ROGNES'S THEORY OF GALOIS EXTENSIONS AND THE CONTINUOUS ACTION OF Gn ON En DANIEL G. DAVIS 1.Introduction In recent years, John Rognes has given various talks introducing his theory of Galois extensions for commutative S-algebras, and several manuscripts about this topic are available at his website. One family of Galois extensions that he has* * dis- cussed are those that arise from the action of the extended Morava stabilizer g* *roup Gn = Sn o Gal(Fpn=Fp) on the Lubin-Tate spectrum En through automorphisms of commutative S-algebras. In considering this new theory, the author has found these En-related Galois extensions quite interesting. In the author's 2003 thesis ([3] - see [2] for a short summary), for a profin* *ite group G, he developed (making explicit ideas that were implicit in the literatu* *re, especially in the foundational work of Bob Thomason and Rick Jardine, and work by Paul Goerss and Steve Mitchell) the notions of continuous action and homotopy fixed points for discrete G-spectra and their towers. For a closed subgroup G * *in Gn, Ethan Devinatz and Mike Hopkins defined EhGn:= LK(n)(colimiEhUiGn) (see [5, Def. 1.5]). The author applied their work [5] and this definition to s* *how that En is a continuous Gn-spectrum with homotopy fixed point spectra, defined using the continuous action, for closed subgroups G in Gn. The above formula for EhGnfollows a convention that is used throughout this paper: EhGnis used to denote both the homotopy fixed point spectra of Devinatz and Hopkins, and the homotopy fixed point spectra defined with respect to the continuous action of G on En (defined by the author in [3]), since the author showed that these constructions are isomorphic in the stable homotopy category (in [3]). Since the author does not possess a detailed account of Rognes's ideas (nor h* *as he had the fortune of hearing Rognes speak), and, believing that the machinery of his thesis could be useful for the theory of Galois extensions, the author w* *rote this paper, to help him precisely understand Rognes's examples and to see more clearly exactly what kind of Galois extensions arise in Lubin-Tate theory. Thus* *, the job of this paper is primarily to study the extensions that arise from En, with* * its Gn-action, and to consider what kinds of definitions of Galois extension are ne* *eded to fit the actual structures. Part of our work in this paper depends on results that are not yet known to be true. Beginning in x3, we assume that the discrete Gn-module ss*(En ^ MI) can be realized by a spectrum, abusively labeled Fn ^ MI, that is a discrete Gn-symmet* *ric ring spectrum, that is, a discrete Gn-spectrum that is a ring object in the cat* *egory ____________ Date: May 14, 2004. 1 of symmetric spectra, and whose discrete Gn-action is by ring maps. We also assume the existence of certain model categories for various categories of disc* *rete G-symmetric spectra. See Remark 3.1 for more details about our assumptions. The author hopes and believes that these assumptions are true. If we only assume what is already known and if we modify Definition 7.2 (for "K(n)-local G-pro-Galois extension") in an obvious way, then our main results, stated as (4) and (6) below, are still true. We make the above assumptions beca* *use they allow us to paint a more coherent and structured picture than would be pos* *sible otherwise, and because we hope that the picture will turn out to be correct. Summary of Main Results. To ease the notation, we write ^Lin place of LK(n). We summarize the types of extensions and examples that are considered in this paper by listing our main results (given the above assumptions): (1) Theorem 2.6: Given an open normal subgroup U of Gn, the map of com- mutative S-algebras ff(U): ^LS0 ! EhUnis a K(n)-local Gn=U-Galois ex- tension. (2) Theorem 3.5: The map fl(U, I): ^LMI ! (Fn ^ MI)hU ~=EhUn^ MI is an associative Gn=U-Galois extension. (3) Theorem 5.4: The map fl(I) = colimifl(Ui, I): ^LMI ! Fn ^ MI is an associative filtered Gn-Galois extension. (4) Theorem 7.5: The map holimIfl(I): ^LS0 ! En is a K(n)-local Gn-pro- Galois extension. Also, we explain why ^LS0 ! En is a strongly K(n)-local filtered Gn-pro-Galois extension. (5) Theorem 8.2: Given any closed subgroup G of Gn, the directed system {fi(G, i, I): EhUiGn^ MI ! EhUin^ MI}i, of associative UiG=Ui-Galois ex- tensions, makes the map fi(G, I) = colimifi(G, i, I): (EhGn^ MI) ! Fn ^ MI ' En ^ MI an associative filtered G-Galois extension. (6) Theorem 8.4: The inverse system {fi(G, I)}I of K(n)-local associative G- Galois extensions makes the map fi(G) = holimIfi(G, I): EhGn! En a K(n)-local G-pro-Galois extension. (7) Theorem 8.5: The map ff(U) = holimIfl(U, I) is a K(n)-local Gn=U-pro- Galois extension. Notation and Conventions. Often, when we use results from [5] and [3], we do not give references. Throughout this paper, U is an open normal subgroup of Gn. Sp is the model category (spectra)stableof Bousfield-Friedlander spectra. We of* *ten use the symbol ~=to denote isomorphism in the stable homotopy category. When- ever necessary, we assume that our commutative S-algebras are cell commutative, and, given an S-algebra R, that our R-modules are cell R-modules. All colimits of spectra are formed in S-modules, Sp, or Sp , the model category of symmetric spectra of simplicial sets; which category is used will be clear from context. * *When- ever necessary, we view an S-module as a (symmetric) spectrum of simplicial set* *s, and vice versa. Given a profinite group G, if a colimit or limit is indexed by a collection {* *N}, then {N} is a cofinal collection of open normal subgroups of G. Also, SpG is the category of discrete G-spectra, and, given X 2 SpG, Xf,G denotes the spectrum 2 obtained from factoring X ! * as X ! Xf,G ! *, a trivial cofibration, followed by a fibration, in SpG. Then, by definition, XhG = (Xf,G)G . If X is a (pointed) discrete G-set, simplicial set, or spectrum, then Map c(G* *, X) is the (pointed) discrete G-set, simplicial set, or spectrum of continuous maps* * from G to X, and oG(X) is the canonical cosimplicial (pointed) discrete G-set, simp* *licial set, or spectrum determined by the triple that is formed from Map c(G, -). We w* *ill use the fact that if the profinite group G has finite virtual cohomological dim* *ension, then holim oG(Xf,G) G is a model for XhG (see [3]). Note to Reader: The author wants to point out that besides the assumption of the validity of certain conjectural remarks, discussed above, the careful reade* *r will notice that there are several other technical problems with this paper, which we now list. (a) We go back and forth between S-modules and spectra of simplicial * *sets frequently, and this movement is less than desirable. An ideal presentation of * *the various Galois extensions considered here would do everything in a single categ* *ory of spectra. (b) Since the colimit in S-modules of S-algebras is not, in general* *, the colimit in the category of S-algebras (see [6, II, Prop. 7.4]), the colimit in * *Definition 5.3 should be handled in a better way. (c) At places where we would like to have point-set level maps of (commutative) S-algebras that are weak equivalences, we often have only isomorphisms in the stable homotopy category. Though the author has not ironed out these technicalities, he still believes that the ideas in th* *is paper are essentially correct and worthwhile. Relationship to Work of Rognes. The inspiration for this paper is the work of John Rognes. Some of the definitions and examples are originally due to him, and those that are not were motivated by his work. At the time of writing, the auth* *or does not know how much of the details of what he has written here is already kn* *own to Rognes. To make the relationship between this work and Rognes's clearer, the author notes the following. (a) Definition 2.1 is from [11, pg. 1]. (b) Definit* *ion 2.4 closely follows Rognes's definition of an E-local G-Galois extension (see [10])* *. (c) Definition 5.3 was motivated by Rognes's notion of a G-pro-Galois extension (see [11, pg. 1]). (d) Our main theorem, Theorem 7.5, is a slight modification of a * *result due to Rognes (see [11, pg. 2, (6)], [9, pg. 6, (4)], and [10]). The author rea* *dily acknowledges that he has perhaps omitted ways that this work is already known by, written up by, or indebted to Rognes. Acknowledgements. First of all, I thank Jeff Smith for his encouragement in my work on continuous G-spectra in chromatic stable homotopy theory. I thank John for sending me [9], which was helpful. I thank Paul Goerss for many helpful conversations in the past several years about homotopy fixed points and various categories of spectra, and for discussions about the idea that ss*(En ^ MI) can be realized by a discrete Gn-symmetric ring spectrum. Also, I thank Paul, Jim McClure, and Clarence Wilkerson for their encouragement. 2.Galois extensions for finite groups Definition 2.1. Let G be a finite group. A map A ! B of commutative S-algebras is a G-Galois extension if the following conditions hold: (1) G acts on B through A-algebra maps. (2) The natural map A ! BhG is a weak equivalence. 3 Q (3) There is a weak equivalence B ^A B ' G B, where B is an A-module and - ^A - is the smash product in the category of A-modules. Remark 2.2. The theorem below is due to Rognes ([9], [10]). Since the author knows of no written proof, he attempts one. The proof below is complete, except* * for the unjustified step marked with a question mark. The unjustified step is askin* *g for a holim to commute with a colimit (specifically, a coequalizer). The author wou* *ld like to know how to complete the proof, and then prove the same result when G is profinite, in which case the result could be used with Galois extensions tha* *t are considered later in this paper. Theorem 2.3. Let G be a finite group and let N be a normal subgroup. If A ! B is a G-Galois extension, then A ! BhN is a G=N-Galois extension. Q Proof.We know that BhG ' A and B ^A B ' G B. Regarding G as a profinite group, with N an open subgroup, (BhN )hG=NQ' BhG ' A (see Lemma 3.2). It remains only to show that BhN ^A BhN ' G=N BhN . We have (where N acts on B and N acts on B): BhN ^A BhN = (holimN B) ^A (holimN B) ?'holimN holimN(B ^A B) Q Q ' holimN holimN( G B) ~=holimN G (holimN B) Q Q hN = holimN G BhN = ( G BhN )hN = Mapc(G, BhN ) . For convenience, we switch to working in Sp, where we use homotopyQfixed pointsQfor a profinite group. Since Map c(G, BhN ) ~= Mapc(N, G=N BhN ) and G=N BhN is a fibrant spectrum, Map c(G, BhN ) is fibrant in SpN . Thus, hN hN hN N Mapc(G, B ) = Mapc(G, B ) f,N hN N Q = Mapc(G, B ) = G=N BhN . We are also interested in an E-local version of the notion of Galois extensio* *n. Definition 2.4. Let G be a finite group. A map A ! B of E-local commutative S-algebras is an E-local G-Galois extension if the following conditions hold: (1) G acts on B through A-algebra maps. (2) The natural map A ! BhG is a weak equivalence.Q (3) There is a weak equivalence LE (B ^A B) ~= G B. Remark 2.5. Definition 2.4 is slightly different from the definition for the sa* *me term given in [10], where Rognes doesQnot assume that A and B are E-local, but he requires A ! BhG and B ^A B ' GB to be E-equivalences. The result below concerns the map ff(U) = F(Gn=U ! Gn=Gn ): EhGnn! EhUn. Following [5, Thm. 1(iii)], we identify EhGnnwith ^LS0. Theorem 2.6. The map ff(U): ^LS0 ! EhUnof commutative S-algebras is a K(n)- local Gn=U-Galois extension. 4 Proof.As stated in [4, pg. 8], the finite group Gn=U acts on EhUnthrough maps of ^LS0-algebra maps. Also, by [5, Thm. 4], ^LS0 ' EhUnhGn=U . Also, by [4, Co* *r. 3.9], there is a weak equivalence ^L(EhUn^LS0 EhUn) ' Map c(Gn=U , EhUn) ' Q Gn=UEhUn. Remark 2.7. Note that (the author believes) EhUn^^LS0EhUnis not K(n)-localQ(see [6, VIII, Cor. 3.5]), so there is no weak equivalence EhUn^LS0 EhUn' Gn=UEhUn. Therefore, ff(U) is not a Gn=U -Galois extension. Remark 2.8. Following [6, Chp. VIII] and [4, x1], let LAEdenote Bousfield local- ization with respect to E for A-modules, where E is an A-module. Note that the K(n)-local spectrum K(n) is a module over ^LS0, the unit in the K(n)-local cate- gory. One can define a strongly E-local G-Galois extension to be as in Definiti* *on 2.4, except that in condition (3), the functor LE is replaced with LAE. Then ff* *(U) is a strongly K(n)-local Gn=U-Galois extension. (To verify this, we only need to show that 0 hU hU hU hU (2.9) L^LSK(n)(En ^^LS0En ) ' ^L(En ^^LS0En ). Let X be an arbitrary ^LS0-module. Then by [4, pg. 4], ^LX ' L^LS0^LS0^K(n)(X* *). Since ^LS0 ^ K(n) is K(n)-local, ^LS0 ^ K(n) ' ^L(^LS0 ^ K(n)) ' ^L(S0 ^ K(n)) ' K(n). Thus, ^LX ' L^LS0K(n)(X), and (2.9) is true.) The lemma below implies that ^L(EhUn^^LS0EhUn), which is associated to the extension ff(U), and ^L(EhUn^ EhUn) are the same as S-modules. Q Lemma 2.10. There is a weak equivalence ^L(EhUn^ EhUn) ' Gn=UEhUn. Proof.The finite product of K(n)-local spectra is K(n)-local, so both spectra u* *n- der consideration are K(n)-local. ThenQit suffices to show that there is a weak equivalence ^L(EhUn^ EhUn^ En) ' ^L(( Gn=U EhUn) ^ En), and this follows from ^L(EhUn^ En) ' Q Gn=UEn [5, Cor. 5.5]. 3. Associative Galois extensions In this section, we consider Galois extensions of S-algebras that are not nec* *es- sarily commutative. We use the fact that if R is just an S-algebra and M and N are right and left R-modules, respectively, then the tensor product M ^R N is s* *till defined, though it need not be an R-module [6, III, Def. 3.1]. Recall that Fn = colimiEhUinis a discrete Gn-spectrum of simplicial sets. Als* *o, [5, Def. 1.5, Thm. 3(i)] shows that En ' ^L(hocolimEiEhUin), where hocolimEis t* *he homotopy colimit in the model category E of commutative S-algebras. Furthermore, by [5, Lem. 6.2], hocolimEiEhUin' colimiEhUin, where the colimit is in the cate* *gory of S-modules. Thus, we can regard Fn as a commutative S-algebra. Remark 3.1. In the next two paragraphs, all statements are unproven, except for the statements that are in italics, which are known to be true. We include the unproven assertions because, if true, they form an integral part of the story o* *f how 5 Galois extensions appear in Lubin-Tate theory, as the rest of this paper shows. Also, the author believes the assertions are probably true, and he has worked on showing that E1 ^ M(pi) can be realized in SpGa1(see below). We assume the unproven statements are true for the remainder of the paper. For G, a profinite group, there is a model category SpGcof discrete G-commuta* *tive symmetric ring spectra, that is, E1 -objects in the category of symmetric spect* *ra of simplicial sets that are also discrete G-spectra, such that the G-action is * *by E1 - maps. Let SpG be the model category of discrete G-symmetric spectra, and let Sp* * c be the model category of commutative symmetric ring spectra. Then the forgetful functor SpGc! SpG and the G-fixed points functor (-)G :SpGc! Sp cpreserve all weak equivalences and fibrations. Also, if X 2 SpGc, then XhG is a commutative symmetric ring spectrum. Now we consider what kind of Galois extension arises for B = Fn ^ MI ' En ^ MI. It is widely believed that Fn ^ MI cannot be a commutative S-algebra. However, it is thought that Fn ^ MI is an S-algebra, since Andrew Baker proved that the closely related spectra E(n)=Iknare S-algebras [1]. Further, we suppos* *e that Fn ^ MI is a discrete Gn-symmetric ring spectrum, that is, Fn ^ MI is an object in SpGan, the model category of A1 -objects in the category of symmetric spectra that are discrete Gn-spectra with an action by A1 -maps. As above, the forgetful functor SpGa! SpG and the G-fixed points functor (-)G :SpGa! Sp apreserve all weak equivalences and fibrations. (Sp ais the model category of symmetric r* *ing spectra.) Thus, if X 2 SpGa, then XhG is a symmetric ring spectrum. The next result is useful for verifying that certain maps are Galois extensio* *ns. Lemma 3.2. Let X be a discrete G-spectrum and let N be an open normal subgroup of G. Then there is a weak equivalence XhG ! (XhN )hG=N is a weak equivalence. Proof.The sheaf of spectra Hom G(-, Xf,G) is a globally fibrant presheaf of spe* *ctra. Then [7, Prop. 6.39] implies that there is a weak equivalence XhG ~=Hom G(*, Xf,G) ! holimG=N Hom G(G=N, Xf,G) ~=holimG=N XhN , since Xf,Gis fibrant in SpN . Note that (Xf,G)N is a G=N-spectrum. Since G=N is finite and the G=N-spectrum XhN is fibrant in Sp, holimG=N XhN = (XhN )hG=N . Assuming the hypothetical picture discussed above, the map fl(U, I): ^LMI ~=(Fn ^ MI)hGn ! (Fn ^ MI)hU ~=EhUn^ MI hGn=U is a map of S-algebras. By Lemma 3.2, (Fn ^ MI)hU ' (Fn ^ MI)hGn. Lemma 3.3. There is a weak equivalence of S-modules Q (Fn ^ MI)hU ^^LMI(Fn ^ MI)hU ' Gn=U(Fn ^ MI)hU. Proof.By Lemma 2.10, there are weak equivalences Q hU Q hU hU hU Gn=U (Fn ^ MI) ' Gn=U (En ^ MI) ' En ^ En ^ MI. Now we consider the left hand side of the desired weak equivalence. Note that (Fn ^ MI)hU ~=FnhU^ MI ~=FnhU^ Ln(MI) ~=FnhU^ ^LMI. 6 Thus, by [6, III, Prop. 3.6], we have: hU hU (Fn ^ MI)hU ^^LMI(Fn ^ MI)hU~=Fn ^ ^LMI ^^LMI(Fn ^ MI) ' FnhU^ FnhU^ MI ~=EhUn^ EhUn^ MI. This lemma motivates us to make the following definition. The theorem below follows immediately from the lemma and the definition. Definition 3.4. Let G be a finite group. A map A ! B of S-algebras is an associative G-Galois extension if the following conditions hold: (1) G acts on B. (2) There is an isomorphism A ~=BhG in theQstable homotopy category. (3) There is a weak equivalence B ^A B ' G B, where B is a left and a right A-module. Theorem 3.5. The map fl(U, I): ^LMI ! (Fn ^ MI)hU ~=EhUn^ MI of S-algebras is an associative Gn=U-Galois extension. Remark 3.6. Without spelling out another definition, it is easyQto see that fl(* *U, I) is also a K(n)-local associative Gn=U-Galois extension, since Gn=U (EhUn^ MI) is K(n)-local. 4.Galois extensions for profinite groups In this section we consider the notion of G-Galois extension for a profinite * *group G. Definition 4.1. Let G be a profinite group, and let A ! B be a map of commuta- tive S-algebras. Also, let B be a discrete G-commutative symmetric ring spectru* *m, that is, B 2 SpGc. Then A ! B is a G-Galois extension if the following conditio* *ns hold: (1) There is a compatible G-action on B that is by A-algebra maps. (2) There is an isomorphism A ~=BhG in the stableQhomotopy category. (3) There is a weak equivalence B ^A B ' colimN G=N B. Remark 4.2. The A-algebra action on the commutative S-algebra B is in the world of S-modules, whereas the discrete G-action is in the world of symmetric spectra of simplicial sets. Since these categories are different, we only ask for compa* *tibility in condition (1) above, instead of requiring that the discrete G-action on B be by A-algebra maps. For an example of what "compatible" means, see the related example mentioned in Remark 7.3 (1). Remark 4.3.QIf B is a spectrum of simplicial sets, then there is an isomorphism colimN G=N B ~= Map c(G, B). We use the former construction in the above definition, since the latter construction, in general, does not give the right * *spectrum, if B is a spectrum of topological spaces. (If B is an S-module and V is a fini* *te dimensional subspace of R1 , then BV , in general, is not a discrete space, and Map c(G, BV ) 6= Map c(G, BVdis), where BVdisis the set BV with the discrete topology.) 7 Remark 4.4. Let G be profinite and let A ! B be a G-Galois extension. Now suppose that G is finite. Then it is known that BhG = (Bf,G)G and Map G(EG+ , B) are weakly equivalent. Also, since B is a discrete G-spectrum, Q Q colimN G=N B ~=Map c(G, B) = GB. Thus, the Galois extension satisfies the conditions of Definition 2.1, so that * *Defini- tion 4.1 includes Definition 2.1 as a special case, as desired. We have the following definition for when A and B are only S-algebras. Definition 4.5. Let G be a profinite group, and let A ! B be a map of S- algebras. Also, let B be a discrete G-symmetric ring spectrum, that is, B 2 SpG* *a. Then A ! B is an associative G-Galois extension if the following conditions hol* *d: (1) There is an isomorphism A ~=BhG in the stableQhomotopy category. (2) There is a weak equivalence B ^A B ' colimN G=N B. 5.Filtered Galois extensions In this section, we introduce the notion of filtered Galois extension, which * *is essentially what Rognes calls a pro-Galois extension [11, pg. 1]. We reserve * *use of the prefix "pro" for later, when we consider Galois extensions that are inve* *rse limits of Galois extensions. Definition 5.1. Let {A ! Bff}ffbe a direct system of Gff-Galois extensions, with {Gff}ffan inverse system of finite groups, such that each map Bff! Bff0is Gff0- equivariant. Let G = limffGffand let B = colimffBff, so that G is a profinite g* *roup and B 2 SpG. Henceforth, whenever we say direct system of Gff-Galois extensions, we are referring to a system with these properties. A direct system of associat* *ive Gff-Galois extensions is a direct system of Gff-Galois extensions, except we on* *ly require the Galois extensions to be associative. Let {A ! Bff}ffbe a direct system of Gff-Galois extensions, such that G has f* *inite virtual cohomological dimension. Recall that if K is profinite with vcd(K) < 1, then, if Z 2 SpK , ZhK ' holim oK(Zf,K) K . Then there are conditionally convergent descent spectral sequences Es,t2(ff) = Hs(Gff; sst(Bff)) ) sst-s(BhGffff), and Es,t2= Hsc(G; sst(B)) ) sst-s(BhG ). Taking a colimit of the spectral sequences E*,*r(ff) yields the spectral sequen* *ce o Gff colimffEs,t2(ff) ~=Hsc(G; sst(B)) ) sst-s holim colimff Gff((Bff)f,Gff) . Thus, the isomorphism of spectral sequences colimffE*,*r(ff) ~= E*,*r, for r * *2, implies that o Gff (5.2) BhG ~=holim colimff Gff((Bff)f,Gff) . Observe that if, in (5.2), the colimit and the holim commute with each other (that is, if the spectral sequence colimffE*,*r(ff) converges to the colimit of* * the abutments ss*(BhGffff)), then BhG ~=colimffBhGffff' colimffA = A. 8 However, a strong hypothesis (e.g. the collection {E*,*2(ff)} is uniformly boun* *ded on the right - see [12, Lem. 5.50]) is needed for this to be true. Thus, in gen* *eral, we believe that it need not be the case that, given a directed system {A ! Bff}* *ff of Gff-Galois extensions, there is a weak equivalence BhG ' A. Thus, A ! B in general, is not automatically a G-Galois extension. This motivates the followi* *ng definition. Definition 5.3. Let {A ! Bff}ffbe a direct system of (associative) Gff-Galois extensions. As before, G = limffGffis profinite and B = colimffBff2 SpG. If A ! B is a (associative) G-Galois extension, then A ! B is called a (associativ* *e) filtered G-Galois extension. Recall that in Theorem 3.5, we showed that {fl(Ui, I)}i= {^LMI ! EhUin^ MI}i is a direct system of associative Gn=Ui-Galois extensions. Theorem 5.4. The map fl(I) = colimifl(Ui, I): ^LMI ! Fn ^MI is an associative filtered Gn-Galois extension. Proof of Theorem 5.4.We only have to show that fl(I) is an associative Gn-Galois extension. Since (Fn ^ MI)hGn ~=QEhGnn^ MI ~=L^MI, it suffices to show that (Fn ^ MI) ^^LMI(Fn ^ MI) ' colimi Gn=Ui(Fn ^ MI). Since Fn is E(n)-local, Fn ^ MI ' Fn ^ ^LMI, so that (Fn ^ MI) ^^LMI(Fn ^ MI)' (Fn ^ ^LMI) ^^LMI(Fn ^ MI) ' Fn ^ Fn ^ MI ' En ^ En ^ MI ' Map c(Gn, Fn ^ MI) Q ' colimi Gn=Ui(Fn ^ MI). Remark 5.5. The map fl(I) is also a K(n)-local associative Gn-Galois extension, where we use the following definition. Definition 5.6. Let G be a profinite group, and let A ! B be a map of E-local S-algebras. Also, let B be a discrete G-symmetric ring spectrum, that is, B 2 S* *pGa. Then A ! B is an E-local associative G-Galois extension if the following condit* *ions hold: (1) There is an isomorphism A ~=BhG in the stable homotopyQcategory. (2) There is a weak equivalence LE (B ^A B) ' colimN G=N B. Let (a) {A ! Bff}ffbe a direct system of (associative) Gff-Galois extensions, and (b) assume that A ! B is a map of commutative S-algebras. Note that if X is a discrete G-set such that X ~=colimN X(N), where each X(N) is a G=N-set, then Map c(G, X) ~=colimNMap c(G=N, X(N)) (see e.g. [8, Lem. 6.5.4(a)]). Similarly, Map c(G, B)~=Map c(limffGff, colimffBff) ~=colimffMapc(Gff, Bff) ~=colimffQGffBff' colimff(Bff^A Bff). Let {ff0} be a copy of the indexing set {ff}, so that ff = ff0. Then the set * *of pairs {(ff, ff)}ffis cofinal in the indexing set {(ff, ff0)}ff,ff0of all pairs, so th* *at colimff(Bff^A Bff) ~=colim(ff,ff)(Bff^A Bff) ~=colim(ff,ff0)(Bff^A Bff0). 9 Since the construction Bff^A Bffis a coequalizer, colim(ff,ff0)(Bff^A Bff0)~=(colimffBff) ^A (colimff0Bff0) ~=B ^A B. Thus, Map c(G, B) ' B ^A B, and we summarize this discussion in the remark below. Remark 5.7. As stated in [11, pg.Q1], (a) and (b) above are enough to imply the weak equivalence B^A B ' colimN G=N B. Thus, Definition 5.3 can be simplified by noting that the last condition in Definitions 4.1 and 4.5 can be ignored. 6.A consequence of Theorem 2.3, when G is profinite In this brief section, we assume that Theorem 2.3 is true, when G is profinit* *e. Thus, we are assuming that if (i) G is profinite; (ii) A ! B is a G-Galois exte* *nsion of commutative S-algebras; and (iii) N is an open normal subgroup of G, then A ! BhN is a G=N-Galois extension. Remark 6.1. In the theorem below, N is an open normal subgroup of G, B is a discrete G-commutative symmetric ring spectrum, and Bf,Gcomes from factoring B ! * in SpGc, as B ! Bf,G ! *, a trivial cofibration followed by a fibration. Since the forgetful functor SpGc! SpG preserves weak equivalences and fibration* *s, B ! Bf,Gis a weak equivalence in SpG , and Bf,Gis fibrant in SpG . Thus, in SpN* * , B ! Bf,G is a weak equivalence and Bf,G is fibrant, so that (Bf,G)N is a model for BhN . Theorem 6.2. Let G ~=limN G=N be profinite. If A ! B is a G-Galois extension, then the direct system {A ! BhN }N , of G=N-Galois extensions, makes the map A ! Bf,G a filtered G-Galois extension in a canonical way. S Proof.Observe that Bf,G = N(Bf,G)N = colimN BhN , as required.Q Also, (Bf,G)hGQ' BhG ' A, and (Bf,G) ^A (Bf,G) ' B ^A B ' colimN G=N B ' colimN G=N Bf,G. Remark 6.3. This theorem says that every G-Galois extension is canonically a filtered G-Galois extension. 7. Pro-Galois extensions In this section, we define a notion of Galois extension for towers of discret* *e G- spectra. We are primarily interested in understanding the structure of the Galo* *is extension ^LS0 ! En, which Rognes has referred to as a "K(n)-local Gn-pro-Galois extension" [10]. We begin by recalling that fl(I): ^LMI ! Fn ^ MI -'!En ^ MI is an associative filtered Gn-Galois extension. Thus, {fl(I)}I is an inverse syste* *m of associative filtered Gn-Galois extensions. Remark 7.1. The definition below is only in the K(n)-local setting because this is all that is needed for our examples. Definition 7.2. Let J = {. .!.i ! i - 1 ! . .!.1 ! 0}. Let {Ai ! Bi}i be a J-shaped tower of K(n)-local G-Galois extensions, such that {Bi} is a tower in SpG, and the isomorphism BhGi~= Ai comes from a natural weak equivalence Ai! BhGi. (Whenever Bi is viewed as an object of SpG, then it is assumed to be 10 fibrant there. Similarly, whenever Aiis viewed as an object of Sp, then it is a* *ssumed to be fibrant.) We allow any or all of the extensions Ai! Bito be associative. * *Let A = holimiAi, B = holimiBi, and let A ! B be the obvious map. Then A ! B is a K(n)-local G-pro-Galois extension if the following conditions hold: (1) The map A ! B is a map of commutative S-algebras. (2) The spectrum B is a continuous G-spectrum, G acts on B by maps of A-algebras, and these two G-actions are compatible. (3) There is a weak equivalence A ' BhG . Q (4) There is a weak equivalence ^L(B ^A B) ' holimi colimN G=N Bi . Remark 7.3. We make some remarks about this definition; in particular, we dis- cuss what is and is not automatically entailed by the hypotheses of the definit* *ion. (1) So that condition (2) above is actually met in practice, we do not requi* *re that the continuous action be by maps of A-algebras; we only require that the continuous action and the A-algebra action be compatible. For exam- ple, Gn acts on En by maps of ^LS0-algebras and this action yields the continuous action described in [3], but the continuous action is only (t* *hus far, known to be) by maps of (unstructured) spectra. (2) Since all the Ai and Bi are K(n)-local, the homotopy limits A and B are also K(n)-local. (3) The hypotheses of the definition imply that B is automatically a continu* *ous G-spectrum. (4) By [3], BhG = holimiBhGi-' holimiAi = A, so that the assumptions automatically imply that condition (3) holds. Remark 7.4. We explain part of our motivation for condition (4) in Definition 7.2. Recall (from [3]) that the functor Map c(G, -): Sp ! SpG is a right Quillen functor.Q Let X 2 SpG be fibrant, so that X is also fibrant in Sp, and hence, colimN G=N X ~= Map c(G, X) is fibrant in SpG. Then . . .! X ! X, the constant tower of fibrations of fibrant spectra in Sp, gives Q holimi colimN G=N X ~=holimiMap c(G, X) ' limiMap c(G, X) ~=Map c(G, X) ~=colimNQ G=NX, where the last spectrum, as desired, has the form of the right-hand side in con- dition (3) of Definition 4.1. Therefore, a K(n)-local G-pro-Galois extension i* *s a generalization of a K(n)-local (associative) G-Galois extension from the settin* *g of discrete G-spectra to that of towers of discrete G-spectra. Theorem 7.5. The map of commutative S-algebras holimIfl(I), ^LS0 ~=holimI^LMI ! holimI(En ^ MI) ~=En, is a K(n)-local Gn-pro-Galois extension. Proof.We only need to verify condition (4) of Definition 7.2: by [4, Cor. 3.9], ^L(En ^^LS0En) ' holimIMap c(Gn, Fn ^ MI) ' holimIcolimiQ Gn=Ui(En ^ MI). Remark 7.6. Since L^LS0K(n)(En ^^LS0En) ' ^L(En ^^LS0En), ^LS0 ! En is a strong* *ly K(n)-local Gn-pro-Galois extension. 11 Though we have shown that ^LS0 ! En is a K(n)-local Gn-pro-Galois extension, this notion still does not capture all of the structure that is present in this* * extension, due to the extra structure that comes from the filtered extension fl(I). We cap* *ture this additional structure in the following way. Let {A -fff!Bff}ffbe a direct system of, possibly K(n)-local, Gff-Galois exte* *n- sions, with {Gff}ffan inverse system of finite groups, such that each map Bff! * *Bff0 is Gff0-equivariant. As usual, let G = limffGff, and let B = ^L(colimffBff). If* * B is regarded as a spectrum of simplicial sets, then, letting (-)f denote functor* *ial fibrant replacement in Sp, B ~=holimI colimff(Bff^ LnMI)f is a continuous G- spectrum, since colimff(Bff^ LnMI)f is a discrete G-spectrum that is fibrant in Sp. Then, if ^L(colimfffff): ^LA ! B is a G-Galois extension, we call ^L(colimf* *ffff) a K(n)-local filtered G-Galois extension. Since the direct system {ff(Ui): ^LS0 ! EhUin}i, of K(n)-local Gn=Ui-Galois extensions, yields the extension ^L(colimiff(Ui)): ^LS0 ! En, we can refer to t* *he map ^LS0 ! En as a K(n)-local filtered Gn-pro-Galois extension. 8. More examples, for closed subgroups of Gn In this section, for any closed subgroup G of Gn (so G is always profinite and not necessarily finite), we give two examples of Galois extensions. First of al* *l, we slightly expand the definition of filtered G-Galois extension. Definition 8.1. Let {Aff! Bff}ffbe a direct system of (associative) Gff-Galois extensions, with {Gff} an inverse system of finite groups, where each map Bff! * *Bff0 is Gff0-equivariant. Let G and B be defined as usual, and let A = colimffAff. If A ! B is a (associative) G-Galois extension, then A ! B is a (associative) filt* *ered G-Galois extension. Theorem 8.2. The direct system {fi(G, i, I): EhUiGn^ MI ! EhUin^ MI}i, of associative UiG=Ui-Galois extensions, makes the map fi(G, I) = colimifi(G, i, I): (EhGn^ MI) ~=(Fn ^ MI)hG ! Fn ^ MI ' En ^ MI an associative filtered G-Galois extension. Proof.To make the notation more manageable, we use X=I to denote the spectrum X ^ MI. Since (we are assuming that) Fn=I 2 SpGan, (Fn=I)hUi ~=EhUin=I and, similarly, EhUiGn=I are S-algebras. Now we show that Q (8.3) (EhUin=I) ^(EhUiGn=I)(EhUin=I) ' UiG=Ui(EhUin=I). Q Note that EhUin=I and UiG=Ui(EhUiGn=I) are K(n)-local. Applying [5, Cor. 5.5], Q Q ss*(En ^ EhUin=I) ~=ss*( Gn=Ui(En=I)) ~= Gn=Uiss*(En=I). Similarly, Q Q Q ss*(En ^ ( UiG=Ui(EhUiGn=I)))~=Gn=UiG UiG=Uiss*(En=I) ~=Q Gn=Uiss*(En=I). Q Thus, ss*(En ^ EhUin=I) ~=ss*(En ^ ( UiG=Ui(EhUiGn=I))), showing that Q EhUin=I ~= UiG=Ui(EhUiGn=I). 12 This implies that Q (EhUin=I) ^(EhUiGn=I)(EhUin=I)~=UiG=Ui(EhUiGn=I) ^(EhUiGn=I)(EhUin=I) i j ~=Q UiG=Ui (EhUiGn=I) ^(EhUiG (EhUi=I) Q n =I) n ' UiG=Ui(EhUin=I), verifying (8.3). This showsQthat fi(G, i, I) is an associative UiG=Ui-Galois ex* *tension. Note that En=I and colimi G=(Ui\G)(EhGn=I) are K(n)-local, and there is an isomorphism ss*(En ^ En=I) ~=Map c(Gn, ss*(En=I)). Also, as abelian groups, Q Q ss*(En ^ (colimi G=(Ui\G)(EhGn=I))) ~=colimi G=(Ui\G)ss*(En ^ EhGn=I), which, by [5, Prop. 6.3], is isomorphic to Q colimi G=(Ui\G)colimjMap c(Gn=UjG , ss*(En=I)). This last abelian group is isomorphic to Q colimi G=(Ui\G)Mapc(Gn=G, ss*(En=I))~=Mapc(G x Gn=G, ss*(En=I)) ~=Mapc(Gn, ss*(En=I)). Q Thus, En=I ~=colimi G=(Ui\G)(EhGn=I), and therefore, Q (En=I) ^(EhGn=I)(En=I)~=colimi G=(Ui\G)(EhGn=I) ^(EhGn=I)(En=I) ~=colimiQ G=(Ui\G)(En=I), completing the proof. Theorem 8.4. The inverse system {fi(G, I)}I of associative G-Galois extensions makes the map fi(G) = holimIfi(G, I): EhGn! En a K(n)-local G-pro-Galois extension. Proof.Using the preceding theorem, it is easy to see that each fi(G, I) is a K(* *n)- local associative G-Galois extension, since En ^ MI, EhGn^ MI, and Q Q colimi G=(Ui\G)(En ^ MI) ~=(colimi G=(Ui\G)En) ^ MI are all K(n)-local. By [4, Cor. 3.9], ss*(^L(En ^EhGnEn)) ~=Map c(G, ss*(En)) ~=limIMap c(G, ss*(En ^ MI)). This implies that ^L(En ^EhGnEn)~=holimIMap c(G, (Fn ^ MI)f,G) ~=holimIcolimiQ G=(Ui\G)(Fn ^ MI), where the second expression only occurs in Sp. Our last result follows from the last line of the proof of Theorem 2.6. Theorem 8.5. The map ff(U) = holimIfl(U, I) is a K(n)-local Gn=U-pro-Galois extension. 13 References [1]Andrew Baker. 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