b) A diagram of A-modules X -f! Y g-!Z, with a preferred null-homotopy of gf, is a cofiber sequence if and only if X ^A M ! Y ^A M ! Z ^A M, with the associated null-homotopy of gf ^ 1, is a cofiber sequence. Proof. (a) Consider the mapping cone Cf of f. (b) Consider the induced map Cf ! Z. Faithful modules and extensions are preserved under base change, and are de- tected by faithful base change. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 21 Lemma 4.3.3. Let A ! B be a map of commutative S-algebras and M a faithful A-module. Then B ^A M is a faithful B-module. Proof. Let N be a B-module such that N ^B (B ^A M) ' *. Then N ^A M ' *, so N ' * since M is faithful over A. Lemma 4.3.4. Let A ! B be a faithful map of commutative S-algebras and M an A-module such that B ^A M is a faithful B-module. Then M is a faithful A-module. Proof. Let N be an A-module such that N ^A M ' *. Then (N ^A B)^B (B^A M) ~= N ^A B ^A M ~=(N ^A M) ^A B ' *, so N ^A B ' * since B ^A M is faithful over B, and thus N ' * since B is faithful over A. Lemma 4.3.5. For each G-Galois extension R ! T of commutative rings, the induced G-Galois extension HR ! HT of commutative S-algebras is faithful. Proof. Recall that T is faithfully flat over R by Proposition 2.3.4(a). For e* *ach HR-module N we have ss*(N ^HR HT ) ~= ss*(N) R T , by the K"unneth spectral sequence E2s,t= TorRs,t(ss*(N), T ) =) sss+t(N ^HR HT ) and the flatness of T . Therefore N ^HR HT ' * implies ss*(N) R T = 0, which * *in turn implies that ss*(N) = 0 by the faithfulness of T . Thus N ' * and HR ! HT is faithful. Question 4.3.6. Is every E-local G-Galois extension A ! B of commutative S- algebras faithful? By Corollary 6.3.4 (or Lemma 6.4.3) the answer is yes when the order of G is invertible in ss0(A), but in some sense this is the less interesting case. In the case E = K(n), it is very easy [HSt99, 7.6] to be faithful over A = L* *K(n)S. Lemma 4.3.7. In the K(n)-local category, every non-trivial S-module is faithful over LK(n)S. Proof. Let M and N be K(n)-local spectra, considered as modules over LK(n)S. From the K"unneth formula K(n)*(M ^LK(n)S N) ~=K(n)*(M) K(n)* K(n)*(N) it follows that if LK(n)(M ^LK(n)S N) ' * then K(n)*(M) = 0 or K(n)*(N) = 0, since K(n)* is a graded field. So if M is non-trivial, we must have N ' *. Th* *us such an M is faithful. 5. Examples of Galois extensions In this chapter we catalog a variety of examples of Galois extensions, some * *global and some local, as indicated by the section headings. 22 JOHN ROGNES 5.1. Trivial extensions. Let E be any S-module and work E-locally. For each cofibrant commutative S-algebra A and stably dualizable group G there is a trivial G-Galois extension from A to B = F (G+ , A), given by the diagonal map : A ! F (G+ , A) that is functionally dual to the collapse map G ! {e}. Here G acts from the left on F (G+ , A) by right multiplication in the source. More precisely, B is* * a functorial cofibrant replacement of F (G+ , A) in the category of commutative A- algebras, which inherits the G-action by functorialityQof the cofibrant replace* *ment. When G is discrete we can write this extension as A ! G A. It is clear that i: A ! BhG = F (G+ , B)hG is a weak equivalence, since (G+* * )hG ' {e}+ , and that h: B ^A B = F (G+ , A)^A F (G+ , A) ! F (G+ ^G+ , A) ~=F (G+ , * *B) is a weak equivalence, since G is stably dualizable. The trivial G-Galois extension admits an A-module retraction F (G+ , A) ! A functionally dual to the inclusion {e} ! G, so : A ! F (G+ , A) is always fait* *hful. For any G-Galois extension A ! B, there is an induced G-Galois extension B ~= B ^A A ! B ^A B (see Proposition 6.2.1 and Lemma 7.1.3 below), and the map h: B ^A B ! F (G+ , B) exhibits an equivalence between this self-induced extension and the trivial G-Galois extension : B ! F (G+ , B). 5.2. Eilenberg-Mac Lane spectra. Let E = S. By Proposition 4.2.1 and Lemma 4.3.5, for each finite G-Galois extension R ! T of commutative rings the induced map of Eilenberg-Mac Lane commutative S-algebras HR ! HT is a faithful G-Galois extension. Proposi- tion 4.2.1 also contains a converse to this statement. 5.3. Real and complex topological K-theory. Let E = S, and let KO and KU be the real and complex topological K-theory spectra, respectively. Their connective versions ko and ku can be realized as * *the commutative S-algebras associated to the bipermutative topological categories of finite dimensional real and complex inner product spaces, respectively [Ma77, VI and VII]. The periodic commutative S-algebras KO and KU are obtained from these by Bousfield localization, in the ko- or ku-module categories, by [EKMM97, VIII.4.3]. The complexification functor from real to complex inner product spaces defin* *es maps c: ko ! ku and c: KO ! KU of commutative S-algebras, and complex conjugation at the categorical level defines a ko-algebra self map t: ku ! ku a* *nd a KO-algebra self map t: KU ! KU. Another name for t is the Adams operation _-1 . Complex conjugation is an involution, so t2 = 1 is the identity in both c* *ases. We therefore have an action by G = {e, t} ~=Z=2 on KU through KO-algebra maps, and can make functorial cofibrant replacements to keep this property, while mak* *ing KO cofibrant as a commutative S-algebra and KU cofibrant as a commutative KO- algebra. Proposition 5.3.1. The complexification map c :KO ! KU is a faithful Z=2- Galois extension, i.e., a global quadratic extension. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 23 See also Example 6.4.4 for more about this extension. Proof. The claim that i: KO ! KUhZ=2 is a weak equivalence is well-known to follow from [At66]. We outline a proof in terms of the homotopy fixed point spe* *ctral sequence E2s,t= H-s (Z=2; sstKU) =) sss+t(KUhZ=2) . Here ss*KU = Z[u 1 ] with |u| = 2, t 2 Z=2 acts by t(u) = -u and E2**= Z[a, u 2 ]=(2a) with a 2 E2-1,2= H1(Z=2; Z{u}) ~= Z=2. A computation with the Adams e- invariant shows that i takes the generator j 2 ss1KO to a class represented by a 2 E1-1,2, so j3 2 ss3KO = 0 implies that a3 2 E2-3,6is a boundary. The only possibility for this is that d3(u2) = a3, up to sign, leaving E4**= E1**= Z[a, b, u 4 ]=(2a, a3, ab, b2 = 4u4) . This abutment is isomorphic to ss*KO, and the graded ring structure implies that ss*(i) is indeed an isomorphism.Q To show that h: KU ^KO KU ! Z=2 KU is a weak equivalence, we use the Bott periodicity cofiber sequence (5.3.2) KO -j!KO -c!KU -@! 2KO of KO-modules and module maps, up to an implicit weak equivalence between the homotopy cofiber of c and 2KO. It is the spectrum level version of the homotopy equivalence (U=O) ' ZxBU, and is sometimes stated as an equivalence KU ' KO^Cj. Here j is given by smashing with the stable Hopf map j :S1 ! S0, and @ is characterized by @ O fi ' 2r : 2KU ! 2KO, where fi : 2KU ! KU is the Bott equivalence and r :KU ! KO is the realification map. We could write @ = 2r O fi-1 in DKO . Inducing (5.3.2) up along c: KO ! KU, we obtain the upper row in the followi* *ng map of horizontal cofiber sequences (5.3.3) KU ^KO KO __1^c_//KU ^KO KU _1^@__//KU ^KO 2KO ~=|| |h| ' |fi| fflffl| Q fflffl| ffi fflffl| KU ____________//_Z=2KU ______________//KU of KU-modules and module maps, up to another implicit identification of the ho- motopy cofiber of with KU. Here h is the canonical map, is the diagonal inclusion (so the lower row contains the trivial Z=2-Galois extension of KU), f* *i is the Bott equivalence KU ^KO 2KO ~= 2KU ! KU,Qand the difference map ffi is the difference of the two projections from Z=2KU, indexed by the elements * *of {e, t} ~=Z=2, written multiplicatively. 24 JOHN ROGNES The left hand square commutes strictly since Z=2 acts on KU through KO- algebra maps. To see that the right hand square commutes up to KU-module homotopy, it suffices to prove this after precomposing with the weak equivalence 1^fi :KU ^KO 2KU ! KU ^KO KU. To show that the two resulting KU-module maps KU ^KO 2KU ! KU are homotopic, it suffices by adjunction to show that the restricted KO-module maps 2KU ! KU are homotopic. This is then the computation fi O 2c O 2r = ffi O h O (c ^ fi) in DKO , which follows directly from ffi O h = ~ - ~ O (1 ^ t) = ~(1 ^ (1 - t))* * and the well-known relations c O r = 1 + t and fi O 2(1 + t) = (1 - t) O fi. Finally, c: KO ! KU is faithful. For if N is a KO-module such that N ^KO KU ' *, then applying N ^KO (-) to (5.3.2) gives a cofiber sequence N -j!N -! N ^KO KU ! 2N . The assumption that N ^KO KU ' * implies that j : N ! N is a weak equiv- alence. But j is also nilpotent, since j4 = 0 2 ss4(S), so we must have N ' *. Therefore KU is faithful over KO. The use of nilpotency in this argument may be suggestive of what could in general be required to answer Question 4.3.6. Q We note that the maps i: ko ! kuhZ=2 and h: ku ^ko kuW! Z=2ku both fail to be weak equivalences. The homotopy cofiber of i is j<0 4jHZ=2, and the homotopy cofiber of h is HZ, as is easily seen by adapting the arguments above. So i: ko ! ku is not Galois. 5.4. The Morava change-of-rings theorem. In this section we fix a rational prime p and a natural number n, and work l* *ocally with respect to the n-th p-primary Morava K-theory K(n). The work of Devinatz and Hopkins [DH04] reinterprets the Morava change-of-rings theorem [Mo85, 0.3.3] as giving a weak equivalence LK(n)S ' EhGnn. We will regard this as a fundamentally important example of a K(n)-local pro- Galois extension LK(n)S ! En of commutative S-algebras. 5.4.1. The Lubin-Tate spectra. Recall that En is the n-th p-primary even periodic Lubin-Tate spectrum, for which ss0(En) = W(Fpn)[[u1, . .,.un-1 ]] (W(-) denotes the ring of p-typical Witt vectors) and ss*(En) = ss0(En)[u 1 ]. Related theories were studied by Morava [Mo79], Rudjak [Ru75] and Baker-W"urgler [BW89], but in this precise form they seem to have been first considered by Hop* *kins and Miller. The height n Honda formal group law n is defined over Fp and is characteriz* *ed by its p-series [p]n(x) = xpn. Its Lubin-Tate deformation e n over Fpn is the * *uni- versal formal group law over a complete local ring with residue field an extens* *ion GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 25 of Fpn, whose reduction to the residue field equals the corresponding extension* * of n. In this case the universal complete local ring equals ss0(En), with maximal* * ideal (p, u1, . .,.un-1 ) and residue field Fpn. The Lubin-Tate spectrum En is (at f* *irst) the K(n)-local complex oriented commutative ring spectrum that represents the r* *e- sulting Landweber exact homology theory (En)*(X) = ss*(En) ss*(MU)MU*(X). More generally, we can consider n as a formal group law over the algebraic closure ~Fpof Fp. Its universal deformation is then defined over the complete l* *ocal ring ss0(Enrn) = W(~Fp)[[u1, . .,.un-1 ]] and there is a similar K(n)-local complex oriented commutative ring spectrum En* *rn with ss*(Enrn) = ss0(Enrn)[u 1 ]. The superscript "nr" is short for "non ramif* *i'ee", indicating that W(~Fp) is the p-adic completion of the maximal unramified exten* *sion colimf W(Fpf) of W(Fp) = Zp. (The infinite product defining p-typical Witt vect* *ors only commutes with the colimit over f after completion.) 5.4.2. The extended Morava stabilizer group. The (profinite) Morava stabilizer group Sn = Aut ( n=Fpn) of automorphisms defined over Fpn of the formal group law n (see [Ra86, xA2.2, x6.2]), and the (finite) Galois group Gal = Gal(Fpn=Fp) ~=Z=n of the extension Fp Fpn, both a* *ct on the universal deformation e n, and thus on ss*(En), by the universal propert* *y. These actions combine to one by the (profinite) semi-direct product Gn = Sn o Gal . By the Hopkins-Miller [Re98] and Goerss-Hopkins theory [GH04, x7] the ring spectrum En admits the structure of a commutative S-algebra, up to a contractib* *le choice. Furthermore, the extended Morava stabilizer group Gn acts on En through commutative S-algebra maps, again up to contractible choice. However, these actions through commutative S-algebras do not take into account the profinite topology on Gn, but rather treat Gn as a discrete group. It is known by recent work of Daniel G. Davis [Da03], that the profinite gro* *up Gn acts continuously on En in the category of K(n)-local S-modules, when En is considered as a pro-object of discrete Gn-module spectra. Presently, this k* *ind of limit presentation is not available in the context of commutative S-algebras. (Hopkins has suggested that a weaker form of structured commutativity, in terms of pro-spectra, may instead be available.) More generally, the Morava stabilizer group Sn and the absolute Galois group Gal (~Fp=Fp) ~=^Z(the Pr"ufer ring) of Fp both act on the universal deformation* * of n over the algebraic closure ~Fp, and thus on ss*(Enrn) by the universal property* *. These combine to an action by the profinite group Gnrn= Sn o ^Z. Note that the (conjugation) action by ^Zon Sn factors through the quotient ^Z! Z=n = Gal , since all the automorphisms of the height n Honda formal group law are already defined over Fpn [Ra86, A2.2.20(a)]. The Goerss-Hopkins theory cited above again implies that Enrnis a commutative S-algebra, and the extended Morava stabilizer group Gnrnacts on Enrn through commutative S-algebra maps, up to contractible choices. 26 JOHN ROGNES 5.4.3. Intermediate S-algebras. In the Galois theory for fields, the intermediate fields F E F~ correspo* *nd bijectively (via E = (F~)K and K = GE ) to the closed subgroups K GF of the absolute Galois group with the Krull topology, and the finite field extensions * *F E correspond to the open subgroups U GF . Note that in this topology, the open subgroups are exactly the closed subgroups of finite index. By analogy, it is d* *esirable to construct intermediate K(n)-local commutative S-algebras EhKn for each closed subgroup K Gn in the profinite topology. If the action were continuous, this could be done by the usual definition EhKn = F (EK+ , En)K , and indeed, for fi* *nite (and thus discrete) subgroups K, the restricted action is continuous and EhKn c* *an be defined in this way. The maximal finite subgroups M Gn were classified by Hewitt [He95, 1.2, 1.4]. When M is unique up to conjugacy, EhMn is known as the n-th higher real K-theory spectrum EOn. Devinatz and Hopkins circumvent this problem by defining EhGnn, and more generally EhUn for each open subgroup U Gn, in a "synthetic" way [DH04, Thm. 1], as the totalization of a suitably rigidified cosimplicial diagram, to * *obtain a K(n)-local commutative S-algebra of the desired homotopy type. In particular, EhGnn' LK(n)S. (See Section 8.2 for further discussion of the kinds of cosimpli* *cial diagrams involved.) For closed subgroups K Gn they then define [DH04, Thm. 2] EhKn = LK(n)(colimiEhUiKn) whereT{Ui}1i=0is a fixed descending sequence of open normal subgroups in Gn with 1 i=0Ui = {e}, and the colimit is the homotopy colimit in commutative S-algebra* *s. For finite subgroups K Gn the synthetic construction agrees [DH04, Thm. 3] with the "natural" definition of EhKn as F (EK+ , En)K . Ethan Devinatz [De05] then proceeds to compare the commutative S-algebras EhKn and EhHn for closed subgroups K and H of Gn with H normal in K. There is a well-defined action by the quotient group K=H on EhHn through commutative S-algebra maps, in the K(n)-local category [De05, x3]. Theorem 5.4.4 (Devinatz-Hopkins). (a) For each pair of closed subgroups H K Gn = Sn o Gal with H normal and of finite index in K, the map EhKn ! EhHn is a K(n)-local K=H-Galois extension. (b) In particular, for each finite subgroup K Gn the map EhKn ! En is a K(n)-local K-Galois extension. (c) Likewise, for each open normal subgroup U Gn (necessarily of finite in* *dex) the map LK(n)S ! EhUn is a K(n)-local Gn=U-Galois extension. (d)TA choice of a descending sequence {Ui}i of open normal subgroups of Gn, with iUi = {e}, exhibits LK(n)S ! En as a K(n)-local pro-Gn-Galois extension, in view of the weak equivalence LK(n)(colimiEhUin) ! En . GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 27 For the notion of a pro-Galois extension, see Section 8.1. Proof. (a) Let A = EhKn, B = EhHn and G = K=H (which is finite and discrete). By [De05, Prop. 2.3, Thm. 3.1 and Thm. A.1] the homotopy fixed point spectral sequence for ss*(BhG ) agrees with a strongly convergent K(n)*-local Adams spec* *tral sequence converging to ss*(A). So i: A ! BhG is a weak equivalence. By [De05, Cor. 3.9] the natural map h: LK(n)(B ^A B) ! F (G+ , B) induces an isomorphism on homotopy groups. Parts (b) and (c) are special cases of (a). Part (d) is contained in [DH04, Thm. 3(i)]. It would be nice to extend the statement of this theorem to the case when H * *is normal and closed, but not necessarily of finite index, in K. For n = 2 and p = 2, the Morava stabilizer group S2 is the group of units in the maximal order in the quaternion algebra Q2{1, i, j, k}, and its maximal fin* *ite subgroup is the binary tetrahedral group ^A4= Q8oZ=3 of order 24, containing the quaternion group Q8 = { 1, i, j, k} and the 16 other elements ( 1 i j k)=2. See [CF67, pp. 137-138], [Ra86, 6.3.27]. The maximal finite subgroup of G2 is G48 = A^4 o Z=2, and EO2 = EhG482 is the K(2)-localization of the connective spectrum eo2 with H*(eo2; F2) ~= A==A2 as a module over the Steenrod algebra, which is related to the topological modular forms spectrum tmf [Ho02, x3.5]. Proposition 5.4.5. At p = 2, the natural map EO2 ! E2 is a K(2)-local faithful G48 = ^A4o Z=2-Galois extension. Proof. This follows from Theorem 5.4.4(b) above and Proposition 5.4.9(b) below, but we would also like to indicate a direct proof of faithfulness, using result* *s of Hopkins and Mahowald [HM98]. There is a finite CW spectrum Cflobtained as the mapping cone of a map fl : 5Cj ^ C ! Cj ^ C , such that H*(Cfl; F2) ~=DA(1) ~=A(2)==E(2) is the "double" of A(1) = . Furthermore, there is a weak equivalence eo2 ^ Cfl' BP <2> that realizes the is* *o- morphism A==A(2) A(2)==E(2) ~= A==E(2) ~= H*(BP <2>; F2). Applying K(2)- localization yields EO2 ^ Cfl' [E(2), in the notation of 5.4.7, using that BP <2> ! v-12BP <2> = E(2) is a K(2)*- equivalence. Since j, and fl are all nilpotent (e.g. by the Devinatz-Hopkins-* *Smith nilpotence theorem [DHS88, Cor. 2]), it follows as in the proof of Proposition * *5.3.1 that EO2 ! [E(2)is faithful. And [E(2)! E2 is faithful by the elementary Propo- sition 5.4.9(a). Z=2 EGal2_____//_E2<> ^A4zzzz | ___ zzz F*4| ___ zz | __ 3 EO2 _____//_[E(2) 28 JOHN ROGNES 5.4.6. Adjoining roots of unity. Including the maximal unramified extensions into this picture, we have the f* *ol- lowing diagram of K(n)-local extensions. The groups label Galois (or pro-Galois) extensions. ^Z ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_________ ______________________&&__________________________* *_______________________________________________________n^Z EGalnO__Gal__//_En________//_EnrnO::uCCOO | OO| | M uuuu | | | uuu | | | uu | | EOnOO Gn |Sn| |Sn| | | | | | | | | ^ | LK(n)S_ _Gal_//_EhSn_nZ__//88_(Enr)hSn __________________________________________________* *___________________________________________________________nn ______________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *__________________________________________________________________________ ^Z The maximal extension LK(n)S ! Enrnis K(n)-locally pro-Gnrn-Galois. The ^Z-extension along the bottom is that obtained by adjoining all roots of* * unity of order prime to p to the p-complete commutative S-algebra LK(n)S. We might write EhSnn= LK(n)S(~pn-1 ) and (Enrn)hSn = LK(n)S(~1,p ), where ~m denotes the group of m-th order roots of unity and ~1,p = colimp-m ~m denotes the group of* * all roots of unity of order prime to p. Note that in the latter case, the infinite * *colimit of spectra must be implicitly K(n)-completed. Similarly, En = EGaln(~pn-1 ) and Enrn= EGaln(~1,p ). The process of adjoining m-th roots of unity makes sense when applied to a p- local commutative S-algebra A, for p - m, following Roland Schw"anzl, Rainer Vo* *gt and Waldhausen [SVW99], since A(~m ) can be obtained from the group A-algebra A[Cm ] = A ^ Cm+ of the cyclic group of order m by localizing with respect to a p-locally defined idempotent. Likewise, adjoining an m-th root of unity to a* * p- complete commutative S-algebra A, for m = pf - 1, can be achieved by localizing with respect to a further idempotent. The situation is analogous to how Qp Q Q(~m ) splits as a product of copies of Qp(~m ), when m = pf - 1. For more on t* *he process of adjoining roots of unity to commutative S-algebras, see [La03, 3.4]. These observations may justify thinking of the projection d: Gnrn= Sn o ^Z! * *^Z as the degree map of a K(n)-local class field theory for structured ring spectra [Ne99, xIV.4]. 5.4.7. Faithfulness. Let [E(n) = LK(n)E(n) be the K(n)-localization of the Johnson-Wilson spec- trum E(n) from 3.2.2, called Morava E-theory in [HSt99]. By [BW89, 4.1] or [HSt99, x1.1, 5.2] it has coefficients ss*E[(n) = Z(p)[v1, . .,.vn-1 , vn1 ]^In, where In = (p, v1, . .,.vn-1 ). The spectrum [E(n) was proved to be an associat* *ive S-algebra in [Ba91], and is in fact a commutative S-algebra by the homotopy fix* *ed point description in Proposition 5.4.9(a) below. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 29 Theorem 5.4.8 (Hovey-Strickland). LK(n)S is contained in the thick subcate- gory of K(n)-local spectra generated by [E(n), so [E(n) is a faithful LK(n)S-mo* *dule in the K(n)-local category. Proof. The first claim is contained in the proof of [HSt99, 8.9], which relies * *heav- ily on the construction by Jeff Smith of a suitable finite p-local spectrum X, * *as explained in [Ra92, x8.3]. The second claim follows from the first, but also m* *uch more easily from Lemma 4.3.7. Proposition 5.4.9. (a) The K(n)-local Galois extension E[(n) ! En and the K(n)-local pro-Galois extension LK(n)S ! En are both faithful. (b) For each pair of closed subgroups H K Gn, with H normal and of finite index in K, the K(n)-local K=H-Galois extension EhKn ! EhHn is faithful. Proof. (a) There is a finite subgroup F*pnof Sn such that for K = F*pno Gal we have E[(n) ' EhKn. In more detail, Sn contains the unit group W(Fpn)* [Ra86, A2.2.17], whose torsion subgroup reduces isomorphically to F*pn. For an element of finite order ! 2 W(Fpn)*, with mod p reduction ~! 2 F*pn, the linear formal power series g(x) = ~!x defines an automorphism of n, i.e., an element g 2 Sn, which acts on ss*(En) by g(u) = !u and g(uuk) = !pkuuk for 1 k < n by [DH95, 3.3, 4.4], leaving vn = u1-pn and vk = u1-pk uk invariant. Thus ss*EGaln= Zp[[u1, . .,.un-1 ]][u 1 ] and ss*EhKn is the In-adic completion of ss*E(n). Then for any spectrum X, (En)_*(X) ~=ss*En ss*[E(n)[E(n)_*(X) with ss*En a free module of rank |K| = (pn - 1)n over ss*E[(n). Here we are usi* *ng the notation (En)_*(X) = ss*LK(n)(En ^ X), and similarly for [E(n), of [HSt99, * *8.3]. It follows easily from this formula that [E(n) ! En is faithful in the K(n)-loc* *al category. In combination with 5.4.8 this shows that the composite extension LK(n)S ! E[(n) ! En is faithful, but Lemma 4.3.7 provides a much easier argument. (b) The second result follows by faithful base change along OE: LK(n)S ! En. There is a commutative diagram (for H and K as in the statement) EhHnO_______//LK(n)(EnO^OEhHn)O _ || |1^_| | 1^OE | hK EhKnO_______//LK(n)(EnO^OEnO ) | | | | | | LK(n)S _____OE_____//_En where the squares are pushouts in the category of K(n)-local commutative S- algebras. By the Morava change-of-rings theorem and [DH04, Thm. 1(iii)], ss*LK(n)(En ^ EhHn) ~=Map (Gn=H, ss*En) 30 JOHN ROGNES and ss*LK(n)(En ^ EhKn) ~=Map (Gn=K, ss*En) . (See also the proof of Theorem 7.2.2 below.) Here Map denotes the (unbased) continuous maps with respect to the profinite topologies on Gn=H, Gn=K and ss*En (in each degree). Note that K=H is a finite group acting freely on the Hausdor* *ff space Gn=H, with orbit space Gn=K, so ss :Gn=H ! Gn=K is a regular K=H- covering space. We claim that it admits a continuous section oe :Gn=K ! Gn=H, so that there is a homeomorphism K=H x Gn=K ! Gn=H, and Y Map (Gn=H, ss*En) ~= Map (Gn=K, ss*En) . K=H Thus ss*LK(n)(En ^ EhHn) is a free module of rank |K=H| over ss*LK(n)(En ^ EhKn* *), so that LK(n)(En ^ EhHn) is faithful over LK(n)(En ^ EhKn). The map 1 ^ OE is obtained by base change from OE, which is faithful by (a), and is therefore fai* *thful by Lemma 4.3.3, so _ :EhKn ! EhHn is faithful by Lemma 4.3.4. It remains to verify the claim. Let {Ui}1i=0be a descending sequence of open normal subgroups of Gn, with trivial intersection as above. Then UiH is normal of finite index in UiK, K=H surjects to UiK=UiH and there is a regular covering space ssi: Gn=UiH ! Gn=UiK, for each i. We have the following commutative diagram for i < j: K=H ______//UjK=UjH _____//_UiK=UiH | | | | | | fflffl| |fflffl fflffl| Gn=H ______//_Gn=UjH_______//Gn=UiH ss|| |ssj| |ssi| fflffl| |fflffl fflffl| Gn=K ______//_Gn=UjK_______//Gn=UiK Since K=H is finite, the surjections UjK=UjH ! UiK=UiH are isomorphisms for all sufficiently large i and j, say for i, j i0, and then ssj is the pullback* * of ssi along Gn=UjK ! Gn=UiK. Thus any choice of section oei to ssi pulls back to a section * *oej of ssj, so that the composite maps Gn=K ! Gn=UiK ! Gn=UiH are compatible for all i i0. Their limit defines the continuous section oe :Gn=K ! Gn=H. 5.5. The K(1)-local case. When n = 1, the discussion in Section 5.4 reduces to various more classical statements about variants of topological K-theory, which we now make explicit, together with a comparison to the even more classical arithmetic theory of abel* *ian extensions of Qp and Q. 5.5.1. p-complete topological K-theory. Mod p complex topological K-theory, with ss*(KU=p) = Fp[u 1 ], splits as p-2` KU=p ' 2iK(1) i=0 GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 31 where ss*K(1) = Fp[v11 ]. Bousfield K(1)-localization equals Bousfield KU=p- localization, which in turn equals Bousfield KU-localization followed by p-adic completion: LK(1)X = LKU=p X = (LKU X)^p[Bo79, 2.11]. The height 1 Honda formal group law over Fp is the multiplicative one: 1(x,* * y) = x + y + xy, its universal deformation e 1is the multiplicative formal group law* * over Zp, and the Morava spectrum E1 equals p-completed complex topological K-theory KU^pwith ss*(KU^p) = Zp[u 1 ]. The Morava stabilizer group G1 = S1 is the group of p-adic units Z*p, with its profinite topology, and k 2 Z*pacts on the commut* *ative S-algebra KU^p by the p-adic Adams operation _k :KU^p! KU^p. On homotopy, _k(u) = ku. 5.5.2. Subalgebras. * The homotopy fixed point spectrum EhGn1= (KU^p)hZp is the p-complete (non- connective) image-of-J spectrum LK(1)S = J^p, defined for p = 2 by the fiber sequence 3-1 J^2! KO^2-_--! KO^2 and for p odd by the fiber sequence r-1 J^p! KU^p-_--! KU^p for r a topological generator of Z*p. These identifications of the p-completed * *KU- localization of S with J^p are basically due to Mark Mahowald and Haynes Miller [Bo79, 4.2], respectively. (Adams-Baird and Ravenel went on to identify the p-l* *ocal KU-localization of S, see [Bo79, 4.3].) The Morava stabilizer group S1 = Z*pis isomorphic to the Galois group of the maximal (totally ramified) p-cyclotomic extension Qp Qp(~p1 ), so the classi- fication of intermediate commutative S-algebras Jp ! C ! KU^p of the form C = (KU^p)hK for K closed in Z*pis identical to the classification of intermed* *iate fields Qp E Qp(~p1 ). In this way J^p ! KU^p provides a K(1)-local "real- ization" of the K(0)-local extension Qp ! Qp(~p1 ). There are similar K(n)-local realizations of the form LK(n)S ! EhKn, when K is the kernel of the determi- nant/abelianization homomorphism Gn ! Gabn! Z*p[Ra86, 6.2.6(b)]. When p = 2, Z*2~= Z2 x Z=2, where Z2 ~= 1 + 4Z2 is open of index 2, and Z=2 ~= { 1} Z*2in closed. There are three different subgroups of index 2, namely the topologically generated subgroups <3>, <5> and <-1, 9>. The first of these corresponds to the complex image-of-J spectrum JU^2= (KU^2)h<3>given by the fiber sequence 3-1 JU^2! KU^2-_--! KU^2, and there is a K(1)-local (quadratic) Z=2-Galois extension c: J^2! JU^2, which * *is compatible with the complexification map c: KO^2! KU^2. See also Example 6.2.2 for more on this quadratic extension. The closed subgroup Z=2 of Z*2corresponds to 2-complete real K-theory: (KU^2)hZ=2 ' KO^2. 32 JOHN ROGNES When p is odd, Z*p~=ZpxF*pis pro-cyclic. Let r 2 Z*pbe a topological generat* *or, chosen to be a natural number. Then Z*phas a unique open subgroup of inde* *x n, for each integer n of the form n = ped with e 0 and d | p - 1. In addition it has the closed subgroups that appear as subgroups of F*p. In particular, Z*phas an open subgroup Zp ~=1 + pZp of index (p - 1), and a closed subgroup F*p Z*p. * The latter corresponds to the p-complete Adams summand L^p= (KU^p)hFp with ss*(L^p) = Zp[v11 ]. There are K(1)-local F*p-Galois extensions J^p ! (KU^p)hZp and L^p! KU^p. Let us write F _rn = (KU^p)h for the homotopy fixed points of _rn, which is equivalent to the homotopy fiber of _rn - 1. Then there is a K(1)-local Z=n-Galois extension n J^p= F _r ! F _r for each integer n = ped with d | p - 1, as above. 5.5.3. Extensions. Incorporating the roots of unity of order prime to p, we have the following diagram KU^pO___^Z_//KU^p(~1,pO) nr sss99 OO Z*p|| G1ssss |Z*p| | ssss | J^p____^Z__//J^p(~1,p ) with Enr1= KU^p(~1,p ). Here the maximal Galois group Gnr1= Z*px ^Zis abelian, since ^Zacts trivially on S1 = Z*p. It provides a K(1)-local realization of the* * Galois group of the maximal abelian extension Qp ! Qp(~1 ). It also appears to be possible to fit the various rational primes together, * *so as to obtain KU-local realizations of the abelian extensions of the rational field Q * *itself. The Galois group G = ^Z*of the maximal abelian extension Q ! Q(~1 ) contains the Galois group of Qp ! Qp(~1 ) as the decomposition group Dp of the prime ideal (p). Let Zp = Q(~1 )Dp be the corresponding decomposition field [Ne99, I.9.2]. Q -G=Dp---!Zp -Dp-!Q(~1 ) . After base change along Q ! Qp there are weak product splittings [Ne99, II.8.3] Y 0 Y 0 Qp Q Zp ~= Qp and Qp Q Q(~1 ) ~= Qp(~1 ) , G=Dp G=Dp i.e., as colimits of the products over the finite quotients of Y G=Dp = ^Z*=(Z*px ^Z) ~=( Z*`)=^Z. `6=p GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 33 In the latter profinite quotient, the unit of ^Zmaps diagonally to the class of* * p in each Z*`. Hence G = GabQis realized as the Galois group of Y 0 Dp Y 0 Qp -G=Dp---! Qp --! Qp(~1 ) , G=Dp G=Dp where the first map is a (pro-)trivial Galois extension. We can realize the same groups in the K(1)-local category, by the two pro-Ga* *lois extensions Y 0 Y 0 J^p-G=Dp---! J^p-Dp-! Enr1. G=Dp G=Dp Here the first is the implicitly K(1)-localized colimit of the trivial Galois e* *xtensions of J^p, indexed over the finiteQquotients of G=Dp. For brevity, let Bp = 0G=DpEnr1. Then J^p! Bp is a K(1)-local realization * *of the maximal abelian extension of Q. It seems plausible to find arithmetic pullb* *ack squares Q Q LKU S ______//_p J^p B ________//_pBp | | | | | | | | | | fflffl| fflffl|Q fflffl| fflffl|Q L0S ______//_L0 pJ^p L0B ______//L0 pBp of commutative S-algebras, so as to get an integral KU-local realization LKU S * *! B of the same Galois group. It would be wonderful if analogous (non-abelian) K(n)- local constructions for n 2 turn out to detect more of the absolute Galois gr* *oup of Qp in Gnrn, or of the absolute Galois group of Q. 5.5.4. p-local topological K-theory. The p-local complex K-theory spectrum KU(p)is also a commutative S-algebra, and admits an action by the Adams operation _r and its powers through commu- tative S-algebra maps [BR:g, 9.2]. However, in this case the E(1)-local extensi* *on n> KUh(p)! KUhp)) Q ~=EQ (irn) is an exterior algebra over Q on one generator, and EQ (ir) ! EQ (irn) is an is* *o- morphism, so {e}-Galois, but not Z=n-Galois. In spite of the relatively rich so* *urce of K(n)-local Galois extensions, there are ramification phenomena that frequent* *ly enter when several chromatic strata arePinvolved. The idempotent operation (p - 1)-1 k2F*p_k on KU^p that defines the p- complete Adams summand L^pis in fact p-locally defined [Ad69, p. 85], so as to split off the p-local Adams summand L(p)in p-2` KU(p)' 2iL(p). i=0 34 JOHN ROGNES However, the p-adic Adams operations _k of finite order, for k in the torsion s* *ub- group F*p Z*p, are not defined over Z(p), since _k(u) = ku on homotopy. Theref* *ore the extension L(p)! KU(p)only becomes Galois after p-adic completion. This pro- vides an example of an E(1)-local 'etale extension (in the sense of Section 9.4* *) that does not extend to a Galois extension. Again, this is an instance of K(0)-local* * (ra- tional) ramification of the E(1)-local prolongation of a, by definition unramif* *ied, K(1)-local Galois extension. These examples are meant as partial justification * *for the last paragraph of Remark 3.2.4. 5.6. Cochain S-algebras. Let G be a topological group and consider a principal G-bundle ss :P ! X. Fix a rational prime p and let A = F (X+ , HFp) and B = F (P+ , HFp) be the mod p cochain HFp-algebras on X and P , respectively. Note that ss*(A) = H-* (X; Fp) and ss*(B) = H-* (P ; Fp). We think of A and B as models for the singular cocha* *in algebras C*(X; Fp) and C*(P ; Fp), in conformance with [DGI:d, x3]. The direct relation between the differential graded E1 structure on C*(X; Fp) and the com- mutative S-algebra structure on A = F (X+ , HFp) seems not to have been made explicit, however. The projection ss induces a map of commutative HFp-algebras A ! B, the right action of G on P induces a left action of G on B through commutative A- algebra maps, and the weak equivalence P xG EG ! X makes its cochain dual i: A ! F ((P xG EG)+ , HFp) ~=BhG a weak equivalence. We now investigate when h: B ^A B ! F (G+ , B) is a weak equivalence. The K"unneth spectral sequence (5.6.1) E2s,t= Torss*(A)s,t(ss*(B), ss*(B)) =) sss+t(B ^A B) can be derived from the skeleton filtration of the (simplicial) two-sided bar c* *on- struction BHFp (B, A, B): [q] 7! B ^ A^q ^ B (with all smash products formed over HFp) [EKMM97, IV.7.7]. Dually, let (P, X, P ): [q] 7! P x Xq x P be the (cosimplicial) two-sided cobar construction, with totalization equal to * *the fiber product P xX P . There is a natural simplicial map ^: BHFp (B, A, B) ! F ( (P, X, P ), HFp) which is a degreewise weak equivalence by the K"unneth formula in mod p cohomol- ogy, under the assumption that H*(X; Fp) and H*(P ; Fp) are finite in each degr* *ee. So the K"unneth spectral sequence equals the one obtained by applying mod p cohomology to the cobar construction, i.e., the mod p Eilenberg-Moore spectral sequence *(X;Fp) (5.6.2) E2s,t= TorHs,t (H*(P ; Fp), H*(P ; Fp)) =) H-(s+t)(P xX P ; Fp) GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 35 [EM66]. By [Dw74], [Sh96, 3.1], the Eilenberg-Moore spectral sequence converges strongly if, for example, ss0(G) is finite, X is path-connected, and ss1(X) act* *s nilpo- tently on H*(G; Fp). The K"unneth spectral sequence is always strongly converge* *nt, so this comparison implies that the upper horizontal map in B ^A B ___^___//F ((P xX P )+ , HFp) h || |,*| fflffl| ~= fflffl| F (G+ , B)_______//F ((P x G)+ , HFp) is a weak equivalence. The right hand vertical map is induced by the homeomor- phism , :P x G ! P xX P , hence is an isomorphism, as is the lower horizontal map. Therefore these hypotheses ensure that the left hand vertical map h is a w* *eak equivalence. Proposition 5.6.3. Let G be a stably dualizable group and P ! X a principal G-bundle. (a) Suppose that ss0(G) is finite, X is path-connected, ss1(X) acts nilpoten* *tly on H*(G; Fp), and H*(X; Fp) and H*(P ; Fp) are finite in each degree. Then the map of cochain HFp-algebras F (X+ , HFp) ! F (P+ , HFp) is a G-Galois extension. (b) In particular, when G is a finite discrete group acting nilpotently on F* *p[G] (this includes all finite p-groups), then there is a G-Galois extension F (BG+ , HFp) ! F (EG+ , HFp) ' HFp that exhibits HFp as a Galois extension by each such group. A similar argument applies for the map of rational cochain algebras F (X+ , HQ) ! F (P+ , HQ) , when H*(X; Q) and H*(P ; Q) are finite dimensional over Q in each degree. For each natural number n the Morava K-theory spectrum K(n) admits un- countably many associative S-algebra structures [Ro89, 2.5], none of which are strictly commutative (cf. Lemma 5.6.4). Therefore F (X+ , K(n)) ! F (P+ , K(n)) is at best a kind of non-commutative G-Galois extension. As a further complicat* *ion, the convergence of the K(n)-based Eilenberg-Moore spectral sequence, analogous to (5.6.2), is not yet well understood. Lemma 5.6.4. K(n) does not admit the structure of a commutative S-algebra. Proof. Suppose that K(n) is a commutative S-algebra. Then so is its connective cover k(n), and there is a 1-connected commutative S-algebra map u: k(n) ! HFp. Then u* :H*(k(n); Fp) ! H*(HFp; Fp) is an injective algebra homomorphism, which commutes with the Dyer-Lashof operations on both sides [BMMS86, III.2.3]. The target equals the dual Steenrod algebra A* = E(Ook | k 0) P (O,k | k * *1), and the image of u* contains Oon-1 , but not Oon. This contradicts the operation Qpk(Ook) = Ook+1 in A*, in the case k = n - 1. 36 JOHN ROGNES 6. Dualizability and alternate characterizations 6.1. Extended equivalences. Let A ! B be a map of E-local commutative S-algebras, and let G be a topolog* *i- cal group acting from the left on B through A-algebra maps, say by ff :G+ ^B ! * *B. For example, A ! B could be a G-Galois extension. The twisted group S-algebra B is defined to be B^G+ (implicitly E-localiz* *ed, like B[G]), with the multiplication B ^ B ! B obtained from the com- posite map G+ ^ B --^1-!G+ ^ G+ ^ B -1^ff-!G+ ^ B ~=B ^ G+ and the multiplications on B and G. As usual, is the diagonal map. The map A ! B and the unit inclusion {e} ! G induce a central map j :A ! B, which makes B an associative A-algebra. Likewise the endomorphism algebra FA (B, B) of B over A is an associative A-algebra, with respect to the composit* *ion pairing. Let (6.1.1) j :B ! FA (B, B) be the canonical map of A-algebras that is right adjoint to the composite map B ^ G+ ^A B -1^ff-!B ^A B -~!B , induced by the (A-linear) action of G on B and the multiplication on B. Note th* *at B and FA (B, B) are left B-modules, with respect to the action on the target in the latter case, and that j is a map of B-modules. There is also a diagonal * *left action by G on B ^ G+ and on the target in FA (B, B), and j is G-equivariant wi* *th respect to these actions. These B- and G-actions do not commute, but combine to a left module action by B. For a map f of spectra, we will write f# and f# for various maps induced by left and right composition with f, respectively. Lemma 6.1.2. Let A ! B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps, such that h :B ^A B ! F (G+ , B) is a weak equivalence. For example, A ! B could be a G-Galois extension. Then: (a) For each B-module M there is a natural weak equivalence hM : M ^A B ! F (G+ , M) . (b) The canonical map j :B ! FA (B, B) is a weak equivalence. (c) For each B-module M there is a natural weak equivalence jM : M ^ G+ ! FA (B, M) . GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 37 Proof. (a) By definition, hM is the composite map M ^A B ~=M ^B B ^A B -1^h-!M ^B F (G+ , B) -! F (G+ , M) , which is a weak equivalence because h is a weak equivalence and G is stably dua* *l- izable. (b) This is the special case of (c) below when M = B. (c) By definition, jM is right adjoint to the composite map M ^ G+ ^A B ! M ^A B ! M induced by the group action of G on B and the module action of B on M. We can factor jM in the stable homotopy category as the following chain * *of weak equivalences: M ^ G+ -1^ae-!M ^ DDG+ -! F (DG+ , M) ~=FB (B ^ DG+ , M) -#- F h# ~ B (F (G+ , B), M) --! FB (B ^A B, M) = FA (B, M) . Here the map h# makes sense because h is a map of B-modules, and similarly for # . Algebraically, m^g lifts over # to the map f 7! f(g).m in FB (F (G+ , B)* *, M), which h# takes to jM (m ^ g). Lemma 6.1.3. Let A ! B be a G-Galois extension. For each B-module M the canonical map 0: M ^A BhG ! (M ^A B)hG is a weak equivalence. Proof. The weak equivalence M ^A A ~=M ! F (G+ , M)hG factors as the composite 0 hhGM M ^A A -1^i-!'M ^A BhG -! (M ^A B)hG - -!' F (G+ , M)hG where i and hM are weak equivalences by hypothesis and the previous lemma, respectively. The G-equivariance of hM follows like that of h. 6.2. Dualizability. For each G-Galois extension R ! T of commutative rings, T is a finitely gen- erated projective R-module. The following is the analogous statement for E-local commutative S-algebras. Proposition 6.2.1. Let A ! B be a G-Galois extension. Then B is a dualizable A-module. Proof. We must show that the canonical map :DA B ^A B ! FA (B, B) is a weak equivalence. To keep the different B's apart, we observe more generally that f* *or 38 JOHN ROGNES each B-module M there is a commutative diagram M ^A FA (B, A) _______________//_FA (B, M ^A A) 1^i#|| |(1^i)#| fflffl| fflffl| M ^A FA (B, BhG ) _____________//FA (B, M ^A BhG ) ~=|| ||0# fflffl| fflffl| M ^A FA (B,OB)hGO _____________//FA (B, MO^AOB)hG 1^jhG || |jhGM^A|B | | M ^A (B ^OG+O)hG _____________//(M ^A BO^OG+ )hG 1^N || |N| | ~= | M ^A (B ^ G+ ^ SadG )hG _____//_(M ^A B ^ G+ ^ SadG )hG where 0#is a weak equivalence by Lemma 6.1.3, the maps induced by i: A ! BhG are weak equivalences by hypothesis, the maps involving j are well-defined by t* *he G-equivariance of j (and jM^A B ), and are weak equivalences by Lemma 6.1.2, and finally the norm maps N from (3.6.1) are weak equivalences because the spectra with G-action in question have the form W ^ G+ , with G acting freely on itself [Ro:s, 5.2.4]. Thus all maps in this diagram are weak equivalences. The special case when M = B then verifies that B is dualizable over A. In the global case, E = S, it follows from Propositions 3.3.3 and 6.2.1 that in a G-Galois extension A ! B, B is a semi-finite A-module, i.e., it is weakly equivalent to a retract of a finite cell A-module. For example, by Proposition * *5.3.1 the complexification map KO ! KU is a global quadratic extension, and indeed, KU ' KO ^ Cj is a finite 2-cell KO-module. However, in the localized cases, the following counterexample shows that dualizability is probably the best one * *can hope for. Example 6.2.2. Let p = 2, recall that LK(1)S = J^2, and consider the K(1)-local quadratic Galois extension c: J^2 ! JU^2 from 5.5.2. We claim that JU^2 is not a semi-finite J^2-module, even if it is a dualizable J^2-module, in the K(1)-lo* *cal category. There is a diagram of horizontal and vertical fiber sequences: _3-1 ^ J^2 ________//KO^2_______________//KO2 c|| c|| c|| |fflffl fflffl| _3-1 fflffl| JU^2 ________//KU^2_______________//KU^2 | | | | @| @| |fflffl fflffl| 2(3-1_3-1) fflffl| 2X3 _____//_ 2KO^2____________//_ 2KO^2 GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 39 The factor 3-1 in the lower row comes from the appearance of the inverse of the Bott equivalence fi : 2KU ! KU in the connecting map @, and the relation _kfi = kfi_k. By definition, following [HMS94, 2.6], but using real K-theory, X* *3 is the homotopy fiber of 3-1 _3 - 1: KO^2! KO^2. We can compute the zero-th E1 = KU^2-cohomology of the spectra in the upper left hand square, as modules over the group S1 = Z*2of stable Adams operations, with k 2 Z*2acting by _k. First, E01(KU^2) ~= Z2[[Z*2]] (see also Example 8.1.* *4), and the remaining modules are the following quotients: Z2Ooo________OZ2[[Z*2=<-1>]]OO c*|| || | | Z2[[Z*2=<3>]]oo______Z2[[Z*2]] Here <3> Z*2is the subgroup topologically generated by 3. The map c* takes E01(JU^2) ~=Z2[[Z*2=<3>]] ~=Z2{1, _-1 } to E01(J^2) ~=Z2{1} by mapping both 1 a* *nd _-1 to the generator. Thus E01( 2X3) = kerc* ~=Z2{1 - _-1 } is such that _3 acts as the identity, but _-1 acts by sign. We claim that there is no semi-finite spectrum with this Morava module, i.e., this E1-cohomology as an S1-module. For each finite CW spectrum X the Atiyah- Hirzebruch spectral sequence Es,t2= Hs(X; ss-t(E1)) =) Es+t1(X) is strongly convergent. After rationalization (inverting 2) it collapses at th* *e E2- term, yielding the Chern character isomorphism M ch: E01(X)[2-1 ] ~= H2i(X; Q2) i2Z in degree zero. Here the i-th summand appears as the eigenspace of weight i, wh* *ere _k acts by multiplication by ki for each k 2 Z*2. By naturality, there is also * *such an eigenspace decomposition of E01(X)[2-1 ] for each semi-finite J^2-module X. (F* *or general spectra X, the Atiyah-Hirzebruch spectral sequence needs not converge.) Now note that E01( 2X3)[2-1 ] ~= Q2 has _3 acting as the identity, and _-1 acting by sign, which means that it should lie both in the weight 0 eigenspace and in an eigenspace of odd weight. This contradicts the possibility that 2X3 * *is semi-finite. It follows that also JU^2 cannot be K(1)-locally semi-finite. Dualizable modules are preserved under base change, and are detected by fait* *hful and dualizable base change. Lemma 6.2.3. Let A ! B be a map of commutative S-algebras and M a dualizable A-module. Then B ^A M is a dualizable B-module. Proof. We must verify that the canonical map :FB (B ^A M, B) ^B (B ^A M) ! FB (B ^A M, B ^A M) 40 JOHN ROGNES is a weak equivalence. It factors as the composite FB (B ^A M, B) ^B (B ^A M) ~=FA (M, B) ^A M -! F ~ A (M, B ^A M) = FB (B ^A M, B ^A M) , where the middle map is a weak equivalence by Lemma 3.3.2(a), since M is a dualizable A-module. Lemma 6.2.4. Let A ! B be a faithful map of commutative S-algebras, with B dualizable over A, and M an A-module such that B ^A M is a dualizable B-module. Then M is a dualizable A-module. Proof. We must verify that :FA (M, A)^A M ! FA (M, M) is a weak equivalence. It suffices to show that the map 1 ^ in the commutative square below is a weak equivalence, since B is assumed to be faithful over A. B ^A FA (M, A) ^A M __1^__//B ^A FA (M, M) ^1 || || fflffl| fflffl| FA (M, B) ^A M ________//_FA (M, B ^A M) Here the lower horizontal map is isomorphic to :FB (B ^A M, B) ^B (B ^A M) ! FB (B ^A M, B ^A M) , which is a weak equivalence because B ^A M is assumed to be dualizable over B. The vertical maps are weak equivalences because B is dualizable over A, in view* * of Lemma 3.3.2(a). Therefore the upper horizontal map 1 ^ is also a weak equiva- lence. Corollary 6.2.5. If A is a commutative S-algebra and G is a stably dualizable group, so S[G] is dualizable over S, then A[G] is dualizable over A. Conversely, if A is a faithful commutative S-algebra, with A dualizable over* * S, and G is a topological group such that A[G] is dualizable over A, then G is sta* *bly dualizable. The following lemma gives the same conclusion as Lemma 6.1.3, but under dif- ferent hypotheses, and will be often used. Lemma 6.2.6. Let A ! B be a map of commutative S-algebras, let G be a topolog- ical group acting on B through A-algebra maps, and let M be a dualizable A-modu* *le. Then the canonical map 0: M ^A BhG ! (M ^A B)hG is a weak equivalence. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 41 Proof. In the commutative diagram ae^1 hG M ^A BhG _______//_DA DA M ^A BhG ______//_FA (DA M, B ) 0|| || ~=|| fflffl|(ae^1)hG fflffl| hG fflffl| (M ^A B)hG ______//(DA DA M ^A B)hG _____//_FA (DA M, B)hG the horizontal maps derived from and ae are weak equivalences because M is dualizable over A, and the right hand vertical map is an isomorphism. Thus the left hand vertical map 0 is a weak equivalence. 6.3. Alternate characterizations. The following alternate characterization of Galois extensions corresponds to* * the Auslander-Goldman definition. Compare Proposition 2.3.2. Implicit cofibrancy and localization at some S-module E is to be understood. Proposition 6.3.1. Let A ! B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Then A ! B is a G-Galois extension if and only if both i :A ! BhG and j :B ! FA (B, B) are weak equivalences and B is a dualizable A-module. Proof. Lemma 6.1.2(b) and Proposition 6.2.1 establish one implication. For the converse, suppose that i and j are weak equivalences and that B is dualizable over A. We must show that h: B ^A B ! F (G+ , B) is a weak equivalence. Again, to keep the B's apart we shall observe that for each B-module M the map hM factors in the stable homotopy category as the following chain of weak equivale* *nces: M ^A B -1^ae-!M ^A DA DA B -! FA (DA B, M) ~=FB (DA B ^A B, M) --# F j# ~ B (FA (B, B), M) -! FB (B, M) = F (G+ , M) . Algebraically, the forward image of m ^ b lifts over # to f 7! f(b) . m, whic* *h maps by j# to hM (m ^ b) = {g 7! g(b) . m}. The hypotheses that B is dualizable ove* *r A and j is a weak equivalence thus imply that hM is a weak equivalence. The spec* *ial case M = B lets us conclude that A ! B is G-Galois. In the presence of faithfulness we have a third characterization of Galois e* *xten- sions. See also Propositions 8.2.6 and 12.1.8. Proposition 6.3.2. Let A ! B be a map of commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Then A ! B is a faithful G-Galois extension if and only if h :B ^A B ! F (G+ , B) is a weak equivalence and B is faithful and dualizable as an A-module. Proof. Proposition 6.2.1 provides one implication. For the converse, suppose th* *at h is a weak equivalence and that B is dualizable and faithful over A. We must s* *how that i: A ! BhG is a weak equivalence, and by faithfulness it suffices to show* * that 42 JOHN ROGNES 1 ^ i: B ~= B ^A A ! B ^A BhG is a weak equivalence. In the stable homotopy category we can identify this map with the chain of weak equivalences hG 0 B -'! F (G+ , B)hG h-- (B ^A B)hG - B ^A BhG . Here 0 is a weak equivalence by Lemma 6.2.6, because B is dualizable over A. We are viewing h as a G-equivariant map with respect to the left G-actions specifi* *ed in Section 4.1. Here is a characterization of faithfulness in terms of the norm map. Proposition 6.3.3. A G-Galois extension A ! B is faithful if and only if the norm map N :(B ^ SadG )hG ! BhG is a weak equivalence, or equivalently, if the Tate construction BtG is contractible. Proof. If the norm map is a weak equivalence, and Z is an A-module so that Z ^A B ' *, then Z ' Z ^A BhG ' Z ^A (B ^ SadG )hG ~= (Z ^A B ^ SadG )hG ' *. Thus A ! B is faithful. For the converse, consider B ^A (-) applied to the norm map, appearing as the left hand vertical map in the following commutative diagram. ~= (h^1)hG B ^A (B ^ SadG )hG _____//_(B ^A B ^ SadG )hG_____//(F (G+ , B) ^ SadG )hG 1^N || |N| |N| fflffl| 0 fflffl| hhG fflffl| B ^A BhG ______________//(B ^A B)hG _____________//_F (G+ , B)hG The map 0 is a weak equivalence because B is dualizable over A, by Lemma 6.2.6. The upper and lower right hand horizontal maps are weak equivalences since h is G-equivariant and a weak equivalence. The right hand vertical map is the norm map for the spectrum with G-action F (G+ , B). In the source, (F (G+ , B) ^ SadG )hG ' (B ^ DG+ ^ SadG )hG ' (B ^ S[G])hG ' B by the stable dualizability of G and the Poincar'e duality equivalence (3.5.2).* * In the target, F (G+ , B)hG ' B. A direct inspection (inducing up from the case B = S, where it suffices to check on ss0) verifies that these identifications * *are compatible under the norm map. Therefore the right hand vertical map N is a weak equivalence, and so the norm map for B must be a weak equivalence, assuming that B is faithful over A. The second equivalence is obvious from the definition of BtG as the homotopy cofiber of the norm map. Corollary 6.3.4. Any finite G-Galois extension A ! B is faithful if the order * *|G| of G is invertible in ss0(B). Proof. Under these hypotheses ss*(BhG ) ~= ss*(B)=G, ss*(BhG ) ~= ss*(B)G and * *the composite ss*(B) ! ss*(B)=G -N*-!ss*(B)G ! ss*(B) GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 43 is multiplication by |G|, so the norm map N must induce an isomorphism in ho- motopy. The same conclusion, under different hypotheses (allowing ramification) appe* *ars in Lemma 6.4.3. 6.4. The trace map and self-duality. In this section we work principally in the derived category, i.e., in the st* *able homotopy category DA,E . The results are not essential for what follows, so we * *will be a bit brief about some of the diagram chases. Let A ! B be a map of E-local commutative S-algebras, and let G be a stably dualizable group acting on B through A-algebra maps. Suppose that i: A ! BhG is a weak equivalence. Definition 6.4.1. The trace map tr :B ^ SadG ! A in DA,E is defined by the natural chain of maps B ^ SadG -in!(B ^ SadG )hG -N! BhG -i'A , where in denotes the inclusion induced by G EG, and the wrong-way map i is a weak equivalence. When G is finite, the dualizing spectrum SadG = S can of course be ignored. Lemma 6.4.2. The trace map tr :B ^ SadG ! A equals the composite map 0)hG i B ^ SadG = B ^ S[G]hG -!' (B ^ S[G])hG -(ff---!BhG -' A where ff0: B ^ S[G] ! B is the right action derived from ff :G+ ^ B ! B by way of the group inverse. Proof. The canonical map :B ^ SadG ! (B ^ S[G])hG can be identified with the chain of weak equivalences hG ' B ^ SadG -'! F (G+ , B ^ SadG )hG --' (B ^ DG+ ^ SadG )hG -! (B ^ S[G])hG , using that G is stably dualizable and the (right G-equivariant) Poincar'e duali* *ty equivalence (3.5.2). In particular, itself is a weak equivalence. The claim is then clear from the commutative diagram __________// hG ____=_____// hG B ^ SadG ' (B ^ S[G]) (B ^ S[G])N NNNN(ff0)hGN in || in || (in)hG|| NNN fflffl| hG fflffl| ~ fflffl| NN''N (B ^ SadG )hG __'___//((B ^ S[G])hG )hG____//_((B ^ S[G])hG )hG_'__//_BhG where ~ is the canonical hocolim/holim exchange map and the bottom row defines the norm map N, as in [Ro:s, 5.2.1]. The right hand triangle uses that the homo* *topy orbits (B ^ S[G])hG are formed with respect to the diagonal left G-action, so * *the identification with B extends the right action map ff0. Algebraically, b^g in B* *^S[G] is identified with g-1 b ^ e in the homotopy orbits, which maps to ff0(b ^ g) =* * g-1 b in B. 44 JOHN ROGNES Lemma 6.4.3. When G is finite the composite B -tr!A ! B is homotopic to the sum over all g 2 G of the group action maps g :B ! B, and the composite A ! B -tr!A is homotopic to the map multiplying by the order |G| of G. Thus, if |G| is invertible in ss0(A) then tr is a split surjective map of A-* *modules, up to homotopy, and B is a faithful A-module. In particular, every G-Galois ex- tension A ! B with |G| invertible in ss0(A) is faithful. Proof. When G is finite, the composite B -tr!A ! B can be expressed by continu- ing the factorization in Lemma 6.4.2 with the map BhG ! B that forgets homotopy invariance, and therefore factors as 0 B -1^--!B ^ S[G] -ff!B , Q where : S ! S[G] ' G S is the diagonal map. Clearly this is the sum over the elements g 2 G of the group action maps g :B ! B, up to homotopy. On the other hand, the composite A ! B -tr!A is the map of G-homotopy fixed points induced by the same composite displayed above. Since the action of each group element is homotopic to the identity when restricted to the homotopy fixed points, their sum equals multiplication by the group order |G|, up to homotopy. Example 6.4.4. In the Z=2-Galois extension c: KO ! KU the trace map tr is homotopic to the realification map r :KU ! KO, as a KO-module map, and therefore also as an S-module map. For c# :DKO (KU, KO) ! DKO (KO, KO) is injective, and both tr O c and r O c are homotopic to the multiplication by 2 m* *ap KO ! KO, by Lemma 6.4.3. To justify the claim just made, that c# is injective, we use the equivalence KU ' KO ^ Cj and adjunction to identify c# with i# in the exact sequence # j# i# ss1(KO) -j-!ss2(KO) -! [Cj, KO] -! ss0(KO) induced by the cofiber sequence S0 -i!Cj -j!S2 -j!S1. Here i# is injective beca* *use j# is well-known to be surjective. In particular, the trace map tr = r :KU ! KO is not split surjective up to homotopy (it is not even surjective on homotopy groups), so the analogue of the algebraic Proposition 2.3.4(b) does not hold in topology. Recall from Section 3.6 the shearing equivalence i :B ^ S[G] ! B ^ S[G] that takes the left action on S[G] to the diagonal left action on B and S[G]. Definition 6.4.5. The trace pairing B ^A B ^ SadG ! A in DA,E is defined as the composite B ^A B ^ SadG -~^1-!B ^ SadG -tr!A . The discriminant map dB=A : B ^SadG ! DA B in DA,E is defined as the composite hG B ^ SadG = B ^ S[G]hG -!' (B ^ S[G])hG -i-!' (B ^ S[G])hG jhG--!F hG ~ hG i# A (B, B) = FA (B, B ) -' FA (B, A) = DA B . Here j is G-equivariant with respect to the left G-action from Section 6.1. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 45 Lemma 6.4.6. The trace pairing B ^A B ^ SadG ! A is left adjoint to the dis- criminant map dB=A : B ^ SadG ! DA B. Thus dB=A is in fact a map in DB,E , and represents a PicE -graded class in ss*DA (B). Proof. The first claim is a chase of definitions. The multiplications by B in * *the two copies of B in the source of the trace pairing get equalized by ~, so the a* *djoint (weak) map dB=A commutes with the obvious B-module actions on B ^ SadG and DA B. Proposition 6.4.7. If A ! B is a G-Galois extension, then the discriminant map dB=A : B ^ SadG ! DA B is a weak equivalence. In particular, B is self-dual as * *an A-module, up to an invertible shift by SadG . Proof. When A ! B is G-Galois, j :B ^ G+ ! FA (B, B) is a weak equivalence by Lemma 6.1.2(b), so the discriminant map is defined as a composite of weak equivalences. In general, we think of the discriminant map dB=A as a measure of the exten* *t to which A ! B is ramified. When it is an equivalence, we think of the trace pairi* *ng as a perfect pairing. 6.5. Smash invertible modules. The K(n)-local Picard group Picn = PicK(n) (S) was introduced in [HMS94]. Here is a slight generalization. Definition 6.5.1. Let A be a commutative S-algebra, and work locally with re- spect to the fixed spectrum E. An A-module M is smash invertible if there exists an A-module N such that N ^A M ' A as (implicitly E-local) A-modules. Let PicE(A) be the class of weak equivalence classes of E-locally smash inve* *rtible A-modules. When PicE(A) is a set we call it the E-local Picard group of A, with* * the group structure induced by the (implicitly E-local) smash product of A-modules. The following proof of the analogue of Proposition 2.3.4(c) is close to one * *found by Andy Baker and Birgit Richter in the case of a finite abelian group G. Proposition 6.5.2. Let A ! B be a faithful abelian G-Galois extension, i.e., o* *ne with G an (E-locally stably dualizable) abelian group. Then B is smash invertib* *le as an A[G]-module. Proof. We consider B as an A[G]-module by way of the given left G-action. The smash inverse of B over A[G] will be its functional dual DA[G](B) = FA[G](B, A[* *G]) in the category MA[G],E. There is a natural counit map ffl: FA[G](B, A[G]) ^A[G] B ! A[G] , that is left adjoint to the identity map on FA[G](B, A[G]) in the category of A* *[G]- modules. In symbols, ffl: f ^x 7! f(x). The claim is that ffl is a weak equival* *ence. By assumption B is faithful over A, so it suffices to verify that ffl becomes an e* *quivalence after inducing up along A ! B. We factor the resulting map 1 ^ ffl as 0 B ^A FA[G](B, A[G]) ^A[G] B -! FA[G](B, B[G]) ^A[G] B ~=FB[G](B ^A B, B[G]) ^B[G] (B ^A B) -ffl0!B[G] . 46 JOHN ROGNES Here 0 is a weak equivalence because B is dualizable over A (cf. Lemma 6.2.6),* * the middle isomorphism is a composite of two standard adjunctions, and ffl0 is a co* *unit of the same sort as ffl, now in the category of B[G]-modules. We have left to p* *rove that ffl0 is a weak equivalence. There is a chain of left B[G]-module maps (B ^A B) ^ SadG -h^1-!F (G+ , B) ^ SadG -^1- B ^ DG adG ' O + ^ S -! B[G] -! B[G] , each of which is a weak equivalence. Here h is a weak equivalence because A ! B* * is G-Galois, is a weak equivalence because G is stably dualizable, and the unnam* *ed weak equivalence is the identity on B smashed with the Poincar'e duality equiva* *lence from (3.5.2). The latter is left G-equivariant with respect to the inverse of t* *he right G-action mentioned in Section 3.5, i.e., with respect to the left action on DG+* * given by right multiplication in the source, the trivial action on SadG , and the inv* *erse of the standard right action on B[G]. The map O is induced by the group inverse in G, and takes the inverse of the standard right action on B[G] to the standard left action on B[G]. When G is finite, the chain simplifies to B ^A B -h!F (G+ , B)- ~ B[G] -O!B[G] , W Q where ~ is the usual inclusion and weak equivalence B[G] ~= G B ! G B = F (G+ , B). Again, the right hand B[G] has the standard left B[G]-module struct* *ure. By [Ro:s, 3.3.4, 3.2.3] the dualizing spectrum SadG is smash invertible (in* * the E-local stable homotopy category), with smash inverse its functional dual S-adG* * = (DG+ )hG , so the counit map ffl0 for the B[G]-module B ^A B is related by a ch* *ain of weak equivalences to the counit map ffl00:FB[G](B[G], B[G]) ^B[G] B[G] ! B[G] , for B[G] considered as a left B[G]-module in the standard way. This map is obvi- ously an isomorphism. So each (implicitly E-local) abelian G-Galois extension A ! B exhibits B as a possibly interesting element in the Picard group PicE (A[G]). The following converse to Proposition 6.5.2 does not require that G is abeli* *an, but for abelian G it follows that the smash invertibility of B over A[G] is equ* *ivalent to B being faithful over A. Lemma 6.5.3. Let A ! B be a (not necessarily abelian) G-Galois extension. If B is smash invertible as an A[G]-module, i.e., if there exists an A[G]-module C a* *nd a weak equivalence B ^A[G] C ' A[G] of A-modules, then B is faithful over A. Proof. If N ^A B ' * then N[G] ~=N ^A A[G] ' N ^A B ^A[G] C ' *, and N is a retract of N[G], so N ' *. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 47 7. Galois theory I We continue to work locally with respect to some S-module E. 7.1. Base change for Galois extensions. Faithful G-Galois extensions A ! C are preserved by base change along arbitr* *ary maps A ! B, COO_____//_BO^AOC | | | | | | A ________//_B and all Galois extensions are preserved by dualizable base change. Conversely, (faithful) Galois extensions are detected by faithful and dualizable base chang* *e. We do not know whether these dualizability hypotheses are necessary. Lemma 7.1.1. Let A ! B be a map of commutative S-algebras and A ! C a faithful G-Galois extension. Then B ! B ^A C is a faithful G-Galois extension. Proof. The action by G on C through A-algebra maps extends uniquely to an action on B ^A C through B-algebra maps, taking g :C ! C to 1^g :B ^A C ! B ^A C on the point set level, for g 2 G. The group G remains stably dualizable, irrespec* *tive of whether it is considered as acting on C or B ^A C. We show that B ! B ^A C is a faithful G-Galois extension by appeal to Propo- sition 6.3.2. We know that C is a dualizable A-module by Proposition 6.2.1, and* * it is faithful by hypothesis. Therefore B ^A C is a dualizable and faithful B-modu* *le by the base change lemmas 6.2.3 and 4.3.3. It remains to verify that the canoni* *cal map h: (B ^A C) ^B (B ^A C) ! F (G+ , B ^A C) is a weak equivalence. It is the lower horizontal map in the commutative square (7.1.2) B ^A C ^A C ____1^h____//B ^A F (G+ , C) ~=|| || fflffl| h fflffl| (B ^A C) ^B (B ^A C) _____//_F (G+ , B ^A C) , where the upper horizontal map 1 ^ h is a weak equivalence because A ! C is G-Galois, and the right hand vertical map is a weak equivalence because G is stably dualizable. This verifies the hypotheses of Proposition 6.3.2, so B ! B * *^A C is a faithful G-Galois extension. Lemma 7.1.3. Let A ! B be a map of commutative S-algebras, with B is dualiz- able over A, and A ! C a G-Galois extension. Then B ! B ^A C is a G-Galois extension. Proof. The group G is stably dualizable, acts on B ^A C through B-algebra maps, and makes the canonical map h: (B ^A C) ^B (B ^A C) ! F (G+ , B ^A C) a weak equivalence, just as in the previous proof. In order to verify the conditions * *in Definition 4.1.3 of a G-Galois extension, it remains to show that the canonical* * map i: B ! (B ^A C)hG is a weak equivalence. But B ~= B ^A A ' B ^A ChG , so we can identify i with 0: B ^A ChG ! (B ^A C)hG , which is a weak equivalence by Lemma 6.2.6 because B is dualizable over A. 48 JOHN ROGNES Lemma 7.1.4. Let A ! B and A ! C be maps of commutative S-algebras, with B a faithful and dualizable A-module, and let G be a stably dualizable group ac* *ting on C through A-algebra maps. (a) If B ! B^A C is a G-Galois extension, then A ! C is a G-Galois extension. (b) If B ! B ^A C is a faithful G-Galois extension, then A ! C is a faithful G-Galois extension. Proof. We must verify that the two maps i: A ! ChG and h: C ^A C ! F (G+ , C) are weak equivalences. For the first map, we factor the weak equivalence i: B ! (B ^A C)hG for the G-Galois extension B ~=B ^A A ! B ^A C as the composite 0 B ^A A -1^i-!B ^A ChG -! (B ^A C)hG . Here the right hand map 0 is a weak equivalence because B is dualizable over A, by Lemma 6.2.6. Therefore the left hand map 1 ^ i is a weak equivalence, and so i: A ! ChG is a weak equivalence because B is faithful over A. For the second map, we use the commutative square (7.1.2) again. The right hand vertical map is a weak equivalence because G is stably dualizable, and t* *he lower horizontal map h is a weak equivalence because B ! B ^A C is assumed to be G-Galois. So the upper horizontal map 1 ^ h is a weak equivalence, and so h: C ^A C ! F (G+ , C) is a weak equivalence because B is faithful over A. Finally, if B ! B ^A C is faithful, then we know that A ! C is faithful by Lemma 4.3.4. 7.2. Fixed S-algebras. Let G be a stably dualizable group and let A ! B be a G-Galois extension. We consider the sub-extensions that occur as the homotopy fixed points C = BhK , f* *or suitable subgroups K of G. Definition 7.2.1. Let K G be a topological subgroup. We say that K is an allowable subgroup if K is stably dualizable and the collapse map c: S[G]hK = S[G xK EK] ! S[G xK *] = S[G=K] is a weak equivalence. We consider two allowable subgroups K and K0 to be equivalent if K K0 and S[K] ! S[K0] is a weak equivalence, or more generally, if K and K0 are related by a chain of such (elementary) equivalences. We say th* *at K is an allowable normal subgroup if, furthermore, K is a normal subgroup of G and the quotient group G=K is stably dualizable. When G is finite and discrete, and E = S, the allowable subgroups of G are just the subgroups of G in the usual sense, for then G is a disjoint union of f* *ree K-orbits. Similarly for the allowable normal subgroups, for G=K is still finite. For A ! B a G-Galois extension, and K G an allowable subgroup, we can form the following maps of commutative A-algebras F (EG+ , B)G ! F (EG+ , B)K ! F (EG+ , B) . GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 49 In view of the natural weak equivalences A ! F (EG+ , B)G and F (EG+ , B) ! B, we will keep the notation simple by writing the maps above as: A ! BhK ! B . So to be precise, we interpret B as F (EG+ , B), which then admits a K-action through BhK -algebra maps. Likewise, if K is normal in G then BhK admits a G=K-action through BhG -algebra maps, which in turn are A-algebra maps. An implicit cofibrant replacement is also necessary at this stage. Here is the forward part of the Galois correspondence for E-local commutative S-algebras. Theorem 7.2.2. Let A ! B be a faithful G-Galois extension and K G any allowable subgroup. Then C = BhK ! B is a faithful K-Galois extension. If furthermore K G is an allowable normal subgroup, then A ! C = BhK is a faithful G=K-Galois extension. Proof. We shall detect that C ! B (resp. A ! C) is faithfully Galois by applying Lemma 7.1.4 to the case of faithful and dualizable base change along C ! B ^A C (resp. A ! B). Here B is faithful and dualizable as an A-module by hypothesis and Proposition 6.2.1, so B ^A C is faithful and dualizable as a C-module by Lemma 4.3.3 and Lemma 6.2.3. In the commutative diagram (7.2.3) BOO_____//_B ^A B___h'__//F (G+ , B)oo=____F_(G+ , B) | OO OO OO | | | | | | | | | | h0 | c# | COO_____//_B ^AOC__'__//_FO(G+O,OB)hKo'o__F (G=K+O,OB) | | | | | | | | | | | | A ________//_B_____=_______//Boo_____=________B the left hand squares are base change pushouts in the category of commutative S-algebras. The middle horizontal maps are weak equivalences. For h is a weak equivalence by the assumption that A ! B is G-Galois. The map h0:B ^A C = B ^A BhK ! F (G+ , B)hK factors as a composite weak equivalence 0 hhK B ^A BhK -!'(B ^A B)hK --!' F (G+ , B)hK using that B is dualizable over A (and Lemma 6.2.6) and that h is a weak equiv- alence. Here K acts from the left on B ^A B and F (G+ , B) by restriction of t* *he actions by G, i.e., on the second copy of B in B ^A B and by right multiplicati* *on in the source in F (G+ , B), so in particular h is K-equivariant. Likewise, the right hand horizontal maps are weak equivalences. For c# is * *the composite map # F (G=K+ , B) -c!'F ((G+ )hK , B) ~=F (G+ , B)hK 50 JOHN ROGNES functionally dual to the collapse map c: (G+ )hK = (G xK EK)+ ! (G xK *)+ = G=K+ , which is a stable weak equivalence by hypothesis on K. Therefore, for K stably dualizable the induced extension B ^A C ! B ^A B is weakly equivalent to the K-fold diagonal inclusion : F (G=K+ , B) ! F (G+ , B), i.e., to the trivial K-Galois extension (Section 5.1) of F (G=K+ , B). In parti* *cular, B ^A C ! B ^A B is faithfully K-Galois, and so by the faithful and dualizable detection result Lemma 7.1.4 it follows that C ! B is faithfully K-Galois. If furthermore K is normal in G, and the quotient group G=K is stably dualiz- able, then the induced extension B ! B ^A C is weakly equivalent to the G=K-fold diagonal inclusion : B ! F (G=K+ , B), i.e., to the trivial G=K-Galois extensi* *on of B. So B ! B ^A C is faithfully G=K-Galois, and so by Lemma 7.1.4 we can conclude that A ! C is faithfully G=K-Galois. The following lemma will be applied in Section 9.1, when we discuss separable extensions. Lemma 7.2.4. Let A ! B be a faithful G-Galois extension and K G an allow- able subgroup. Suppose also that S[G=K] is dualizable. Then C = BhK is faithful and dualizable over A, and the canonical map ~ :BhK ^A BhK ! (B ^A B)h(KxK) is a weak equivalence. Proof. It is formal that A ! C is faithful when the composite A ! C ! B is faithful. For if N 2 MA has C ^A N ' * then B ^A N ~= B ^C C ^A N ' *, so N ' *. The extension A ! B is faithful with B dualizable over A by Proposition 6.2.* *1, and B ^A C ' F (G=K+ , B) as in (7.2.3) is dualizable over B since S[G=K] is dualizable over S. Thus C is dualizable over A by Lemma 6.2.4. The map ~ factors as the composite of two weak equivalences 0 ( 0)hK BhK ^A BhK -!'(B ^A BhK )hK ----!' (B ^A B)h(KxK) derived from Lemma 6.2.6, where the first uses that C = BhK is dualizable over* * A, and the second uses that B is dualizable over A. Question 7.2.5. When does G and K stably dualizable and GhK ' G=K imply that S[G=K] is dualizable? This is obvious when G is finite and discrete, but n* *ot in general. If there are allowable, non-normal K G with S[G=K] not dualizable, then should these BhK take part in the Galois correspondence, or not? 8. Pro-Galois extensions and the Amitsur complex We continue to let E be a fixed S-module and to work entirely in the E-local category. 8.1. Pro-Galois extensions. Definition 8.1.1. Let A be a cofibrant commutative S-algebra, and consider a directed system of finite Gff-Galois extensions A ! Bff, such that Bff! Bfiis a cofibration of commutative A-algebras for each ff fi. Suppose further that ea* *ch GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 51 A ! Bffis a sub-Galois extension of A ! Bfi, so/such that there is a preferred surjection Gfi! Gffwith kernel Kfffi, and a natural weak equivalence Bff' BhKff* *fifi. Let B = colimffBff, where the colimit is formed in CA,E , and let G = limffG* *ff, with the (profinite) limit topology. Then, by definition, A ! B is a pro-G-Galo* *is extension. More generally, one might consider a directed system of (E-local) Galois ext* *en- sions with stably dualizable (rather than finite) Galois groups Gff, arranging * *that each normal subgroup Kfffiis stably dualizable. We prefer to wait for some rele* *vant examples before discussing the analogue of the Krull topology on the resulting * *limit group G, but compatibility with the "natural topology" on E-local Hom -sets (s* *ee [HPS97, x4.4] and [HSt99, x11]) is certainly desirable. For each ff the weak equivalence hff:Bff^A Bff ! F (Gff+, Bff) extends by Lemma 6.1.2(a) to a weak equivalence hff,B:B ^A Bff! F (Gff+, B). The colimit of these over ff is a weak equivalence (8.1.2) h: B ^A B ! F ((G+ , B)) , where by definition F ((G+ , B)) = colimffF (Gff+, B) is the "continuous" map- ping spectrum with respect to the Krull topology, and colimffB ^A Bff= B ^A colimffBff= B ^A B, since pushout with B commutes with colimits in the category of commutative A-algebras. Likewise, for each ff the weak equivalence jff:Bff ! FA (Bff, Bff) exte* *nds by Lemma 6.1.2(c) to a weak equivalence jff,B:B ! FA (Bff, B). The limit of these over ff is a weak equivalence (8.1.3) j :B<> ! FA (B, B) , where by definition B<> = lim ffB is the "completed" twisted group A- algebra, and limffFA (Bff, B) ~=FA (colimffBff, B) = FA (B, B). Example 8.1.4. In the case of the K(n)-local pro-Gn-Galois extension LK(n)S ! En, these weak equivalences induce the isomorphism : En_*(En) ~=Map (Gn, ss*(En)) that is implicit in [Mo85] and explicit in [St00, Thm. 12] and [Ho04, 4.11], an* *d the isomorphism : En*<> ~=E*n(En) from [St00, p. 1029] and [Ho04, 5.1]. The appearance of the continuous mapping space and the completed twisted group ring corresponds to the spectrum level colimits and limits above, combined with the In-adic completion at the level of homotopy groups induced by the implicit K(n)-localization [HSt99, 7.10(e)]. The pro-Galois formalism thus accounts for the first steps in a proof of Gro* *ss- Hopkins duality [HG94], following [St00]. The next step would be to study the K(n)-local functional dual of En as the continuous homotopy fixed point spectrum LK(n)DEn = F (En, LK(n)S) ' F (En, En)hGn ' (En<>)hGn , but here technical issues related to the continuous cohomology of profinite gro* *ups arise, which are equivalent to those handled by Strickland. 52 JOHN ROGNES 8.2. The Amitsur complex. As usual, let A be a cofibrant commutative S-algebra and B a cofibrant com- mutative A-algebra. Definition 8.2.1. The (additive) Amitsur complex [Am59, x5], [KO74, xII.2] is the cosimplicial commutative A-algebra Co(B=A): [q] 7! B A [q] = B ^A . .^.AB ((q + 1) copies of B), coaugmented by A ! B = C0(B=A). Here B A [q] refers to the tensored structure in CA,E , and the cosimplicial structure is derived from* * the functoriality of this construction. In particular, the i-th coface map is induc* *ed by smashing with A ! B after the i first copies of B, and the j-th codegeneracy map is induced by smashing with B ^A B ! B after the j first copies of B. Let the completion of A along B be the totalization A^B= Tot Co(B=A) of this cosimplicial resolution. The coaugmentation induces a natural completion map j :A ! A^B of commutative A-algebras. Gunnar Carlsson has considered this form of completion in his work on the descent problem for the algebraic K-theory of fields [Ca:d, x3]. More generally* *, for each functor F from commutative A-algebras to a category of spaces or spectra, * *like the units functor U = GL1, the Amitsur complex Co(B=A; F ) is the cosimplicial object [q] 7! F (B A [q]). It is natural to consider the colimit of its total* *ization, as B ranges over a class of A-algebras. When F is the identity functor, this is* * the completion defined above. When A ! B is Galois, or ranges through all Galois extensions, we obtain forms of Amitsur cohomology [Am59] and Galois cohomology [CHR65, x5]. Note that if Spec B is thought of as a covering of Spec A, then Spec (B ^A B) consists of the covering of Spec A by double intersections, or fi* *ber products, from the first covering, and likewise for Spec Cq(B=A) and (q + 1)-fo* *ld intersections. We are therefore recovering a form of Ce~ch cohomology. In gener* *al, the appropriate context for what classes of extensions A ! B to consider is that of a Grothendieck model topology on the category of commutative A-algebras, or a model site. We simply refer to [TV:h] for a detailed exposition on this matter. The following is a form of faithfully projective descent. Lemma 8.2.2. If B is faithful and dualizable over A, then j :A ! A^B is a weak equivalence, i.e., A is complete along B. Proof. It suffices to prove that 1 ^ j :B ^A A ! B ^A A^B is a weak equivalence. Here B ^A A^B ' FA (DA B, TotCo(B=A)) ~= TotFA (DA B, Co(B=A)) ' Tot B ^A Co(B=A), and B ^A Co(B=A): [q] ! B ^A (B A [q]) ~=B A [q]+ admits a cosimplicial contraction to B, so 1 ^ j is indeed a weak equivalence. Let G be a topological group acting from the left on an S-module M, and let EGo = B(G, G, *): [q] 7! Map ([q], G) ~=Gq+1 be the usual free contractible simplicial left G-space. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 53 Definition 8.2.3. The (group) cobar complex for G acting on M is the cosimplic* *ial S-module Co(G; M) = F (EGo+ , M)G :[q] 7! F (Gq+1+, M)G ~= F (Gq+, M) . Its totalization is the homotopy fixed point spectrum MhG = Tot Co(G; M). Here the standard identification F (Gq+1+, M)G ~= F (Gq+, M) takes the left * *G-map f :Gq+1+! M to the map OE: Gq+! M that satisfies f(g0, . .,.gq)= g0 . OE([g-10g1| . .|.g-1q-1gq]) OE([h1| . .|.hq])= f(e, h1, . .,.h1 . .h.q) (adapted as needed to make sense when the target is a spectrum). In the presence of a left G-action on B through commutative A-algebra maps, these two cosimplicial constructions can be compared. Definition 8.2.4. There is a natural map of cosimplicial commutative A-algebras ho :Co(B=A) ! Co(G; B) given in codegree q by the map hq: B ^A . .^.AB ! F (Gq+1+, B)G ~= F (Gq+, B) given symbolically by b0 ^ . .^.bq7! f :(g0, . .,.gq) 7! g0(b0) . . ...gq(bq) ~= OE: [h1| . .|.hq] 7! b0 . h1(b1) . . ...(h1 . .h.q)(bq)* * . On totalizations, ho induces a natural map of commutative A-algebras h0: A^B ! BhG . In codegree 1, we can recognize h1: B ^A B ! F (G+ , B) as the canonical map h from (4.1.2). It is not hard to give a formal definition of hq as the right a* *djoint of a G-equivariant map B A [q] ^ Map ([q], G)+ ! B. Lemma 8.2.5. Let G be a stably dualizable group acting on B through A-algebra maps, and suppose that h :B ^A B ! F (G+ , B) is a weak equivalence. Then ho is a codegreewise weak equivalence that induces a weak equivalence h0: A^B! BhG . Proof. In each codegree q, the map hq factors as a composite of weak equivalenc* *es of the form B^A i^A F (Gj+, B) -'!B^A (i-1)^A F (Gj+, B ^A B) '-!B^A (i-1)^ j ~ ^A (i-1) (j+1) A F (G+ , F (G+ , B)) = B ^A F (G+ , B) with j = 0, . .,.q - 1 and i + j = q. Here the first map is a weak equivalence because G, and thus Gj, is stably dualizable, and the second map is a weak equi* *v- alence because h: B ^A B ! F (G+ , B) is assumed to be one. The claim follows by induction. The following is close to Proposition 6.3.2. See also Proposition 12.1.8. Proposition 8.2.6. Let G be a stably dualizable group acting on B through com- mutative A-algebra maps, and suppose that h :B ^A B ! F (G+ , B) is a weak equivalence. Then A ! B is G-Galois if and only if A is complete along B Proof. We have i = h0O j, with h0 a weak equivalence, so i: A ! BhG is a weak equivalence if and only if j :A ! A^B is a weak equivalence. 54 JOHN ROGNES 9. Separable and 'etale extensions We now address structured ring spectrum analogues of the unique lifting prop- erties in covering spaces, continuing to work implicitly in some E-local catego* *ry. Throughout, we let A be a cofibrant commutative S-algebra and B a cofibrant associative (or cofibrant commutative) A-algebra. Starting in Section 9.4, only commutative B are considered. Our main observations are that G-Galois extensions A ! B with G discrete are necessarily separable and dualizable, hence symmetrically 'etale (= thh-'et* *ale) and 'etale (= taq-'etale). In most cases of current interest, including E = S * *and E = K(n) for 0 n 1, a discrete group G is stably dualizable if and only if * *it is finite. 9.1. Separable extensions. The algebraic definition [KO74, p. 74] of a separable extension of commutati* *ve rings can be adapted to stable homotopy theory as follows. Definition 9.1.1. We say that A ! B is separable if the A-algebra multiplicati* *on map ~: B ^A Bop ! B, considered as a map in the stable homotopy category DB^A Bop of B-bimodules relative to A, admits a section oe :B ! B ^A Bop. Equiv- alently, there is a map oe :B0 ! B ^A Bop of B-bimodules relative to A, such th* *at the composite ~oe :B0 ! B is a weak equivalence. Here Bop is B with the opposite A-algebra multiplication ~fl :B ^A B ~= B ^A B ! B. It equals B precisely when B is commutative. Since B will rarely be cofibrant as a B-bimodule relative to A, it is only reasonable to ask for the e* *xistence of a bimodule section oe in the stable homotopy category. The condition for A !* * B to be separable only involves the bimodule structure on B, so it is quite acces* *sible to verification by calculation. For example, it is equivalent to the condition* * that the algebra multiplication ~ induces a surjection ~# :THH0A(B, B ^A Bop) ! THH0A(B, B) of zero-th topological Hochschild cohomology groups. Lemma 9.1.2. Let A ! B be a G-Galois extension, with G a discrete group. Then A ! B is separable. Proof. Let d: G+ ! {e}+ be the continuous (Kronecker delta) map given by d(e) = e (the unit element in G) and d(g) = * (the base point) for g 6= e. Its functi* *onal dual ine = d# :B ~=F ({e}+ , B) ! F (G+ , B) and the canonical weak equivalence h define the required weak B-bimodule section oe = h-1 O ine to ~, as a morphism in the stable homotopy category. (9.1.3) B F` oe`//`B ^A B__~__//_B;; FF xxx FFF ' |h xxx ineFFF##fflffl||Qprexxx G B GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 55 Proposition 9.1.4. Let A ! B be a faithful G-Galois extension, with G a discre* *te group and K G an allowable subgroup. Then A ! C = BhK is separable. Proof. The canonical map G ! G=K admits a continuous section, because G is discrete, so S[G] dualizable implies that S[G=K] is dualizable by Lemma 3.3.2(c* *). We are therefore in theQsituation of Lemma 7.2.4. The map h: B ^A B ! G B is (K x K)-equivariant with respect to the action (k1, k2).(b1^b2) = k1(b1)^k2(b2) in the source, and the action that takes a seq* *uence {g 7! OE(g)} to the sequence {g 7! k1(OE(k-11gk2))} in the target. There are ma* *ps Y inK Y prK Y B --! B --! B K G K functionally dual to a (characteristic) map dK : G+ ! K+ (taking G\K to the bas* *eQ point) and the inclusion K+ G+ , whose composite is the identity. We give K* * B the (K x K)-action that takes {k 7! OE(k)} to {k 7! k1(OE(k-11kk2))},Qso that i* *nK and prK are (K xK)-equivariant. TheQweak equivalence B ! ( K B)hK induces a natural weak equivalence BhK ! ( K B)h(KxK) that makes the following diagram commute: ~ C ` ` ` ` ` ` `//BhK ^A BhK ______________//C = || ' |~| |=| fflffl| fflffl| |fflffl BhK ` ` ` ` `//(B ^A B)h(KxK) ___________//BhK ' || ' |h#| |'| Q fflffl| inK# Q fflffl| prK# Q |fflffl ( K B)h(KxK) ______//_( G B)h(KxK)______//_( K B)h(KxK) The vertical map ~ is a weak equivalence by Lemma 7.2.4, and the maps h# and prK# OinK# are obtained from weak equivalences by passage to (K xK)-homotopy fixed points, so a little diagram chase shows that ~: C ^A C ! C does indeed ad* *mit a weak bimodule section. Remark 9.1.5. It is easy to see that separable extensions are preserved by base change. To detect separable extensions by faithful base change will require so* *me additional hypotheses, as in [KO74, III.2.2]. 9.2. Symmetrically 'etale extensions. The topological Hochschild homology THHA (B) of B relative to A is the geome* *tric realization of a simplicial A-module oo____ ______//oo_oo_ B _____//_Bo^AoB_oo___////_Bo^AoB_^AoBo___////_.o.o._oo_ with the smash product of (q + 1) copies of B in degree q. See [EKMM97, IX.2]. Alternatively, THHA (B) can be computed in the stable homotopy category as op L Tor B^A B (B, B) = B ^B^A Bop B . 56 JOHN ROGNES In the case A = S, it agrees with the topological Hochschild homology THH(B) introduced by Marcel B"okstedt [BHM93]. The inclusion of 0-simplices defines a * *nat- ural map i :B ! THHA (B). When B is commutative, THHA (B) can be expressed in terms of the topologically tensored structure on CA as B A S1. It is also possible to define THHA (B) for non-commutative A, by analogy with the definition of Hochschild homology over a non-commutative ground ring [Lo98, 1.2.11], but we have found no occasion to make use of this more general definit* *ion. Definition 9.2.1. We say that A ! B is formally symmetrically 'etale (= formal* *ly thh-'etale) if the map i :B ! THHA (B) is a weak equivalence. If furthermore B is dualizable as an A-module, then we say that A ! B is symmetrically 'etale (= thh-'etale). Remark 9.2.2. This definition of an (symmetrically) 'etale map does not quite c* *on- form to the algebraic case, in that it may be too restrictive to ask that B is * *dualiz- able as an A-module. Instead, it is likely to be more appropriate to only impos* *e the dualizability condition locally with respect to some Zariski open cover of Spec* * A. This may be taken to mean that for some set of (smashing, Bousfield) localizati* *on functors {LEi}i, such that the collection {A ! LEiA}i is a faithful cover in the sense of Definition 4.3.1, each localization LEiB is dualizable as an LEiA-modu* *le. The author is undecided about exactly which localization functors to allow. How- ever, for Galois extensions the stronger (global) dualizability hypothesis will* * always be satisfied, and this may permit us to leave the issue open. Example 9.2.3. Note that Definition 9.2.1 implicitly takes place in an E-local category. By [MS93, 5.1], the inclusion i :` ! THH(`) is a K(1)-local equivalen* *ce, where ` = BP <1> is the p-local connective Adams summand of topological K- theory, so S ! ` is K(1)-locally formally symmetrically 'etale. It also follo* *ws that the localization of this map, J^p = LK(1)S ! LK(1)` = L^p is K(1)-locally formally symmetrically 'etale. Here L^pis the p-complete periodic Adams summand, as in 5.5.2. These maps are not K(1)-locally symmetrically 'etale, because L^pis not dual- izable as a J^p-module. More globally, S ! L^pfails to be E(1)-locally formally symmetrically 'etale. For by [MS93, 8.1], THH(L^p) ' L^p_ L0( L^p), so i has a rationally non-trivial cofiber. Similarly, i :ku ! THH(ku) is a K(1)-homology equivalence by Christian Au- soni's calculation [Au:t, 6.5], so the map S ! ku to connective topological K- theory, and its K(1)-localization J^p ! KU^p, are K(1)-locally formally symmet- rically 'etale. The map L^p ! KU^p is K(1)-locally F*p-Galois, as noted in 5.5* *.2, so by Lemma 9.2.6 below, L^p ! KU^p is K(1)-locally symmetrically 'etale. In ^ ^ other words, i :ku ! THH`(ku) and i :KU^p ! THHLp (KUp ) are K(1)-local equivalences. The terminology "thh-'etale" is that of Randy McCarthy and Vahagn Minasian [MM03, 3.2], except that for brevity they suppress the distinction between the formal and non-formal cases. The author's lengthier term "symmetrically 'etale" was motivated by the following definitions and result. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 57 Definition 9.2.4. Let M be a B-bimodule relative to A, i.e., a B ^A Bop-module. The space of associative A-algebra derivations of B with values in M is defined* * to be the derived mapping space ADer A (B, M) := (AA =B)(B, B _ M) in the topological model category of associative A-algebras over B, where pr1: * *B _ M ! B is the square-zero A-algebra extension of B with fiber M. We say that a B-bimodule relative to A is symmetric if it has the form ~!N for some B-module N, i.e., if the bimodule action is obtained by composing with the A-algebra mul* *ti- plication map ~: B ^A Bop ! B. Proposition 9.2.5. A ! B is formally symmetrically 'etale if and only if the space of associative derivations ADer A(B, M) is contractible for each symmetr* *ic B-bimodule M. Proof. Let B=A be a cofibrant replacement of the homotopy fiber of ~: B^A Bop ! B in the category of B-bimodules relative to A. There is a cofiber sequence B ^B^A Bop B=A ! B -i!THHA (B) and for each B-module N, with associated symmetric B-bimodule M = ~!N, there is an adjunction equivalence MB^A Bop( B=A , M) ' MB (B ^B^A Bop B=A , N) . Furthermore, there is an equivalence (for each B ^A Bop-module M) ADer A (B, M) = (AA =B)(B, B _ M) ' MB^A Bop( B=A , M) obtained by Lazarev [La01, 2.2]. So i is an equivalence if and only if B ^B^A B* *op B=A ' *, which is equivalent to ADer A (B, M) ' MB (B ^B^A Bop B=A , N) being contractible for each symmetric B-bimodule M = ~!N. In the E-local context, this argument shows that E*(i) is an isomorphism if * *and only if ADer A (B, M) ' * for each E-local symmetric B-module M. For AA,E =B is a full subcategory of AA =B, and likewise for the homotopy categories. Lemma 9.2.6. Each separable extension A ! B of commutative S-algebras is formally symmetrically 'etale. In particular, each G-Galois extension A ! B with G discrete is symmetrically 'etale. Proof. By assumption there is a bimodule section oe so that the composite B -oe! B ^A Bop -~!B is homotopic to the identity. Smashing with B over B ^A Bop tells us that the composite THHA (B) -oe^1-!B -i!THHA (B) is an equivalence. Furthermore, there is a retraction ae: THHA (B) ! B given in simplicial degree q by the iterated multiplication map ~(q):B ^A . .^.AB ! B, since we are assuming that B is commutative. Therefore i admits a right and a left inverse, up to homotopy, and is therefore a weak equivalence. When A ! B is G-Galois with G discrete, we showed in Lemma 9.1.2 that A ! B is separable and in Proposition 6.2.1 that B is a dualizable A-module. The above argument then implies that A ! B is symmetrically 'etale. 58 JOHN ROGNES 9.3. Smashing maps. Maps A ! B having the corresponding property to the conclusion of Propo- sition 9.2.5 for associative derivations into arbitrary (not necessarily symmet* *ric) B-bimodules relative to A, also have a familiar characterization. This material* * is not needed for our Galois theory, but nicely illustrates the relation of smashi* *ng localizations (and Zariski open sub-objects) to 'etale and symmetrically 'etale* * maps. Definition 9.3.1. We say that A ! B is smashing if the algebra multiplication map ~: B ^A Bop ! B is a weak equivalence. In view of the following proposition, smashing maps could also be called for* *mally associatively 'etale extensions. Proposition 9.3.2. A ! B is smashing if and only if ADer A (B, M) is contracti* *ble for each B-bimodule M relative to A. Proof. This is immediate from the equivalence ADer A (B, M) ' MB^A Bop( B=A , M) from [La01], since A ! B is smashing if and only if B=A ' *. The terminology is explained by the following result, one part of which the * *author learned from Mark Hovey. Proposition 9.3.3. A ! B is smashing if and only if LM = B ^A M defines a smashing Bousfield localization functor on MA , in which case B = LA. In particular, B will be a commutative A-algebra. Proof. Let BA*(-) be the homotopy functor on MA defined by BA*(M) = ss*(B ^A M). The natural map M ! B ^A M is a BA*-equivalence, since A ! B is smashing, and B ^A M is BA*-local by the prototypical ring spectrum argument of Adams [Ad71]: If B ^A Z ' * then any map f :Z ! B ^A M factors as Z ! B ^A Z -1^f-!B ^A B ^A M -~^1-!B ^A M and is therefore null-homotopic. So LM = B^A M defines a (Bousfield) localizati* *on functor L on MA . Conversely, a smashing localization functor L on MA produces an associative A-algebra B = LA, by [EKMM97, VIII.2.1], such that LM ' B^A M (since L is as- sumed to be smashing). The idempotency of L then ensures that the multiplication map B ^A Bop ! B is a weak equivalence. Lemma 9.3.4. Each smashing map A ! LA is separable, hence formally sym- metrically 'etale. Proof. If A ! B = LA is smashing, then ~: B ^A Bop ! B is an equivalence. It therefore admits a bimodule section oe up to homotopy, so A ! B is separable. In general, LA is not dualizable as an A-module, as easy algebraic examples illustrate (Z Z(p)). Instead, the local dualizability of Remark 9.2.2 is more appropriate. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 59 9.4. E'tale extensions. We keep on working implicitly in an E-local category, now with B a cofibrant commutative A-algebra. For a map A ! B of commutative S-algebras, the topological Andr'e-Quillen homology TAQ(B=A) is defined in [Ba99, 4.1] as TAQ(B=A) := (LQB )(RIB )(B ^LAB) , i.e., as the B-module of (left derived) indecomposables in the non-unital B-alg* *ebra given by the (right derived) augmentation ideal in the augmented B-algebra defi* *ned by the (left derived) smash product B ^LAB, augmented over B by the A-algebra multiplication ~. Definition 9.4.1. Let A ! B be a map of commutative S-algebras. We say that A ! B is formally 'etale (= formally taq-'etale) if TAQ(B=A) is weakly equivale* *nt to *. If furthermore B is dualizable as an A-module, then we say that A ! B is 'etale (= taq-'etale). Like in Remark 9.2.2, the condition that B is dualizable over A is likely to* * be stronger than necessary for B to qualify as 'etale over A, and should eventuall* *y be replaced with a local condition over each subobject in an open cover of A. The discussion from the associative/symmetric case applies unchanged here. The terminology is justified by the following definition and result from [Ba* *99]. The vanishing of TAQ(B=A) gives a unique infinitesimal lifting property, up to contractible choice, for geometric maps into the affine covering represented (i* *n the opposite category) by a formally 'etale map A ! B. BO___=____//GBOOO | G | | G G | | G##| A _____//_B _ M Compare [Mi80, I.3.22]. Definition 9.4.2. Let A ! B be a map of commutative S-algebras and let M be a B-module. The space of commutative A-algebra derivations of B with values in M is defined to be the derived mapping space CDerA (B, M) := (CA =B)(B, B _ M) in the topological model category of commutative A-algebras over B, where pr1: * *B_ M ! B is the square-zero extension of B with fiber M. Proposition 9.4.3. A map A ! B of commutative S-algebras is formally 'etale if and only if CDer A(B, M) is contractible for each B-module M. Proof. There is an equivalence CDer A (B, M) = (CA =B)(B, B _ M) ' MB (TAQ(B=A), M) 60 JOHN ROGNES for each B-module M, by [Ba99, 3.2]. By considering the universal example M = TAQ(B=A), we conclude that TAQ(B=A) ' * if and only if CDer A(B, M) ' * for each B-module M. In the implicitly local context only E-local M occur, so we can conclude that TAQ(B=A) is E-acyclic, i.e., E-locally weakly equivalent to *. For a finite commutative R-algebra T , the two conditions T ~=HHR*(T ) and D*(T=R) = AQ*(T=R) = 0 are logically equivalent [Gr67, 18.3.1(ii)]. In the con- text of commutative S-algebras this is only true subject to a connectivity hypo* *thesis [Mi03, 2.8], due to a convergence issue in the analogue of the Quillen spectral* * se- quence from Andr'e-Quillen homology (D* = AQ*) to Hochschild homology (HH*). However, one implication (from symmetrically 'etale to 'etale) does not depend * *on the connectivity hypothesis stated there. In other words, if i :B ! THHA (B) is a weak equivalence, then TAQ(B=A) ' *. We discuss a proof below, based on [BM:h]. There is a counterexample to the opposite implication, due to Mike Mandell, which is discussed in [MM03, 3.5]. For n 2 let X = K(Z=p, n) be an Eilenberg- Mac Lane space and let B = F (X+ , HFp) be its mod p cochain HFp-algebra, with ss*(B) = H-* (K(Z=p, n); Fp). Then HFp ! B is formally 'etale, but not symmet- rically (=thh-)'etale. So, any converse statement deducing that an 'etale map * *is symmetrically 'etale must contain additional hypotheses to exclude this example. Lemma 9.4.4. Each (formally) symmetrically 'etale extension A ! B of commu- tative S-algebras is (formally) 'etale. In particular, each G-Galois extension * *A ! B with G discrete is 'etale, and each smashing localization A ! LA = B is formally 'etale. Proof. Recall that THHA (B) ' B A S1 as commutative A-algebras. Here A denotes the tensored structure on CA over unbased topological spaces. To descri* *be the commutative B-algebra structure on THHA (B) in similar terms, and to relate it to the B-module TAQ(B=A), we will need a tensored structure over based topo- logical spaces. This makes sense when we replace CA by the pointed category CB * *=B of commutative B-algebras augmented over B. There is then a (reduced) tensor structure (-)e B X on CB =B over based topological spaces X, with (CB =B)(C eB X, C0) ~=Map *(X, (CB =B)(C, C0)) , where Map * denotes the base-point preserving mapping space. It follows that (C eB X)e B Y ~=C eB (X ^ Y ). The unbased and based tensored structures are related by C eB X ~=B ^C (C B X) and C B T ~= C eB (T+ ), for unbased spaces * *T . There is a pointed model structure on CB =B, and the associated Quillen susp* *en- sion functor E is given on cofibrant objects by the reduced tensor E (C) = C eB* * S1 with the based circle. For each n 0 we can form the n-fold iterated suspension En (C) = C eB Sn in CB =B, so that E(E n(C)) ~=E n+1(C), and these objects assemble to a sequent* *ial suspension spectrum E 1(C), in this category. By [BM:h, Thm. 3], the homotopy category of such spectra, up to stable equivalence, is equivalent to the homoto* *py category DB of B-modules, up to weak equivalence. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 61 Base change along A ! B takes B to B ^A B, which is a cofibrant commutative B-algebra, augmented over B by the multiplication map ~: B ^A B ! B. Here- after, write C = B ^A B for brevity. By [BM:h, Thm. 4], the cited equivalence takes E1 (C) to the topological Andr'e-Quillen homology spectrum TAQ(B=A). So E 1 (C) is stably trivial if and only if TAQ(B=A) ' *, i.e., if and only if A !* * B is formally 'etale. On the other hand, E (C) = C eB S1 ' B ^C THHB (C) ~=THHA (B) , now as commutative B-algebras. So E (C) is weakly trivial, i.e., weakly equival* *ent to the base point B in CB =B, if and only if i :B ! THHA (B) is a weak equivale* *nce. The proof of the lemma is now straightforward. If A ! B is formally symmetri- cally 'etale, then E (C) is weakly trivial, and therefore so is each of its sus* *pensions E n(C) = E n-1(E (C)) for n 1. Thus the suspension spectrum E 1(C) is stably trivial (in a very strong sense), and so TAQ(B=A) is weakly equivalent to the t* *rivial B-module. In the notation of the above proof: C = B ^A B is weakly trivial in CB =B if and only if A ! B is smashing, E (C) = THHA (B) is weakly trivial if and only if A ! B is formally symmetrically 'etale, and E 1(C) is stably trivial if and onl* *y if A ! B is formally 'etale. 9.5. Henselian maps. By definition, an 'etale map A ! B has the unique lifting property up to con- tractible choice for each square-zero extension of commutative A-algebras B _M ! B, and satisfies a finiteness condition. In this chapter we conversely ask whic* *h ex- tensions D ! C of commutative A-algebras are such that each 'etale map A ! B, with B mapping to C, has this homotopy unique lifting property with respect to D ! C. BO@_____//_COOO || @ @ || | @__| A ______//D We shall refer to such D ! C as Henselian maps. Section 9.6 will exhibit some interesting examples of Henselian maps. In the opposite category to that of commutative A-algebras, of affine algebr* *o- geometric objects in a homotopy-theoretic sense [TV:h, x5.1], we can view the square-zero extensions as infinitesimal thickenings of a special kind, forming * *a gen- erating class of acyclic cofibrations. The 'etale extensions then correspond to* * smooth and unramified covering maps, and constitute a class of fibrations characterize* *d by their right lifting property with respect to these generating acyclic cofibrati* *ons, together with a finiteness hypothesis. The Henselian maps, in turn characteriz* *ed by their left lifting property with respect to these fibrations, then form a cl* *ass of thickenings that contains all composites of the generating acyclic cofibrati* *ons of the theory, i.e., all infinitesimal thickenings, but which also encompasses * *many 62 JOHN ROGNES other maps. By comparison, in the algebraic context Hensel's lemma applies to a complete local ring mapping to its residue field, but also to many other cases. For a fixed commutative S-algebra A, this discussion could take place as abo* *ve in the context of commutative A-algebras, with maps from (taq-)'etale extensions A ! B, but also in the alternative context of associative A-algebras, with maps from symmetrically (= thh-)'etale extensions. To be concrete we shall focus on * *the commutative case, although all of the formal arguments carry over to the associ* *ative category and extensions by symmetric bimodules. Throughout this section we continue to work E-locally, and let A be a cofibr* *ant commutative S-algebra, B ! C a map of commutative A-algebras and M any C- module. We sometimes consider M as a B-module by pull-back along B ! C. We always make the cofibrant and fibrant replacements required for homotopy invari- ance, implicitly. The proofs of the following lemmas are routine for (topologi* *cal) model categories. Lemma 9.5.1. The square-zero extension B _ M ! B is the pull-back in CA of the square-zero extension C _ M ! C, along B ! C, BOO___=___//_GTTB_______//COOOO | G T T T| | | GG | T T T |pr1 | G##| T** | A ______//B _ M _____//_C _ M so there is a weak equivalence (CA =B)(B, B _ M) ' (CA =C)(B, C _ M) . In particular, both of these spaces are contractible whenever A ! B is formally 'etale. Lemma 9.5.2. The commutative diagram BO________________//_QQ@CO;;wOO | @ Q Q = www | | @ QQwww pr1| | OO@ww Q Q(( | A _____//_C_in1_//C _ M yields a homotopy fiber sequence (CA =C _ M)(B, C) ! (CA =C)(B, C) ! (CA =C)(B, C _ M) for which the middle space is contractible. In particular, all three spaces ar* *e con- tractible whenever A ! B is formally 'etale. The following definition is the commutative analogue of that in [La01, 3.3]. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 63 Definition 9.5.3. A map ss :D ! C of commutative A-algebras is a singular ex- tension if there is an A-linear derivation of C with values in M, i.e., a commu* *tative A-algebra map d: C ! C _ M over C, and a homotopy pull-back square COO__d__//_CO_OM ss|| |in1| | | D _______//_C of commutative A-algebras. For example, the square-zero extension C _ M ! C is the singular extension pulled back from the trivial derivation in1: C ! C _ M. So the class of singul* *ar extensions contains the class of square-zero extensions. Lemma 9.5.4. For each singular extension ss :D ! C the commutative diagram BOO_____//_@RRCd_//CO_OMOO | R | | | @ @ sRs|R R in1| | __@| R R((| A _____//_D_______//C induces a weak equivalence (CA =C)(B, D) ' (CA =C _ M)(B, C) . In particular, both of these spaces are contractible whenever A ! B is formally 'etale. In view of [Mi80, I.4.2(d)] we make the following definition. Definition 9.5.5. Let D ! C be a map of commutative S-algebras. We say that D ! C is Henselian if for each 'etale map A ! B, with B and D commutative A-algebras over C, BO@_____//_COOO || @ @ || | @__| A ______//D the derived mapping space (CA =C)(B, D) ' * is contractible, i.e., if A ! B has the unique lifting property up to contracti* *ble choice with respect to D ! C. If D is a commutative S-algebra and C is an associative D-algebra, we say th* *at D ! C is symmetrically (= thh-)Henselian if for each symmetrically (= thh-)'eta* *le map A ! B, in a diagram as above, the associative A-algebra mapping space (AA =C)(B, D) is contractible. By the following lemma it suffices (in the commutative case) to verify the h* *omo- topy unique lifting property for the 'etale maps A ! B with A = D. For A ! B 'etale implies D ! B ^A D 'etale by the base change formula TAQ(B ^A D=B) ' TAQ(B=A) ^A D [Ba99, 4.6] and Lemma 6.2.3. 64 JOHN ROGNES Lemma 9.5.6. Let B ! C and D ! C be maps of commutative A-algebras, with pushout B ^A D ! C. The commutative diagram BO______//SSSBH^A_D___//_COOOOO | S S| H H | | |S S S H H | A|_________//D_____S_S))##//_|| = D induces a weak equivalence (CA =C)(B, D) ' (CD =C)(B ^A D, D) . Proposition 9.5.7. The class of Henselian maps D ! C contains the square-zero extensions C _ M ! C and the singular extensions ss :D ! C. It is closed under weak equivalences, compositions, retracts and filtered homotopy limits (for dia* *grams of maps to a fixed C). Proof. The first claims follow from Lemma 9.5.4 and the remark that square-zero extensions are trivial examples of singular extensions. The closure claims are * *clear, perhaps except for the the last one. If ff 7! (Dff! C) is a diagram of Henseli* *an maps to C, then let D = holim ffDff. For each 'etale map A ! B (mapping to D ! C as above) there is a weak equivalence (CA =C)(B, D) ' holimff(CA =C)(B, Dff) ' * , since each Dff! C is Henselian and the limit category is assumed to be filter- ing. In fact, the Henselian maps that we will encounter in the following section * *are sequential homotopy limits of towers of singular extensions, and thus of a rath* *er special form. If desired, the reader can view them as the residue maps of compl* *ete local rings, and refer to them as formal thickenings, rather than as general He* *nselian maps. 9.6. I-adic towers. For the duration of this section, let R be a commutative S-algebra and R=I an associative (or commutative) R-algebra. Define the R-module I by the cofiber sequence I ! R ! R=I, and let I(s)= I ^R . .^.RI (s copies of I) be its s-fold smash power over R, for each s 1. Define the R- module R=I(s) by the cofiber sequence I(s)! R ! R=I(s). There is then a tower of R-modules (9.6.1) R ! . .!.R=I(s)! . .!.R=I GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 65 that Baker and Lazarev [BL01, x4] refer to as the external I-adic tower. Earlier on, Lazarev [La01, 7.1] proved that this diagram can be given the st* *ruc- ture of a tower of associative R-algebras, and that each cofiber sequence I(s)=I(s+1) ! R=I(s+1) ss-!R=I(s) is a singular extension of associative R-algebras. It remains an open problem * *to decide when the diagram (9.6.1) can be realized as a tower of commutative R- algebras, and whether each map R=I(s+1) ! R=I(s) can be taken to be a singular extension in the commutative context. See [La04, 4.5] for a remark on a similar problem for square-zero extensions. The homotopy limit L^RR=IR = holim R=I(s) s of the tower is the Bousfield R=I-nilpotent completion of R, formed in the cate* *gory of R-modules (or R-algebras). It is in general not the same as the Bousfield R=* *I- localization of R, formed in the category of R-modules, which we denote by LRR=* *IR. However, the case when R is even graded, i.e., the homotopy ring ss*(R) is concentrated in even degrees, and R=I is a commutative regular quotient, i.e., R=I is a homotopy commutative R-algebra and ss*(I) is an ideal in ss*(R) that is generated by a regular sequence, is manageable. Baker and Lazarev [BL01, 6.3] use an internal I-adic tower to prove that for such R and I the R=I-nilpotent completion and the R=I-localization of R, both formed in R-modules, do in fact agree LRR=IR ' ^LRR=IR and have the expected homotopy ring ss*LRR=IR ~=ss*(R)^ss*(I). Proposition 9.5.7 therefore has the following consequence, which admits some fairly obvious localized generalizations that we shall also make use of. Proposition 9.6.2 (Baker-Lazarev). Let R be an even graded commutative S- algebra, and R=I a homotopy commutative regular quotient R-algebra. Then the limiting map LRR=IR ' holimsR=I(s)! R=I is symmetrically Henselian, and induces the canonical surjection ss*(R)^ss*(I)! ss*(R)=ss*(I) of homotopy rings. In particular, if ss*(R) is already ss*(I)-adically complete* *, then R ! R=I is symmetrically (=thh-)Henselian. We now claim that the complex cobordism spectrum MU can be viewed as a global model, up to Henselian maps, of each of the commutative S-algebras E[(n) = LK(n)E(n) that occur as fixed S-algebras in the p-primary K(n)-local 66 JOHN ROGNES pro-Galois extensions LK(n)S ! En ! Enrn. So, even if there is ramification between the expected maximal unramified Galois extensions (covering spaces) over the different chromatic strata, reflected in the changing pro-Galois groups Gn * *and Gnrnfor varying n and p, these can all be compensated for by appropriate Hensel* *ian maps (formal thickenings), and unified into one global model, namely MU. For the sphere spectrum S, the chromatic stratification we have in mind is f* *irst branched over the rational primes p, and then S(p) is filtered by the Bousfield localizations LnS = LE(n)S for each n 0. The associated (Zariski) stack has the category MS,E(n) of E(n)-local S-modules over the n-th open subobject in the filtration, and the n-th monochromatic category of E(n)-local E(n - 1)-acyc* *lic S-modules over the n-th half-open stratum. The latter category is equivalent to the category MS,K(n) of K(n)-local S-modules, at least in the sense that their homotopy categories are equivalent [HSt99, 6.19]. The latter K(n)-local module category is in turn equivalent to the category of K(n)-local LK(n)S-modules, and we propose to understand it better by way of Galois descent from the related categories MB,K(n) of K(n)-local B-modules, for the various K(n)-local Galois extensions LK(n)S ! B. The limiting case of pro-Galois descent from K(n)-local modules over B = Enrn, or over the separable closure B = E~n (cf. Section 10.3), can optimistically be hoped to be particula* *rly transparent. This decomposition of the sphere spectrum, appearing in the lower row in the diagram below, can be paralleled for MU by applying the same Bousfield local- ization functors. However, the proposition above indicates that it may be more appropriate to localize MU in the category of MU-modules, rather than in the category of S-modules. In other words, we are led to focus attention on the upp* *er row, rather than the middle row, in the following commutative diagram. (9.6.3) MUOO _____//_LMUE(n)MU____//_LMUK(n)MU | OO OO = || || || | | | MUOO _____//_LE(n)MU _____//_LK(n)MU | OO OO | | | | | | | | | S ________//_LE(n)S________//_LK(n)S The coefficient rings of the various localizations of MU occurring in the di* *agram above are mostly understood. See [Ra92, 8.1.1] for ss*LE(n)MU (or rather, its B* *P - version). Let Jn ss*MU(p)be the kernel of the ring homomorphism ss*MU(p)! ss*E(n), i.e., the ideal generated by the kernel of ss*MU(p)! ss*BP and the cla* *sses vk for k > n. Let In = (p, . .,.vn-1 ), also considered as an ideal in ss*MU(p* *), so that the sum of ideals In + Jn is the kernel of the ring homomorphism ss*MU(p)! ss*K(n). Then ss*LK(n)MU = ss*MU(p)[v-1n]^In by [HSt99, 7.10(e)]. By [HSa99, Thm. B], LK(n)BP splits as the K(n)-localization of an explicit countable wedge sum of suspensions of [E(n). It follows that LK(* *n)MU splits in a similar way. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 67 By Proposition 9.6.2, applied to R = MU(p)[v-1n] and R=I = E(n), we find that LMUE(n)MU ' LRR=IR ! E(n) is symmetrically Henselian, with ss*LMUE(n)MU = ss*MU(p)[v-1n]^Jn. By the same proposition applied to R = MU(p)[v-1n] and R=I = K(n), at least for p 6= 2 (to ensure that K(n) is homotopy commutative), we also find that LMUK(n)MU ' LRR=IR ! K(n) is symmetrically Henselian, with (9.6.4) ss*LMUK(n)MU = ss*MU(p)[v-1n]^In+Jn. This differs from the K(n)-localization of MU in S-modules by the additional completion along Jn. This K(n)-local part, in MU-modules, of the global commutative S-algebra MU, can now be related by a symmetrically Henselian map to the K(n)-local, nearly pro-Galois, extension LK(n)S ! [E(n). (9.6.5) EnOO____'?____//LMUK(n)En OO | | | | | | E[(n) ___'?___//LMUK(n)E(n) OO ffMM OO | M qM | | M M | | | LK(n)MUOO ______//LMUK(n)MU OO | | | | | | LK(n)S ___=____//_LK(n)S Here the horizontal map [E(n) = LK(n)E(n) ! LMUE(n)E(n), and its analogue for En, are both plausibly weak equivalences. For instance, the corresponding map of localizations of MU induces completion along Jn at the level of homotopy groups, and ss*E[(n) and ss*En are already Jn-adically complete in a trivial way. We shall now apply Proposition 9.6.2 with R = LMUK(n)MU, which is an even graded commutative S-algebra. Considering Jn as an ideal in ss*R, which is give* *n by formula (9.6.4), it is still generated by a regular sequence, and (ss*R)=Jn ~=s* *s*E[(n). So we can form R=I ' E[(n) as a homotopy commutative regular quotient R- algebra. Then ss*(I) = Jn, and ss*(R) is Jn-adically complete, so by the last c* *lause of Proposition 9.6.2 the map q :R ! R=I, labeled q in the diagram (9.6.5) above, is symmetrically Henselian. Corollary 9.6.6. Each K(n)-local pro-Galois extension LK(n)S ! En factors as the composite map of commutative S-algebras LK(n)S ! LMUK(n)MU -q!E[(n) ! En , 68 JOHN ROGNES where the first map admits the global model S ! MU, the second map is symmet- rically (= thh-)Henselian, and the third map is a K(n)-local pro-Galois extensi* *on. In other words, each K(n)-local stratum of S is related by a chain of pro-Galois covers LK(n)S ! En [E(n) to a formal thickening q :LMUK(n)MU ! [E(n) of the corresponding K(n)-local stratum of MU, formed in MU-modules. We shall argue in Section 12.2 that there is a Hopf-Galois structure on this global model S ! MU that also encapsulates all the known Galois symmetries over LK(n)S, at least up to the adjunction of roots of unity, i.e., up to the p* *assage from [E(n) to En (or to Enrn). The question remains whether q is (commutatively) Henselian, which would follow if the diagram (9.6.1) could be realized by singu* *lar extensions of commutative S-algebras. After this discussion of the K(n)-local category in MU-modules, we make some remarks on the chromatic filtration in MU-modules. The study of the chromatic filtration and the monochromatic category of S-modules relies on the basic fact [JY80, 0.1] that E(n)*(X) = 0 implies E(n - 1)*(X) for S-modules X, so that there is a natural map LE(n)X ! LE(n-1) X. The analogous claim in the context of MU-modules is false, i.e., that E(n)MU* (X) = 0 implies E(n - 1)MU* (X) = 0, as the easy example X = MU(p)=(vn) illustrates. Thus there is no natural map LMUE(n)X ! LMUE(n-1)X. For brevity, let K[0, n] = K(0) _ . ._.K(n). It is well-known that LK[0,n]= LE(n) in the category of S-modules [Ra84, 2.1(d)], even if the example above sh* *ows that the two localization functors LMUK[0,n]and LMUE(n)in MU-modules cannot be equivalent. For brevity, again, let (9.6.7) LMUn X = LMUK[0,n]X = LMUK(0)_..._K(n)X . It is obvious that K[0, n]MU* (X) = 0 implies K[0, n - 1]MU* (X) = 0, so there * *is a natural map LMUn X ! LMUn-1X. We therefore think that it will be more appropria* *te to filter the category of MU-modules by the essential images MMU . . .MMUMU,K[0,n] MMUMU,K[0,n-1] . . . of these Bousfield localization functors LMUn , i.e., the full subcategories of* * K[0, n]- local MU-modules, within MU-modules, and to consider the MU-chromatic tower (9.6.8) X ! . .!.LMUn X ! LMUn-1X ! . . . for each MU-module X. We then expect that LMUn is a smashing localization, and that there is an equivalence of homotopy categories between the n-th monochro- matic category of MU-modules and the K(n)-local category of MU-modules, like that of [HSt99, 6.19], but we have not verified this expectation. To be precise* *, the monochromatic category in question has objects the MU-modules that are LMUn - local and LMUn-1-acyclic. The K(n)-local category has objects the MU-modules th* *at are LMUK(n)-local. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 69 The thrust of Corollary 9.6.6 is now that the chromatic filtration on S-modu* *les is related to a chromatic filtration on MU-modules, by a chain of pro-Galois ex- tensions and Henselian maps with geometric content. The chromatic filtration on MU-modules is likely to be much easier to understand algebraically, in terms of* * the theory of formal group laws. Taken together, these two points of view may clari* *fy the chromatic filtration on S-modules. 10. Mapping spaces of commutative S-algebras We turn to the computation of the mapping space CA (B, B) for a G-Galois ex- tension A ! B, and related mapping spaces of commutative S-algebras, using the Hopkins-Miller obstruction theory in the commutative form presented by Goerss and Hopkins [GH04]. For the more restricted problem of the classification of co* *m- mutative S-algebra structures, the related obstruction theory of Alan Robinson [Ro03] is also relevant. 10.1. Obstruction theory. Let A be a cofibrant commutative S-algebra and let E be an S-module. We shall need an extension of the Goerss-Hopkins theory to the context of (simplic* *ial algebras over simplicial operads in) the category MA,E of E-local A-modules. The base change to A-modules is harmless, but in working E-locally we may loose the identification of the dualizable A-modules with the (homotopy retracts of) fini* *te cell A-modules, recalled in Proposition 3.3.3 above. It seems clear that only the fo* *rmal properties of dualizable modules are important to the Goerss-Hopkins theory, so that the whole extension can be carried through in full generality. However, f* *or our specific purposes the only dualizable A-modules we must consider will in fa* *ct be finite cell A-modules, so we do not actually need to carry the generalization through. Next, consider a fixed (cofibrant, E-local) commutative A-algebra B. The Goerss-Hopkins spectral sequence [GH04, Thm. 4.3 and Thm. 4.5] for the com- putation of the homotopy groups of commutative A-algebra mapping spaces like CA (C, B), for various commutative A-algebras C, is based on working with a fix* *ed homology theory given by a commutative A-algebra that they call E, but which we will take to be B. In particular, the target B in the mapping space is then equivalent to its completion along the given homology theory (cf. Definition 8.* *2.1), as required for the convergence of the spectral sequence. This commutative A-algebra B is required to satisfy the so-called Adams cond* *i- tions [Ad69, p. 28], [GH04, Def. 3.1], which in our notation asks that B is wea* *kly equivalent to a homotopy colimit of finite cell A-module spectra Bff, satisfyin* *g two conditions. For our purposes it will suffice that B itself satisfies the two co* *nditions, i.e., that there is only a trivial colimit system. The conditions are then: Adams conditions 10.1.1. The commutative A-algebra B is weakly equivalent to a finite cell A-module, such that (a) BA*(DA B) is finitely generated and projective as a B*-module. (b) For each B-module M the K"unneth map [DA B, M]A*! Hom B* (BA*(DA B), M*)* 70 JOHN ROGNES is an isomorphism. In the E-local situation we expect that it suffices to assume that B is a du* *alizable A-module, but in our applications the stronger finite cell hypothesis will alwa* *ys be satisfied. Lemma 10.1.2. The Adams conditions (a) and (b) are satisfied when A ! B is an E-local G-Galois extension, with G a finite discrete group. Proof. From Lemma 6.1.2 we know that j :B ! FA (B, B) is a weak equiv- alence, and that hM : B ^A M ! F (G+ , M) is a weak equivalence for each B- module M. By Proposition 6.2.1, B is dualizable over A, so B^A DA B ' FA (B, B). So BA*(DA B) ~=ss*FA (B, B) ~=B* is a finitely generated free B*-module, and BA*M ~=[DA B, M]A*is isomorphic to Y Hom B*(BA*(DA B), M*) ~=Hom B*(B*, M*) ~= M* ~=ss*F (G+ , M) . G A diagram chase verifies that the K"unneth map equals the composite of this cha* *in of isomorphisms. The more general situation, with G an indiscrete stably dualizable group, wi* *ll lead to much more complicated spectral sequence calculations, which we will not try to address here. Goerss and Hopkins proceed to consider an E2- or resolution model structure * *on spectra, which is suitably generated by a class P of finite cellular spectra. * *This class is required to satisfy a list of conditions [GH04, Def. 3.2.(1)-(5)]. Fo* *llowing the proof of [BR:r, 2.2.4], by Baker and Richter, we take P to be the smallest * *set of dualizable A-modules that contains A and B, and is closed under (de-)suspensions and finite wedge sums. This immediately takes care of conditions (3) and (4). Lemma 10.1.3. The resolution model category conditions [GH04, Def. 3.2.(1)-(5)] are satisfied when A ! B is a finite E-local G-Galois extension. Proof.Q(1) BA*(X) is a finite sum of shifted copies of BA*(A) = B* and BA*(B) ~= G B*, for each A-module X 2 P, hence is projective as a B*-module. (2) DA B is represented in P, since B is self-dual as an A-module by Proposition 6.4.7. * *(5) The K"unneth map [X, M]A*! Hom B* (BA*(X), M*)* is an isomorphism for all B-module spectra M when X = DA B, by the Adams condition (b), and trivially for X = A, so the same follows for all X 2 P by passage to (de-)suspensions and finite wedge sums. To sum up, a finite Galois extension A ! B satisfies the Adams conditions and has an associated resolution model structure on A-modules, as required by [GH04, x3], whenever B is weakly equivalent to a finite cell A-module. It seems likely* * that the cited theory also extends to cover all finite Galois extensions, by replaci* *ng all references to finite cell objects by dualizable objects. However, in the follo* *wing applications we shall always make use of the identification CA (C, B) ~=CB (B ^A C, B) GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 71 and only apply the Goerss-Hopkins spectral sequence in the case of commutative B-algebra maps to B. This is the very special case of Lemmas 10.1.2 and 10.1.3 when A = B and G is the trivial group, in which case B is certainly a finite cell A-module. So we are only using the straightforward extension of [GH04] to a more general (cofibrant, commutative) ground S-algebra, namely B. Note also that BB*(B ^A C) ~=BA*(C), so the two equivalent mapping spaces above will have the same associated spectral sequences, which we now review. Goerss and Hopkins define Andr'e-Quillen cohomology groups Ds of algebras and modules over a simplicially resolved E1 -operad [GH04, (4.1)], as non-abelian r* *ight derived functors of algebra derivations. They then construct a convergent spect* *ral sequence of Bousfield-Kan type [GH04, Thm. 4.5], which in our notation appears as (10.1.4) Es,t2=) sst-sCA (C, B) (based at a given commutative A-algebra map C ! B), with E2-term E0,02= AlgB* (BA*(C), B*) and Es,t2= DsB*T(BA*(C), tB*) for t > 0. Here tB* is the t-th desuspension of the module B*. As usual for Bousfield-Kan spectral sequences, this spectral sequence is concentrated in the wedge-shaped region 0 s t. The subscript B*T refers to a (Reedy cofibrant, etc.) simplicial E1 operad* * T that resolves the commutative algebra operad in the sense of [GH04, Thm. 2.1], and B*T is the associated simplicial E1 operad in the category of B*-modules. The Goerss-Hopkins Andr'e-Quillen cohomology groups Ds are the right derived functors of derivations of B*T -algebras in B*-modules, in the sense of Quillen* *'s homotopical algebra. As surveyed by Basterra and Richter [BR04, 2.6], these gro* *ups Ds do not depend on the choice of resolving simplicial E1 operad T , and agree with the Andr'e-Quillen cohomology groups AQssE1 defined by Mandell in [Ma03, 1.1] for simplicial E1 B*-algebras. These do in turn agree with the Andr'e-Qui* *llen cohomology groups AQsdgE1 defined by Mandell for E1 differential graded B*- algebras [Ma03, 1.8], and with Basterra's topological Andr'e-Quillen cohomology groups TAQs of the Eilenberg-Mac Lane spectra associated to these algebras and modules [Ma03, x7]. By the comparison result of Basterra and McCarthy [BM02, 4.2], these are finally isomorphic to the -cohomology groups H s of Robinson a* *nd Sarah Whitehouse [RW02], when BA*(C) is projective over B*, or more generally, when BA*(C) is flat over B* and the universal coefficient spectral sequence from homology to cohomology collapses. So in these cases the Goerss-Hopkins groups can be rewritten as DsB*T(BA*(C), tB*) = H s,-t(BA*(C)|B*; B*) . It is not quite obvious from the above references that this chain of identif* *ica- tions preserves the internal t-grading of these cohomology groups, since this g* *rading 72 JOHN ROGNES could be lost by the passage through Eilenberg-Mac Lane spectra. However, Birgit Richter has checked that both gradings are indeed respected, up to the sign ind* *i- cated above. In our applications all of these cohomology groups will in fact be* * zero, so the finer point about the internal grading is not so important. If BA*(C) is an 'etale commutative B*-algebra (thus flat over B*), then by [* *RW02, 6.8(3)] all -homology and -cohomology groups of BA*(C) over B* are zero, so by the sequence of comparison results above (and the universal coefficient spec- tral sequence for TAQ), all the Goerss-Hopkins Andr'e-Quillen cohomology groups DsB*T(BA*(C), tB*) vanish. Therefore one can conclude: Corollary 10.1.5. Let C ! B be a map of commutative A-algebras. If BA*(C) is 'etale over B*, then the Goerss-Hopkins spectral sequence for ss*CA (C, B) ~=ss*CB (B ^A C, B) collapses to the origin at the E2-term, so CA (C, B) is homotopy discrete (each* * path component is weakly contractible) with ss0CA (C, B) ~=Alg B*(BA*(C), B*) . 10.2. Idempotents and connected S-algebras. The converse part of the Galois correspondence, begun in Theorem 7.2.2, shou* *ld intrinsically characterize the intermediate extensions A ! C ! B that occur as K-fixed S-algebras C = BhK by allowable subgroups K G. Already in the algebraic case of a G-Galois extension R ! T of discrete rings there are additi* *onal complications (compared to the field case) when T admits non-trivial idempotent* *s, i.e., when the spectrum of T is not connected in the sense of algebraic geometr* *y. See [Ma74] for a general treatment of these complications. We do not expect that th* *ese issues are so central to the extension of the theory from discrete rings to S-a* *lgebras, so we prefer to focus on the analogue of the situation when T is connected. We can identify the idempotents E(T ) of a commutative ring T with the non- unital T -algebra endomorphisms T ! T , taking an idempotent e (with e2 = e) to the homomorphism t 7! et. The forgetful functor from T -algebras to non-unital T -algebras has a left adjoint, taking a non-unital T -algebra N to T N, with the multiplication (t1, n1) . (t2, n2) = (t1t2, t1n2 + n1t2 + n1n2) and unit (1* *, 0). In particular, we can identify the set of idempotents E(T ) with the set of T -alg* *ebra maps E(T ) ~=Alg T(T T, T ) . Here T T ~= T [x]=(x2 - x) is finitely generated and free as a T -module. It * *is 'etale as a commutative T -algebra by [Mi80, I.3.4], since (x2 - x)0 = 2x - 1 is its o* *wn multiplicative inverse in T [x]=(x2 - x). This leads us to the following definitions. Definition 10.2.1. Let B be a (cofibrant) commutative S-algebra. Let the space of idempotents E(B) = NB (B, B) GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 73 be the mapping space of non-unital commutative B-algebra [Ba99, x1] endomor- phisms B ! B. We say that B is connected if the map {0, 1} ! E(B) taking 0 and 1 to the constant map and the identity map B ! B, respectively, is a weak equivalence. We shall not have need to do so, but if we wanted to express that the spectr* *um B has the property that ss*(B) = 0 for all * 0, we would say that B is 0-connec* *ted, reserving the term "connected" for the algebro-geometric interpretation just in* *tro- duced. A spectrum B with ss*(B) = 0 for * < 0 will be called (-1)-connected or connective. There is a homeomorphism E(B) ~=CB (B _ B, B) , where B _B is defined as the split commutative S-algebra extension of B with fi* *ber the underlying non-unital commutative S-algebra of B. Its unit B ! B _ B is the inclusion on the first wedge summand, and its multiplication is the composite (B _ B) ^B (B _ B) ~=B _ (B _ B _ B) -1_r--!B _ B where r folds the last three wedge summands together. Proposition 10.2.2. Let B be any commutative S-algebra. The space of idem- potents E(B) is homotopy discrete, with ss0E(B) ~= E(ss0(B)). In particular, t* *he commutative S-algebra B is connected if and only if the commutative ring ss0(B)* * is connected. Proof. We compute the homotopy groups of E(B) ~= CB (B _ B, B) by means of the Goerss-Hopkins spectral sequence (10.1.4), in the almost degenerate case wh* *en A = B and C = B _ B. Here A ! B is of course a G-Galois extension, in the trivial case G = 1, so our discussion in Section 10.1 justifies the use of this* * spectral sequence. It specializes to Es,t2=) sst-sE(B) with E0,02= AlgB* (B* B*, B*) = E(B*) and Es,t2= DsB*T(B* B*, tB*) for t > 0. Here B* B* = B*[x]=(x2 - x) is 'etale over B*, so all the Andr'e-Q* *uillen cohomology groups Ds = H s vanish [RW02, 6.8(3)], and we deduce that E(B) is homotopy discrete, with ss0E(B) ~= E(B*) equal to the set of idempotents in the graded ring B*, which of course are the same as the idempotents in the ring ss0* *(B). In short, we have applied Corollary 10.1.5. The following argument, explained by Neil Strickland, shows that the above definition of connectedness for structured ring spectra is equivalent to anothe* *r def- inition originally proposed by the author. We say that an S-algebra B is trivia* *l if it is weakly contractible, i.e., if ss*(B) = B* = 0, and non-trivial otherwise. 74 JOHN ROGNES Lemma 10.2.3. A non-trivial commutative S-algebra B is either connected, or weakly equivalent to a product B1 x B2 of non-trivial commutative B-algebras, b* *ut not both. Proof. If B is non-trivial and not connected then there exists an idempotent e 2 ss0(B) different from 0 and 1. Let f1 and f2: B ! B be B-module maps inducing multiplication by e and 1 - e on ss*(B), respectively. (These could also be ta* *ken to be non-unital commutative B-algebra maps by the previous proposition.) For i = 1, 2 let B[f-1i] be the mapping telescope for the iterated self-map fi, and* * let Bi = LBB[f-1i]B be the Bousfield B[f-1i]-localization of B in the category of B-modules. Then t* *here are commutative B-algebra maps B ! B1 and B ! B2 inducing isomorphisms ess*(B) ~= ss*(B1) and (1 - e)ss*(B) ~= ss*(B2), of nontrivial groups, and their product B ! B1 x B2 is the asserted weak equivalence. Conversely, if B ' B1 x B2 as commutative B-algebras (or even just as ring spectra), with B1 and B2 non-trivial, then ss0(B) is not connected as a commuta* *tive ring, so B is not connected as a commutative S-algebra. 10.3. Separable closure. The following terminology presumes, in some sense, that each finite separable extension can be embedded in a finite Galois extension, i.e., a kind of normal * *closure. We will not prove this in our context, but keep the terminology, nonetheless. Definition 10.3.1. Let A be a connected commutative S-algebra. We say that A is separably closed if there are no G-Galois extensions A ! B with G finite and non-trivial and B connected, i.e., if each finite G-Galois extension A ! B * *has G = {e} or B not connected. A separable closure of A is a pro-GA -Galois extension A ! A~ such that A~ is connected and separably closed. The pro-finite Galois group GA of A~over A is t* *he absolute Galois group of A. The existence of a separable closure follows from Zorn's lemma. However, we have not yet proved that two separable closures of A are weakly equivalent, so talking of "the" absolute Galois group is also a bit presumptive. By Minkowski's theorem on the discriminant [Ne99, III.2.17], for every number field K different from Q the inclusion Z ! OK is ramified at one or more prime* *s. In particular, there are no Galois extensions Z ! OK other than the identity. * *The following inference appears to be well-known. Proposition 10.3.2. The only connected Galois extension of the integers is Z itself, so Z = ~Zis separably closed. Proof. Let Z ! T be a G-Galois extension of commutative rings, so T is a finite* *ly generatedQfree Z-module. Then Q ! Q T is also a G-Galois extension, so Q T ~= iKi is a product of number fields [KO74, III.4.1].Q Then T is contained * *in the integral closure ofQZ in Q T , which is a product iOKi of number rings. The condition T T ~= GT and an index count imply, in combination, that GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 75 Q T = iOKi and that each OKi is unramified overQZ. By Minkowski's theorem, this only happens when each Ki = Q, so T = G Z. If T is connected, this impli* *es that G is the trivial group and that T = Z. In other words, to have interesting Galois extensions of Z one must localize away from one or more primes. We have the following analogue in the context of commutative S-algebras. The examples in Section 5.4 demonstrate that after localization there are indeed interesting examples of (local) Galois extensions* * of S. Theorem 10.3.3. The only (global, finite) connected Galois extension of the sp* *here spectrum S is S itself, so S = ~Sis separably closed. Proof. Let S ! B be any finite G-Galois extension of global, i.e., unlocalized,* * com- mutative S-algebras (Definition 4.1.3). Then B is a dualizable S-module (Propo- sition 6.2.1), hence of the homotopy type of (a retract of) a finite CW spectrum (Proposition 3.3.3). Thus H*(B) = H*(B; Z) is finitely generated in each degre* *e, and non-trivial only in finitely many degrees. Let k be minimal such thatQHk(B) 6= 0 and let ` be maximal such that H`(B) 6= 0. The condition B ^ B ' G B implies thatQk = ` = 0. For if k < 0 then Hk(B) Hk(B) is isomorphic to H2k(B ^ B) ~= G H2k(B) = 0, which contradicts Hk(B) 6= 0 andQfinitely generated. If ` > 0 then H`(B) H`(B) injects into H2`(B ^ B) ~= G H2`(B) = 0, which again contradicts H`(B) 6= 0 and finitely generated. Thus H*(B) = T is concentrated in degree 0. By the Hurewicz theorem, B is a connective spectrumQwith ss0(B) ~=H0(B) = T . The K"unneth formula then implies that T T ~= GT and TorZ1(T, T ) = 0, so t* *he unit map Z ! T makes T a free abelian Z-module of rank equal to the order of G. In particular, T is a faithfully flat Z-module. The result of inducing B up along the Hurewicz map S ! HZ has homotopy ss*(HZ ^ B) = H*(B) = T concentrated in degree 0, so there is a pushout square BOO_____//_HTOO | | | | | | S _____//_HZ of commutative S-algebras. By a variation on the proof of Lemma 7.1.1, we shall now show that HZ ! HT is G-Galois.Q The mapQHT ^HZ HT ! G HT is induced up from the weak equivalence B ^ B ! G B, cf. diagram (7.1.2), and is therefore a weak equivalence. Next, S ! B is dualizable, so HZ ! HT is dualizable (Lemma 6.2.3). Finally, T is faithfully flat over Z and so HT is faithful over HZ by the proof of Lemma 4.3.* *5. Thus HZ ! HT is a faithful G-Galois extension (Proposition 6.3.2). From Proposition 4.2.1 we deduce that Z ! T is a G-Galois extension of commu- tative rings. By the classical theorem of Minkowski, this is only possible if G* * = {e} is the trivial group or T is not connected. And ss0(B) ~=T , so either G is tri* *vial or B is not connected (Proposition 10.2.2). Thus S is separably closed. Note that we did not have to (possibly) restrict attention to faithful G-Gal* *ois extensions S ! B in this proof. 76 JOHN ROGNES Question 10.3.4. Can the absolute Galois group GA , or its maximal abelian quo- tient GabA, be expressed in terms of arithmetic invariants of A, such as its al* *gebraic K-theory K(A)? This would constitute a form of class field theory for commutati* *ve S-algebras. The author expects that there is a better hope for a simple answer * *in the maximally localized category of K(n)-local commutative S-algebras, than for general commutative S-algebras. Question 10.3.5. If an E-local commutative S-algebra A is an even periodic Landweber exact spectrum, and A ! B is a finite E-local G-Galois extension, does it then follow that B is also an even periodic Landweber exact spectrum, a* *nd that ss0(A) ! ss0(B) is a G-Galois extension of commutative rings? In the case of E = K(n) and A = Enrn, for which ss0(A) = W(~Fp)[[u1, . .,.un* *-1 ]] is separably closed, there are no non-trivial such algebraic extensions to a co* *nnected ring ss0(B), so it would follow that Enrnis K(n)-locally separably closed. In * *par- ticular, Enrnwould be the K(n)-local separable closure of LK(n)S, with absolute Galois group Gnrn= Sn o ^Z. This amounts to Conjecture 1.3 in the introduction. Baker and Richter [BR:r] have partial results in this direction, in the glob* *al category. They are able to show that Enrndoes not admit any non-trivial connect* *ed faithful abelian G-Galois extensions. So Enrn= Eabnis the maximal faithful abel* *ian extension of En. 11. Galois theory II As before, we are implicitly working E-locally, for some spectrum E. 11.1. Recovering the Galois group. The space of commutative A-algebra endomorphisms of B in a G-Galois exten- sion A ! B can be rewritten as CA (B, B) ~=CB (B ^A B, B) ' CB (F (G+ , B), B) , in view of the weak equivalence h: B ^A B ! F (G+ , B). When G is finite and discrete, and B admits no non-trivial idempotents, we can compute the homotopy groups of this mapping space by the Goerss-Hopkins spectral sequence. When G is not discrete, these spectral sequence computations appear to be mu* *ch harder, and we will not attempt them. We are therefore principally working in t* *he context of the separable/'etale extensions from Chapter 9. Theorem 11.1.1. Let A ! B be a finite G-Galois extension of commutative S- algebras, with B connected. Then the natural map G ! CA (B, B) , giving the action of G on B through commutative A-algebra maps, is a weak equiv- alence. In particular, CA (B, B) is a homotopy discrete grouplike monoid, so e* *ach commutative A-algebra endomorphism of B is an automorphism, up to a con- tractible choice. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 77 Q Proof. This time we compute the homotopy groups of CA (B, B) ' CB ( G B, B) by means of (10.1.4),Qonce again in the almost degenerate case when A = B and C = F (G+ , B) = G B. The E2-term has Y E0,02= AlgB* ( B*, B*) ~=G G since B* = ss*(B) is connected in the graded sense, or equivalently, ss0(B) has* * no non-trivial idempotents. The remainder of the E2-term is Y Es,t2= DsB*T( B*, tB*) = 0 , G Q since G B* is 'etale over B*. We are therefore in the collapsing situation of* * Corol- lary 10.1.5, and CA (B, B) ' G follows. The extension to profinite pro-Galois extensions is straightforward. Proposition 11.1.2. Let A ! B = colimffBffbe a pro-G-Galois extension, with each A ! Bff a finite Gff-Galois extension and G = limffGff. Suppose that B is connected. Then CA (B, B) is homotopy discrete, and the natural map G ! ss0CA (B, B) is a group isomorphism. Proof. Using (8.1.3), we rewrite the commutative A-algebra mapping space as CA (B, B) ~=CB (B ^A B, B) ' CB (colimffF (Gff+, B), B) ' holimffCB (F (Gff+, B),* * B) . By the finite case, each CB (F (Gff+, B), B) is homotopy discrete with Y ss0CB (F (Gff+, B), B) ~=Alg B*( B*, B*) ~=Gff, Gff when B is connected. So CA (B, B) ' holimffGff is homotopy discrete, with ss0CA (B, B) ~=limffGff~=G. 11.2. The brave new Galois correspondence. We now turn to the converse part of the Galois correspondence. The proper r^ole of the separability condition in the following result was found in a conv* *ersation with Birgit Richter. Proposition 11.2.1. Let A ! B be a G-Galois extension, with B connected and G finite and discrete, and let A ! C ! B be a factorization of this map through a separable commutative A-algebra C. Then CC (B, B) is homotopy discrete, and the natural map CC (B, B) ! CA (B, B) ident* *i- fies K = ss0CC (B, B) with a subgroup of G = ss0CA (B, B). Furthermore, the act* *ion of CC (B, B) ' K on B induces a weak equivalence Y h :B ^C B ! B . K 78 JOHN ROGNES Proof. By assumption A ! C is separable, so there are maps C0 oe-!C ^A C -~!C of C-bimodules relative to A such that ~oe :C0 ! C is a weak equivalence. Induc* *ing these maps and modules up along C ! B, both as left and right modules, we get maps B ^C C0 ^C B -~oe!B ^A B -~~!B ^C B of B-bimodules relative to A, such that the composite is a weak equivalence. We consider C^A C as a commutative C-algebra via the left unit C ~=C^A A ! C^A C, and similarly for B ^A B over B. Then ~ is a map of commutative C-algebras and ~~is a map of commutative B-algebras. At the level of homotopy groups, we get a diagram BC*(B) -~oe*!BA*(B) -~~*!BC*(B) of BA*(B)-module homomorphisms, whose composite is the identity. Furthermore, ~~*is a B*-algebra homomorphism. It follows from the BA*(B)-linearity of ~oe*th* *at the latter map is also a B*-algebra homomorphism. In detail, if x, y 2 BC*(B) t* *hen ~oe*x 2 BA*(B) acts on y through multiplication by its image ~~*~oe*x = x, and * *on ~oe*y 2 BA*(B) by multiplication by ~oe*x. The BA*(B)-linearity of ~oe*now asse* *rts ~oe*x . ~oe*y = ~oe*(~~*~oe*x . y) = ~oe*(x . y) . Therefore BC*(B) is a retract of BA*(B), both in the category of BA*(B)-modules and, more importantly to us, in the category of commutative commutative B*- algebras. Q Q L Recall that BA*(B) ~= G B*, since A ! B is G-Galois. Here G B* ~= G B* * *is a finitely generated free B*-module, since G is finite, so the retraction above* * implies that BC*(B) is a finitely generated projective B*-module. We may therefore once more consider the Goerss-Hopkins spectral sequence (10.1.4), now for the mapping space CC (B, B) ~=CB (B ^C B, B) . The E2-term has Es,t2= DsB*T(BC*(B), tB*) for t > 0. The commutative B*-algebra retraction ~~*:BA*(B) ! BC*(B) induces a split injection from each of these cohomology groups to DsB*T(BA*(B), tB*) , Q which we saw was zero in the proof of Theorem 11.1.1, since BA*(B) = G B* is 'etale over B*. We therefore have Es,t2= 0 away from the origin, also in the Goerss-Hopkins spectral sequence for sst-sCC (B, B). GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 79 Thus CC (B, B) is homotopy discrete, in the sense that each path component is weakly contractible, with set of path components K = ss0CC (B, B) ~=Alg B*(BC*(B), B*) . The B*-algebra retraction ~~*induces a split injection from this set to Alg B*(BA*(B), B*) ~=G . It is clear that the natural map CC (B, B) ! CA (B, B), viewing a map B ! B of commutative C-algebras as a map of commutative A-algebras, is a monoid map with respect to the composition of maps. Therefore the injection K ! G identifi* *es K as a sub-monoid of G. But G is a finite group, so K is in fact a subgroup of * *G. This completes the proof of the first claims of the proposition. The tautological action by CC (B, B) on B through commutative C-algebra maps can be converted to an action by K on a commutative C-algebra B0 weakly equiv- alent to B. We hereafter implicitly make this replacement, so as to have K acti* *ng directly on B over C, and turn to the proof of the final claim. The composite ~oe*~~*:BA*(B)Q! BA*(B) is an idempotent B*-algebra map. Under the isomorphism BA*(B)Q~= G B* it corresponds to an idempotent B*-algebra endomorphismQof G B*. SinceQB* is connected, it must be the retraction of G B* onto theQsubalgebra K0 B*, for some subset K0 G (containing e 2 G). Thus BC*(B) ~= K0 B*, which implies Y K = AlgB* (BC*(B), B*) ~=Alg B*( B*, B*) ~=K0. K0 Q Thus K = K0 as subsets of G, and the weak equivalence B ^A B ' G B retracts to a weak equivalence Y h: B ^C B ! B . K It is quite clearly given by the action of K on B through commutative C-algebra maps, as in (4.1.2). This leads us to the converse part of the Galois correspondence for E-local * *com- mutative S-algebras, in the case of finite, faithful Galois extensions. Theorem 11.2.2. Let A ! B be a G-Galois extension, with B connected and G finite and discrete. Furthermore, let A ! C ! B be a factorization of this map through a separable commutative A-algebra C such that C ! B is faithful, and let K = ss0CC (B, B) G. If A ! B is faithful, or more generally, if B is dualizable over C, then C '* * BhK as commutative C-algebras, and C ! B is a faithful K-Galois extension. Proof. We first prove that A ! B faithful implies that B is dualizable over C. 80 JOHN ROGNES By hypothesis, A ! C is separable, so the multiplication map ~ and its weak section oe make C a retract up to homotopy of C ^A C, as a C ^A C-module. Therefore ~ makes C a dualizable C ^A C-module, by Lemma 3.3.2(c). Similarly, for each g 2 G the twisted multiplication map ~(1 ^ g): C ^A C ! C and its weak section (1 ^ g-1 )oe make C a dualizable C ^A C-module. Inducing up along C ! B, Lemma 6.2.3 implies that each map B ^A C ! B, given algebraically as b ^ c 7! b . g(c), makes B a dualizable B ^A C-module. By LemmaQ3.3.2(c)Waga* *in, it follows that the natural map B ^A C ! B ^A B makes B ^A B ' G B ' G B a dualizable B ^A C-module. By Proposition 6.2.1 and hypothesis, B is dualizable and faithful over A, so by Lemma 6.2.3 and Lemma 4.3.3 we know that B ^A C is faithful and dualizable over C. Thus by Lemma 6.2.4 it follows that the natu* *ral map C ! B makes B a dualizable C-module. By Proposition 11.2.1, K = ss0CC (B, B) G acts on (a weaklyQequivalent re- placement for) B through C-algebra maps, so that h: B ^C B ! K B is a weak equivalence. By hypothesis (and the argument above), C ! B is faithful and dualizable. Then by Proposition 6.3.2 the natural map i: C ! BhK is a weak equivalence, and so C ! B is a faithful K-Galois extension. 12. Hopf-Galois extensions in topology In this final chapter we work globally, i.e., not implicitly localized at an* *y spec- trum (other than at E = S). 12.1. Hopf-Galois extensions of commutative S-algebras. Let A ! B be a G-Galois extension of commutative S-algebras, with G stably dualizable, as usual. The right adjoint f"f:B ! F (G+ , B) of the group action map ff :G+ ^ B ! B can be lifted up to homotopy through the weak equivalence fl :B ^ DG+ ! F (G+ , B), to a map fi :B ! B ^ DG+ . The group multiplication G x G ! G induces a functionally dual map DG+ ! D(G x G)+ , which likewise can be lifted up to homotopy through the weak equivalence ^: DG+ ^ DG+ ! D(G x G)+ , to a coproduct _ :DG+ ! DG+ ^ DG+ . We shall require rigid forms of these structure maps. Definition 12.1.1. A commutative Hopf S-algebra is a cofibrant commutative S- algebra H equipped with a counit ffl: H ! S and a coassociative and counital coproduct _ :H ! H ^S H, in the category of commutative S-algebras. Note that we are not assuming that the coproduct _ is (strictly) cocommutati* *ve, nor that it admits a strict antipode/conjugation O: H ! H. This would severely limit the number of interesting examples. Example 12.1.2. Let X be an infinite loop space. The E1 structure on X makes S[X] = S ^ X+ an E1 ring spectrum. The diagonal map : X ! X x X and X ! * induce a coproduct _ :S[X] ! S[X x X] ~= S[X] ^ S[X] and counit ffl: S[X] ! S, which altogether can be rigidified to make H ' S[X] a commutative Hopf S-algebra. GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 81 Definition 12.1.3. Let A be a cofibrant commutative S-algebra, let B be a cofi- brant commutative A-algebra and let H be a commutative Hopf S-algebra. We say that H coacts on B over A if there is a coassociative and counital map fi :B ! B ^ H of commutative A-algebras. In this situation, let h: B ^A B ! B ^ H be the composite map (~ ^ 1)(1 ^ fi) (of commutative B-algebras). Definition 12.1.4. The (Hopf) cobar complex Co(H; B), for H coacting on B over A, is the cosimplicial commutative A-algebra with Cq(H; B) = B ^ H ^ . .^.H (q copies of H) in codegree q. The coface maps are d0 = fi^1^q, di = 1^i^_^1^(q* *-i) for 0 < i < q and dq = 1^q ^ j, where j :S ! H is the unit map. The codegeneracy maps involve the counit ffl: H ! S. Let C(H; B) = Tot Co(H; B) be its totalizat* *ion. The algebra unit A ! B induces a coaugmentation A ! Co(H; B), and a map i: A ! C(H; B) . Definition 12.1.5. A map A ! B of commutative S-algebras is an H-Hopf-Galois extension if H is a commutative Hopf S-algebra that coacts on B over A, so that the maps i: A ! C(H; B) and h: B ^A B ! B ^ H are both weak equivalences. Note that there is no finiteness/dualizability condition on H in this defini* *tion. See [Ch00] for a recent text on Hopf-Galois extensions in the algebraic setting. Example 12.1.6. Let G be a stably dualizable topological group. The weak co- product on DG+ = F (G+ , S), derived from the group multiplication, can be rigi* *d- ified to give H ' DG+ the structure of a commutative Hopf S-algebra. If G acts on B over A, then the weak coaction of DG+ on B can be rigidified to a coaction of H on B over A. Then the (Hopf) cobar complex Co(H; B) maps by a degreewise weak equivalence to the (group) cobar complex Co(G; B) from Definition 8.2.3. In codegree q it is weakly equivalent to the composite natural map B ^ DG+ ^ . .^.DG+ -^!'B ^ DGq+-fl!~DGq+^ B -! F (Gq+, B) . = ' On totalizations, we obtain a weak equivalence C(H; B) ' BhG . In this case, the definition of an H-Hopf-Galois extension A ! B generalizes that of a G-Galois extension A ! B, since i: A ! BhG factors as A -i!C(H; B) -'!BhG , Q and h: B ^A B ! G B factors as B ^A B -h!B ^ H -'! F (G+ , B) . Recall the Amitsur complex Co(B=A) from Definition 8.2.1. 82 JOHN ROGNES Definition 12.1.7. There is a natural map of cosimplicial commutative A-algebr* *as ho :Co(B=A) ! Co(H; B) given in codegree q by the map hq: B ^A B ^A . .^.AB ! B ^ H ^ . .^.H that is the composite of the maps B^A (i+1)^ H^j ~= B^A (i-1)^A (B ^A B) ^ H^j -1^(i-1)^h^1^j--------!B^A (i-1)^ ^j ~ ^A i ^* *(j+1) A (B ^ H) ^ H = B ^ H for j = 0, . .,.q - 1 and i + j = q. Upon totalization, it induces a map h0:A^B! C(H; B) of commutative A-algebras. The diagram chase needed to verify that ho indeed is cosimplicial uses the s* *trict coassociativity and counitality of the Hopf S-algebra structure on H. Proposition 12.1.8. Suppose that H coacts on B over A, as above, and that h :B ^A B ! B ^ H is a weak equivalence. Then h0: A^B ! C(H; B) is a weak equivalence. As a consequence, A ! B is an H-Hopf-Galois extension if and only if A is complete along B. Proof. The cosimplicial map ho is a weak equivalence in each codegree, so the induced map of totalizations h0 is a weak equivalence. Therefore the composite i = h0O j, of the two maps 0 A -j!A^B-h! C(H; B) , is a weak equivalence if and only if j is one. 12.2. Complex cobordism. Let A = S be the sphere spectrum, B = MU the complex cobordism spec- trum and H = S[BU] = 1 BU+ the unreduced suspension spectrum of BU. Bott's infinite loop space structure on BU makes H a commutative S-algebra, and the diagonal map : BU ! BU x BU induces the Hopf coproduct _ :S[BU] ! S[BU] ^ S[BU]. The Thom diagonal fi :MU ! MU ^ BU+ defines a coaction by S[BU] on MU over S. The induced map h: MU ^ MU ! MU ^ BU+ is the weak equivalence known as the Thom isomorphism. The Bousfield-Kan spectral sequence associated to the cosimplicial commutative S-algebra Co(MU=S) is the Adams-Novikov spectral sequence Es,t2= Exts,tMU*MU(MU*, MU*) =) sst-s(S) . The convergence of this spectral sequence is the assertion that the coaugmentat* *ion i: S ! S^MU = Tot Co(MU=S) is a weak equivalence. In view of Proposition 12.1.8 we can summarize these fac* *ts as follows: GALOIS EXTENSIONS OF STRUCTURED RING SPECTRA 83 Proposition 12.2.1. The unit map S ! MU is an S[BU]-Hopf-Galois extension of commutative S-algebras. Remark 12.2.2. There is no topological group G such that S ! MU is a G-Galois extension, but S[BU] is taking on the r^ole of its functional dual DG+ , as in * *Ex- ample 12.1.6. Note that there is no bimodule section to the multiplication map ~: MU ^ MU ! MU, since the left and right units jL , jR :MU* ! MU*MU are really different, so S ! MU is not separable in the sense of Section 9.1. Remark 12.2.3. There are similar S[X]-Hopf-Galois extensions S ! T h(fl) to the Thom spectrum induced by any infinite loop map fl :X ! BGL1(S). For example, there is such an extension S ! MUP to the even periodic version MUP of MU, which is the Thom spectrum of the tautological virtual bundle over X = Z x BU = 1 ku. More generally, for any commutative S-algebra R and infinite loop map fl :X ! BGL1(R) there is an R-based Thom spectrum T hR (fl), i.e., a fl-twisted form of R[X] = R ^ X+ , and an R[X]-Hopf-Galois extension R ! T hR (fl). Remark 12.2.4. The extension S ! MU is known not to be faithful, since by [Ra84, x3] or [Ra92, 7.4.2] MU*(cY ) = 0 for every finite complex Y with trivial rati* *onal cohomology. Here cY denotes the Brown-Comenetz dual of Y . This faithlessness leaves the telescope conjecture [Ra84, 10.5] or [Ra92, 7.5.5] a significant cha* *nce to be false. Recall that if F (n) is a finite complex of type n (with a vn-self m* *ap), and T (n) = v-1nF (n) its mapping telescope, the conjecture is that the natural map ~: T (n) ! LnF (n) is a weak equivalence. After inducing up to MU, 1 ^ ~: MU ^ T (n) ! MU ^ LnF (n) is an equivalence, by the localization theorem v-1nMU ^ F (n) ' LnMU ^ F (n) [Ra92, 7.5.2]. Positive information about the faithfulness of Galois- or Hopf-Galois extensions (Question 4.3.6) might concei* *vably reflect back on this conjecture. To conclude this paper, we wish to discuss how the Hopf-Galois extension S ! MU provides a global, integral object whose p-primary K(n)-localization LK(n)S ! LMUK(n)MU governs the pro-Galois extensions LK(n)S ! En, for each rational prime p and integer n 0. (12.2.5) MUOO LMUK(n)MU ____t____//En88 | OO qqqqq S[BU] || || qqqqGq | | qq n S LK(n)S This suggests that S ! MU is a kind of near-maximal ramified Galois exten- sion, and that its "weak" Galois group (only realized through its functional du* *al DG+ = S[BU]) is a kind of near-absolute ramified Galois group of the sphere. Mo* *re precisely, the maximal extension may be the one obtained from the even periodic theory MUP by tensoring with the ring O~Qof algebraic integers. Even if S ! MU does not admit many Galois automorphisms, the Hopf coaction fi :MU ! MU ^ BU+ still determines the Galois action of each element g 2 Gn on En. By the Hopkins-Miller theory, each commutative S-algebra map g :En ! 84 JOHN ROGNES En is uniquely determined by the underlying map of (commutative) ring spectra, so it is this description that we shall review. Recall from 5.4.2 that n is the Honda formal group law over Fpn and e n its universal deformation, defined over ss0(En). By the Lubin-Tate theorem [LT66, 3.1], each automorphism g 2 Sn Gn of n determines a unique pair (OE, "g), wh* *ere OE: ss0(En) ! ss0(En) is a ring automorphism and "g:e n! OE*e n is an isomorphi* *sm of formal group laws over ss0(En), whose expansion "g(x) 2 ss0(En)[[x]] ~=E0n(C* *P 1) reduces to the expansion g(x) 2 Fpn[[x]] of g. Then OE = ss0(g), when g is cons* *idered as a self-map of En. Furthermore, "g(x) 2 E0n(CP 1)~=Hom En*(En*(CP 1), En*) ~=Alg En*(En*(BU), En*) E0n(BU) corresponds to a unique map of ring spectra "g:S[BU] ! En . Let t: MU ! En be the usual complex orientation, corresponding to the graded version of e n. T* *hen the following diagram commutes up to homotopy: (12.2.6) MU __fi_//_MU ^ BU+ t|| ~(t^"g)|| fflffl|g fflffl| En __________//_En The composite g O t = ~(t ^ "g)fi determines g, in view of t* :E*n(En) ! E*n(MU) being nearly injective. Only the Galois automorphisms in Gal Gn are missing, but these may be ignored if we are focusing on [E(n), or can be detected by pas* *sing to MUP and adjoining some roots of unity. By analogy, for number fields K L and primes p 2 OK , a factorizationQpOL = Pe11. . ...Perrleads to a splitting of completions Kp ! L K Kp ~= i LPi . If * *the field extension K ! L is G-Galois, then each local extension Kp ! LPi is GPi - Galois, where GPi G is the decomposition group of Pi, and G acts transitively* * on the finite set of primes over p. Thus when the global extension K ! L is locali* *zed (i.e., completed), it splits as a product of smaller local extensions, in a way* * that depends on the place of localization. (12.2.7) LOO L K KpOO _pri_//_LPi:: uuu G || || uuGuu | | uuu Pi K Kp In the algebraic case of a pro-Galois extension K ! K~ there is a profinite set* * of places over each prime p, still forming a single orbit for the action by the ab* *solute Galois group GK . 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