MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS MARK HOVEY Abstract.Comodules over Hopf algebroids are of central importance in al- gebraic topology. It is well-known that a Hopf algebroid is the same thi* *ng as a presheaf of groupoids on Aff, the opposite category of commutative rin* *gs. We show in this paper that a comodule is the same thing as a quasi-coher* *ent sheaf over this presheaf of groupoids. We prove the general theorem that* * in- ternal equivalences of presheaves of groupoids with respect to a Grothen* *dieck topology T on Affgive rise to equivalences of categories of sheaves in t* *hat topology. We then show using faithfully flat descent that an internal eq* *uiv- alence in the flat topology gives rise to an equivalence of categories o* *f quasi- coherent sheaves. The corresponding statement for Hopf algebroids is th* *at weakly equivalent Hopf algebroids have equivalent categories of comodule* *s. We apply this to formal group laws, where we get considerable generaliza* *tions of the Miller-Ravenel [MR77 ] and Hovey-Sadofsky [HS99] change of rings * *the- orems in algebraic topology. Introduction A commutative Hopf algebra is a (commutative) ring A together with a lift of * *the functor SpecA: Rings -! Setto a functor Rings -! Groups . Here Rings is the category of commutative rings with unity, Setis the category of sets, Groups is* * the category of groups, and (SpecA)(R) = Rings(A, R). So a Hopf algebra is the same thing as an affine algebraic group scheme, or a representable presheaf of group* *s on Aff, the opposite category of Rings . In the same way, a Hopf algebroid (A, ) is an affine algebraic groupoid scheme, or a representable presheaf of groupoids (SpecA, Spec ) on Aff. Here, given a ring R, SpecA(R) is the set of objects of the groupoid corresponding to R, and Spec (R) is the set of morphisms of that groupoid. Hopf algebroids are very important in algebraic topology, because for many im- portant homology theories E, the ring of stable co-operations E*E is a (graded) Hopf algebroid over E* but not a Hopf algebra. In particular, this is true for complex cobordism MU and complex K-theory. In this case, E*X is a (graded) comodule over the Hopf algebroid E*E. Of course, not all schemes are affine. One of the essential contributions of Grothendieck was the realization that it is necessary to study all schemes even if one is only interested in affine schemes. In the same way, to understand Ho* *pf algebroids, one should study more general groupoid schemes. ____________ Date: May 16, 2001. 2000 Mathematics Subject Classification. Primary 14L15; Secondary 14L05, 16W* *30, 18F20, 18G15, 55N22. The author was supported in part by NSF grant DMS 99-70978. 1 2 MARK HOVEY One of the difficulties is that the standard approach to schemes, involving c* *overs by open affine subschemes, is not the right one for the algebraic topology sett* *ing. Instead, it is better to use the functorial approach hinted at above in our def* *inition of SpecA. This approach is well-known in algebraic geometry [DG70 ]. It was introduced to algebraic topology by Hopkins and Neil Strickland. Strickland has written an excellent exposition of this point of view in [Str99]. In this appro* *ach, we study arbitrary presheaves of sets (or groupoids) on Aff. Demazure and Gabriel [DG70 ] show that the category of A-modules is equivalent to the category of quasi-coherent sheaves over the presheaf of sets SpecA on Af* *f. Our first goal in this paper is to extend this theorem as follows. Let T denote* * a Grothendieck topology on Aff, and let AffT denote the resulting site (we put a cardinality restriction on rings to make Aff a small category). Given a preshe* *af of groupoids (X0, X1) on Aff, we define the category Sh T(X0,X1)of sheaves over (X0, X1) with respect to T and we define the category Shqc(X0,X1)of quasi-coher* *ent sheaves over (X0, X1). Our first main result is then the following theorem, pro* *ved as Theorem 2.2. Theorem A. Suppose (A, ) is a Hopf algebroid. Then there is an equivalence of categories between -comodules and quasi-coherent sheaves over (SpecA, Spec ). There is a natural notion of an internal equivalence of presheaves of groupoi* *ds on AffT, studied by Joyal and Tierney [JT91 ] and other authors as well. A map : (X0, X1) -! (Y0, Y1) of presheaves of groupoids is an internal equivalence w* *ith respect to T if (R) is fully faithful for all R and if is essentially surjec* *tive in a sheaf-theoretic sense, related to T . This is really the natural notion of i* *nter- nal equivalence for sheaves of groupoids on AffT ; there is a more general noti* *on appropriate for presheaves, introduced by Hollander [Hol01], but we do not need* * it. Our second main result is that the category of sheaves is invariant under int* *ernal equivalence. The following theorem is proved as Theorem 3.2. Theorem B. Suppose : (X0, X1) -! (Y0, Y1) is an internal equivalence of pre- sheaves of groupoids on AffT. Then *: ShT(Y0,Y1)-!ShT(X0,X1)is an equivalence of categories. What we really care about is the category of quasi-coherent sheaves. Faithful* *ly flat descent shows that a quasi-coherent sheaf is a sheaf in the flat topology * *on Aff. This is often called the fpqc topology; in it,Qa cover of a ring R is a f* *inite family {R -!Si} of flat extensions of R such that Si is faithfully flat over * *R. A strengthening of faithfully flat descent then leads to the following theorem, p* *roved as Theorem 4.5. Theorem C. Suppose : (X0, X1) -! (Y0, Y1) is an internal equivalence of pre- sheaves of groupoids on AffT, where T is the flat topology. Then *: Shqc(Y0,Y1* *)-! Shqc(X0,X1)is an equivalence of categories. In order to apply this theorem to Hopf algebroids, we need to characterize th* *ose maps of Hopf algebroids that induce internal equivalences in the flat topology * *of the corresponding presheaves of groupoids. The following theorem is proved as Theorem 5.5. Theorem D. Suppose f = (f0, f1): (A, ) -!(B, ) is a map of Hopf algebroids. Then f* :(SpecB, Spec ) -! (SpecA, Spec ) is a internal equivalence in the flat MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 3 topology if and only if jL f1 jR :B A A B -! is an isomorphism and there is a ring map g :B A -! C such that g(f0 jR ) exhibits C as a faithfully flat extension of A. This condition has appeared before, in [Hop95 ] and [HS99 ]. We point out that if we used the more general notion of internal equivalence mentioned above, The* *o- rem D would remain unchanged, since SpecA is already a sheaf in the flat topolo* *gy by faithfully flat descent. Finally, we apply our results to the Hopf algebroids relevant to algebraic to* *pology. The following theorem is proved as Theorem 6.2 (and the terminology is defined * *in Section 6). Theorem E. Fix a prime p and an integer n > 0. Let (A, ) denote the Hopf algebroid (v-1nBP*=In, v-1nBP*BP=In). Suppose B is a ring equipped with a homo- geneous p-typical formal group law of strict height n, classified by f :A -!B. * *Then the functor that takes an (A, )-comodule M to B A M defines an equivalence of categories from graded (A, )-comodules to graded (B, B A A B)-comodules. As an immediate corollary, we recover a strengthening of the change of rings theorem of [HS99 ], which itself is a strengthening of the well-known Miller-Ra* *venel change of rings theorem [MR77 ]. The precise change of rings theorem is prove is stated below. Theorem F. Let p be a prime and m n > 0 be integers. Suppose M and N are BP*BP -comodules such that vn acts isomorphically on N. If either M is finitely presented, or if N = v-1nN0 where N0 is finitely presented and In-nilpotent, th* *en Ext**BP*BP(M, N) ~=Ext**E(m)*E(m)(E(m)* BP* M, E(m)* BP* N). This theorem implies that the chromatic spectral sequence based on E(m) is the truncation of the chromatic spectral sequence based on BP consisting of the fir* *st n + 1 columns, as pointed out in [HS99 , Remark 5.2]. There are several ways in which the results in this paper might be generalize* *d. Most substantively, we do not recover the Morava change of rings theorem [Mor85* * ] from our result. The Morava change of rings theorem is about complete comodules over a complete Hopf algebroid, so one would need to account for the topology in some way. Secondly, our results will probably hold if we replace Aff by the opposite category of rings in some topos, as suggested by Rick Jardine. In fact* *, we already have to replace Aff by the opposite category of graded rings in order to cope with the graded Hopf algebroids that arise in algebraic topology. Lastly, * *there is the aforementioned generalization of the notion of internal equivalence, due* * to Hollander [Hol01]. In this generalization, one would replace äf ithful" by "she* *af- theoretically faithfulä nd üf ll" by "sheaf-theoretically full". We are confid* *ent our results will hold for this generalization, but we would not get any new example* *s of equivalences of categories of comodules. Nevertheless, this generalization migh* *t be useful in other circumstances. This paper arose from trying to understand comments of Mike Hopkins, and I thank him deeply for sharing his insights. The one-line summary of this paper is "The category of comodules over a Hopf algebroid only depends on the associated stack", and the author first heard this summary from Hopkins. It is certain that 4 MARK HOVEY Hopkins has proved some of the theorems in this paper. As far as I know, how- ever, Hopkins approached these theorems by using stacks, which I have completely avoided. In particular, my definition of sheaves and quasi-coherent sheaves ov* *er presheaves of groupoids is quite different from the definition I have heard from Hopkins, though the two definitions are presumably equivalent. The author would also like to thank Dan Christensen and Rick Jardine, both of whom thought that the original version of this paper, dealing as it did with on* *ly quasi-coherent sheaves, was much too specific and must be a corollary of a simp* *ler, more general theorem. Notation We compile the notations and conventions we use in this paper. All rings are assumed commutative, and of cardinality less than some fixed infinite cardinal * *~. Rings denotes the category of such rings, and Aff denotes its opposite category. We think of Aff as the category of representable functors SpecA: Rings -! Set, where (SpecA)(R) = Rings (A, R). We will also want to consider Rings *, the category of graded rings (of cardinality less than ~) that are commutative in t* *he graded sense, and its opposite category Aff*. If x, y :A -! R are ring homomorphisms, the symbol xRy denotes R with its A-bimodule structure, where A acts on the left through x and on the right throu* *gh y. This is especially useful for the tensor product; the symbol Rx A yS indica* *tes the bimodule tensor product, where A acts on the right on R via x and on the le* *ft on S via y. We use this same notation in the graded case as well, where x and y are tacitly assumed to preserve the grading and the tensor product is the graded tensor product. The symbols (A, ) and (B, ) denote (possibly graded) Hopf algebroids. We follows the notation of [Rav86 , Appendix 1] for the structure maps of a Hopf a* *lge- broid. So we have the counit ffl: -! A, the left and right units jL, jR :A -!* * , the diagonal : -! jR A jL , and the conjugation c: jL jR -!jR jL. Capital letters at the end of the alphabet, such as X, Y , and Z, will denote functors from Rings to Set, or functors from Rings* to Set in the graded case. The symbol Yf xX gZ will denote the pullback of the diagram Y -f!X -g Z. The symbols (X0, X1) and (Y0, Y1) will denote functors from Rings (or Rings*) to Gpds , the category of small groupoids. Here X0(R) is the object set of the groupoid corresponding to R, and X1(R) is the morphism set of that groupoid. There are structure maps id:X0 -!X1 dom, codom:X1 -!X0 O:(X1)dom xX0 codom(X1)X1 -!X1 inv:X1 -!X1 satisfying the relations necessary to make (X0(R), X1(R)) a groupoid. 1. Sheaves over functors The object of this section is to define the notion of a sheaf of modules M ov* *er a sheaf of sets X on Aff. We will generalize this in the next section to sheave* *s of modules over sheaves of groupoids (X0, X1) on Aff. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 5 We will assume given a Grothendieck topology T on Aff, and denote the result- ing site consisting of Afftogether with T by AffT. For us, the two most importa* *nt Grothendieck topologies on Aff will be the trivial topology, where the only cov* *ers are isomorphisms, and the the fpqc, or flat, topology, which will be discussed * *later. Now suppose X :Rings -! Setis a functor. We think of X as a presheaf of sets on AffT . We need to define the category of sheaves over X. We first define the overcategory AffT=X. An object of AffT=X is a map of presheaves SpecR x-!X, and the morphisms are the commutative triangles. We call the opposite category * *of AffT=X the category of points of X following [Str99]; it is called the category* * of X- models in [DG70 ]. A point of X is a pair (R, x), where R is a ring and x 2 X(R* *), and a morphism from (R, x) to (S, y) is a ring homomorphism f :R -! S such that X(f)(x) = y. We often abuse notation and write f(x) for X(f)(x). As an overcategory, AffT=X inherits the Grothendieck topology T . A cover of (R, x) is a family {(R, x) -! (Si, xi)} such that {R -! Si} is a cover of R. The category AffT=X also comes equipped with a structure presheaf O :(AffT =X)op -!Rings , where O(R, x) = R. Definition 1.1.Suppose X :Rings -! Set is a presheaf of sets on AffT. Then a sheaf of modules over X, often called just a sheaf over X, is a sheaf of O-modu* *les on AffT=X. More concretely, a sheaf M is a functorial assignment of an R-module Mx to each point (R, x), satisfying the sheaf condition. Functoriality means that a m* *ap (R, x) f-!(S, y) induces a map of R-modules Mx `M-(f,x)----!My, where My is tho* *ught of as an R-module by restriction. We often abbreviate `(f, x) to `(f). We must have `(gf) = `(g)O`(f) and `(1) = 1. The sheaf condition means that if {(R, x) * *-! (Si, xi)} is a cover, then the diagram Y Y Mx -! Mxi' Mxjk i jk is an equalizer of R-modules, where xjk is the image of x in X(Sj R Sk). The maps in this diagram are all maps of R-modules. We have an evident definition of a map of sheaves over X. To be concrete, a map ff: M -! N of sheaves over X assigns to each point (R, x) of X a map ffx: Mx -! Nx of R-modules, natural in (R, x). This gives us a category ShTX of sheaves over X. A map of sheaves X -! Y induces a functor *: ShTY-! Sh TX. Here, if M is a sheaf over Y and (R, x) is a point of X, we define ( *M)x = M x. Note that all of these definitions work perfectly well in the graded case as * *well. We would have a Grothendieck topology T on Aff*, and a functor X :Aff* -!Set . A point of X would be a graded ring R and a point x 2 X(R). A sheaf M over X would be as assignment of a graded R-module Mx to each point (R, x) of X(R), satisfying the functoriality and sheaf conditions. We now consider quasi-coherent sheaves. We only need quasi-coherent sheaves in the trivial topology, so we will stick to that case. A quasi-coherent sheaf* * is supposed to be a sheaf that acts like a free sheaf in an appropriate sense. The salient property of the free sheaf O is that, if (R, x) -! (S, y) is a map of p* *oints, then Oy = S R Ox. We therefore make the following definition. Definition 1.2.Suppose X :Rings -! Set is a functor. A quasi-coherent sheaf M over X is a sheaf over X in the trivial topology such that, given a map 6 MARK HOVEY (R, x) -! (S, y) of points of X, the adjoint æM (f): S R Mx -! My of `M (f) is an isomorphism. This is the same definition given in [DG70 ] and [Str99]. We get a category S* *hqcX, which is the full subcategory of sheaves over X in the trivial topology consist* *ing of the quasi-coherent sheaves. Given a map : X -! Y of functors, *: ShTY-! ShTX restricts to define *: ShqcY-!Sh qcX. The value of this definition of quasi-coherence is shown by the following lem* *ma. Lemma 1.3. Suppose A 2 Rings , and let SpecA: Rings -! Set be the rep- resentable functor (SpecA)(R) = Rings (A, R). Then the category of A-modules is equivalent to the category of quasi-coherent sheaves over SpecA. The equiva- lence takes an A-module M to the quasi-coherent sheaf fM over SpecA defined by fMx= Rx A M for x: A -! R, and its inverse takes a quasi-coherent sheaf N to its value at 1: A -!A. This lemma is due to Demazure and Gabriel [DG80 , p. 61], who actually show that the category of quasi-coherent sheaves over a scheme when defined this way agrees (up to equivalence) with the usual notion of quasi-coherent sheaves on a scheme. A direct proof can be found in [Str99]. Once again, we note that Lemma 1.3 will work in the graded case as well. The definition of a quasi-coherent sheaf over a functor X :Rings *-! Set is similar to the ungraded case, and the same argument used to prove Lemma 1.3 shows that, if A is a graded ring, the category of quasi-coherent sheaves over SpecA * *(now defined by (SpecA)(R) = Rings *(A, R)) is equivalent to the category of graded A-modules. It will be useful later to note that, if f :A -! B is a ring homomorphism and Specf : SpecB -! SpecA is the corresponding map of functors, then the induced map (Specf)*: ShqcSpecA-!Sh qcSpecBtakes the A-module M to the B-module B A M. 2. Sheaves over groupoid functors The object of this section is to prove Theorem A, showing that a comodule over a Hopf algebroid is a special case of the more general notion of a quasi-cohere* *nt sheaf over a presheaf of groupoids. This will require us to define the notion o* *f a sheaf M of modules over a presheaf of groupoids (X0, X1) on AffT. We will consider a presheaf of groupoids (X0, X1) on AffT . This means that X0 and X1 are presheaves of sets on AffT, and that (X0(R), X1(R)) is a groupoid for all R, naturally in R. So we have structure maps as defined in the notation section. A presheaf of groupoids (X0, X1) is called a sheaf of groupoids when X0 and X1 are sheaves of sets on AffT; we would be happy to assume our presheaves of groupoids are in fact sheaves of groupoids, but that assumption is unnecessa* *ry. Sheaves of groupoids have been much studied in the literature; a stack is a special kind of sheaf of groupoids, and stacks are essential in modern algebraic geometry [FC90 ]. The homotopy theory of sheaves of groupoids has been studied by Joyal and Tierney [JT91 ], Jardine [Jar00], and Hollander [Hol01]. Definition 2.1.Suppose (X0, X1) is a presheaf of groupoids on AffT . A sheaf over (X0, X1) is a sheaf M over X0 together with an isomorphism _ :dom *M -! codom* M of sheaves over X1 satisfying the cocycle condition. To explain the cocycle condition, note that, if ff is a morphism of X1(R), _ffis an isomorphis* *m of MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 7 R-modules _ff:Mdomff-! Mcodomff. The cocycle condition says that if fi and ff a* *re composable morphisms, then _fiff= _fiO _ff. A quasi-coherent sheaf over (X0, X1) is a sheaf M over (X0, X1) in the trivial topology such that M is quasi-coheren* *t as a sheaf over X0. We also get a notion of a map ø :M -! N of sheaves over (X0, X1). Such a map is a map of sheaves over X0 such that the diagram M Mdomff --_ff--!Mcodomff ? ? ødomff?y ?yøcodomff Ndomff ----! Ncodomff _Nff commutes for all points (R, ff) of X1(R). We then get categories Sh T(X0,X1)and Shqc(X0,X1). Note that a map : (X0, X1) -! (Y0, Y1) induces a functor *: ShT(Y0,Y1)-! ShT(X0,X1)and *: Shqc(Y0,Y1)-!Shqc(X0,X1). Indeed, we define _f*Mf= _M ff. Also note that all of the comments above work perfectly well for presheaves of groupoids on Aff*. In this case, _ff:Mdomff-! Mcodomffwill be an isomorphism of graded R-modules. As originally noted by Haynes Miller, a Hopf algebroid [Rav86 , Appendix 1] is just a pair of commutative rings (A, ) such that (SpecA, Spec ) is a sheaf * *of groupoids (in the trivial topology). The structure maps of a Hopf algebroid (li* *sted in the notation section) are therefore dual to the structure maps of a presheaf* * of groupoids; for example, the diagonal : -! jR A jL is dual to the composition map (X1)dom xX0 codomX1. It is useful to recall the composition in the groupoid (SpecA, Spec )(R) from this point of view. Suppose fi, ff: -! R are ring homomorphisms with ffjL = x, ffjR = fijL = y, and fijR = z, so that ff is a morphism from x to y and fi is a morphism from y to z. The composition fi O ff: -! R is defined to be the composite -! jL jR A jL jR ff-fi--!xRy A yRz ~-!xRz. Just as a quasi-coherent sheaf over SpecA is the same thing as a module over A, so a quasi-coherent sheaf over (SpecA, Spec ) is the same thing as a comodule over (A, ). The following theorem is Theorem A of the introduction. Theorem 2.2. Suppose (A, ) is a Hopf algebroid. Then there is an equivalence * *of categories between -comodules and quasi-coherent sheaves over (SpecA, Spec ). This theorem will also hold in the graded context: if (A, ) is a graded Hopf algebroid, then the category of graded -comodules is equivalent to the category of quasi-coherent sheaves over the presheaf of groupoids (SpecA, Spec ) on Aff*. The proof is the same as the proof below. Proof.We first construct a functor from quasi-coherent sheaves over (SpecA, Spe* *c ) to (A, )-comodules. Suppose that fMis a quasi-coherent sheaf over (SpecA, Spec* * ). Then fM is in particular a quasi-coherent sheaf over SpecA, so corresponds to an 8 MARK HOVEY A-module M. Then if ff: -!R is a point of Spec defined over R, with ffjL = x and ffjR = y, (dom *fM)ff= Rx A M and (codom *fM)ff= Ry A M. Let us denote by e_the isomorphism of sheaves dom *fM -! codom*fM. Then, e_ defines an isomorphism e_ff:Rx A M -! Ry A M. of R-modules. Taking ff to be the identity map 1 of , we define _ :M -! jR A M to be the composite e_1 M = A A M -jL-1-! jL A M -! jR A M. We must show that _ is counital and coassociative. Note first that ffl: -! A, thought of as a morphism in the groupoid (SpecA, Spec )(A), is the identity morphism of the object 1A :A -! A, and so in particular is idempotent. The cocycle condition implies that e_fflis also idempotent, and since it is an isom* *orphism, it follows that e_fflis the identity of M. Now, ffl defines a map from the poin* *t ( , 1) to the point (A, ffl) of Spec . Since e_is a map of sheaves over Spec , we conc* *lude that 1 e_1:A ( jL A M) -!A ( jR A M) is the identity map. From this it follows easily that _ is counital. To see that _ is coassociative, let ff: -! A denote the map that takes* * t to t 1. Let fi denote the map that takes t to 1 t. Then we have yjR (a) = jR a 1 = 1 jLa = xjL(a), and so fi O ff makes sense. A calculation shows that fi O ff = , the diagonal * *map. If (R, fl) is an arbitrary point of Spec with fljL = x and fljR = y, there is * *a map from ( , 1) to (R, fl). Since e_is a map of sheaves, we find that e_flis the co* *mposite _e1 Rx A M ~=Rfl jL A M -1--! Rfl jR A M ~=Ry A M. This description allows us to compute e_fiand e_ff, and so also their composite* *. We find that e_fiO e_fftakes 1 1 m to (1 _)_(m). Similarly e_ takes 1 1 m to ( 1)_(m). The cocycle condition forces these to be equal, and so _ is coassociative. We have now constructed a comodule M associated to any quasi-coherent sheaf fMover (SpecA, Spec ). We leave to the reader the striaghtforward check that th* *is is functorial. Our next goal is to construct a functor from (A, )-comodules to quasi-cohere* *nt sheaves over (SpecA, Spec ). Suppose M is a -comodule with structure map _ :M -! jR A M. Then, in particular, M is an A-module, so there is an associated quasi-coherent sheaf fM over SpecA, defined by fMx= Rx A M, where x: A -! R is a ring homomorphism. Given a point ff: -! R of Spec with ffjL = x and ffjR = y, we have (dom *fM)x = Rx A M and (codom *fM)x = Ry A M We define e_:dom *fM -!codom *fM by letting e_ffbe the composite Rx A M -1-_-!Rx A jL jR A M -1-ff-1-!Rx A xRy A M -~-1!Ry A M. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 9 We leave to the reader the check that e_is a map of sheaves. It remains to show that e_satisfies the cocycle condition and is an isomorphi* *sm. We begin with the cocycle condition. Suppose that ff, fi : -! R are ring homo- morphisms with ffjL = x, ffjR = fijL = y, and fijR = z. Consider the following commutative diagram, in which all tensor products that occur are taken over A, and = jL jR. Rx M -1-_---!Rx M -1-ff-1--!Rx xRy M --~-1--! Ry M ? ? ? ? 1 _?y 1 1 _?y 1 1 _?y 1 1 _?y Rx M 1---1--!Rx -1M-ff-1-1---!Rx xRy -~-M1-1--!Ry M ? ? 1 1 fi?1y 1 fi?1y Rx xRy yRz M -~-1-1--!Ry yRz M ? ? 1 ~ 1?y ~ 1?y Rx xRz M --~-1--! Rz M The outer clockwise composite in this diagram is e_fiO e_ff, and the outer co* *un- terclockwise composite is e_fiOff, using the description of fi O ff given above* *. Thus e_ satisfies the cocycle condition. We must still show that e_ffis an isomorphism for all ff: -!R. Since e_sati* *sfies the cocycle condition and ff is itself an isomorphism, it suffices to show that* * e_1xis an isomorphism, where 1x is the identity morphism of x: A -! R. That is, 1x is the composite -ffl!A x-!R. But one can check, using the fact that _ is counital, that e_1xis the identity * *of Rx A M. This completes the proof that fMis a quasi-coherent sheaf over (SpecA, Spec * *). We leave to the reader the check that it is functorial in M. We also leave to the reader the check that these constructions define inverse_ equivalences of categories. |__| Maps of Hopf algebroids (f0, f1): (A, ) -! (B, ) are defined in [Rav86 , De* *fi- nition A1.1.7]; they are, of course, maps such that = (Specf0, Specf1) is a m* *ap of sheaves of groupoids. According to Theorem 2.2, (f0, f1) will induce a map * from (A, )-comodules to (B, )-comodules. This maps takes the -comodule M to B A M. In order to define the structure map of B A M, recall from [Rav86 , Definition A1.1.7] that the definition of a map of Hopf algebroids requires jLf0 = x = f1jL and jR f0 = y = f1jR . We then define the structure map of B A M to be the composite Bf0 A M 1-_--!B A jL jR A M -jL-f1-1---! x A x y A M ~-1-! ~ y A M = jR B (Bf0 A M). 3.Internal equivalences yield equivalences The object of this section is to prove Theorem B, showing that if : (X0, X1)* * -! (Y0, Y1) is an internal equivalence of presheaves of groupoids on AffT, then *: ShT(Y0,Y1)-!ShT(X0,X1) 10 MARK HOVEY is an equivalence of categories. This statement essentially says that the categ* *ory of sheaves is a homotopy-invariant construction. We begin by defining an internal equivalence. Internal equivalences are the w* *eak equivalences in the model structure on sheaves of groupoids considered by Joyal and Tierney in [JT91 ]. Definition 3.1.Suppose : (X0, X1) -!(Y0, Y1) is a map of presheaves of group- oids on AffT . The essential image of is the subfunctor of Y0 consisting of * *all points (R, y) of Y0 such that there exists a point (R, x) of X0 and a morphism ff 2 Y1(R) from x to y. The sheaf-theoretic essential image of is the subfun* *ctor of Y0 consisting of all points (R, y) such that there exists a cover {R -fi!Si}* * of R in the topology T such that yi = fiy is in the essential image of for all i. * *The map is called an internal equivalence if (R) is full and faithful for all R,* * and if the sheaf-theoretic essential image of is Y0 itself. For example, is an internal equivalence in the trivial topology if and only if (R) is full, faithful, and essentially surjective for all R, so that (R) i* *s an equivalence of groupoids for all R. Our goal is then to prove the following theorem, which is Theorem B of the introduction. Theorem 3.2. Suppose : (X0, X1) -!(Y0, Y1) is an internal equivalence of pre- sheaves of groupoids on AffT. Then *: ShT(Y0,Y1)-!ShT(X0,X1)is an equivalence of categories. As usual, our proof of this theorem will work in the graded case as well. We point out that there should be a model structure on presheaves of groupoids extending the Joyal-Tierney model structure. The weak equivalences in this model structure would be the maps which are sheaf-theoretically fully faithful and * *whose sheaf-theoretic essential image is all of Y0. Theorem 3.2 should then be a spec* *ial case of the more general theorem that a weak equivalence of presheaves of group* *oids induces an equivalence of their categories of sheaves. We have not considered t* *his more general case, because SpecA is already a sheaf in the flat topology, and S* *pecA is our main object of interest. We will prove this theorem by showing that * is full, faithful, and essentia* *lly surjective. The proof of each such step will be long, but divided into discrete* * steps very much like a diagram chase. In general, we are trying in each case to const* *ruct something for every point (R, y) of Y0. So first we do it for points (R, y) in* * the essential image of . This generally involves choosing a point (R, x) of X0 and* * a morphism ff: x -!y, so we generally have to prove that which choice one makes is immaterial. Then we show that every property we hope for in the construction is true on the essential image of . Next we extend the definition to all poin* *ts (R, y) in the sheaf-theoretic essential image of by using a cover. Once again* *, this depends on the choice of cover, so we have to show the choice is immaterial. For this, it is enough to show that refining the cover makes no difference, since a* *ny two covers have a common refinement. Finally, we show that the properties we want are sheaf-theoretic in nature, so that since they hold already on the essential* * image of , they also hold on the sheaf-theoretic essential image of . MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 11 Proposition 3.3.Suppose : (X0, X1) -! (Y0, Y1) is an map of presheaves of groupoids on AffT whose sheaf-theoretic essential image is all of Y0. Then *: ShT(Y0,Y1)-!ShT(X0,X1) is faithful. Proof.Suppose ø :M -! N is a map of sheaves on (Y0, Y1) such that *ø = 0. This means that ø x = 0 for all points (R, x) of X0. We must show that øy = 0 f* *or all points (R, y) of Y0. We first show that øy = 0 for all y in the essential i* *mage of . Indeed, suppose ff is a morphism from x to y. Then, since ø commutes with the structure map _, we get the commutative diagram below. M M x --_ff--!My ? ? ø x?y ?yøy N x ----! Ny _Nff It follows that øy = 0. Now suppose (R, y) is a general point of Y0. Since y is in the sheaf-theoret* *ic essential image of , we can choose a covering {R fi-!Si} such that yi= Y0(fi)(* *y) is in the essential image of for all i. Thus øyi = 0 for all i. We then hav* *e a commutative diagram Q My ----! Myi ? ? øy?y ?yQøyi Q Ny ----! Nyi The horizontal arrows are monomorphisms, since M and N are sheaves in T ,_so øy = 0 as well. |__| Note that we have actually shown, more generally, that if ø :M -! N is a morphism of sheaves over (Y0, Y1) such *ø = 0, then ø restricted to the sheaf- theoretic essential image of is also 0. Proposition 3.4.Suppose : (X0, X1) -! (Y0, Y1) is an map of presheaves of groupoids on AffT whose sheaf-theoretic essential image is all of Y0 and such t* *hat (R) is full for all R. Then *: ShT(Y0,Y1)-!ShT(X0,X1)is full. Proof.Suppose we have a map ø : *M -! *N. This means we have maps øx: M x -! N x for all points (R, x) of X0. We need to construct maps oey: My -! Ny for all points (R, y) of Y0 such that oe x = øx. Suppose first that y is in* * the essential image of , so that there is a morphism ff from x to y for some point (R, x) of X0. If oe were to exist, then we would have the commutative diagram below, M M x --_ff--!My ? ? øx?y ?yoey N x ----! Ny _Nff so we define oey = _Nfføx(_Mff)-1. 12 MARK HOVEY We claim that this definition of oey is independent of the choice of ff. Ind* *eed, suppose fi 2 Y1(R) is a morphism from x0to y. Then fi-1ff is a morphism from x to x0, and so, since is full, there is a morphism fl 2 X1(R) from x to x0* *such that fl = fi-1ff. Since ø is a map of sheaves, øx0_M fl= _N fløx. On the other* * hand, by the cocycle condition we have _ fl= (_fi)-1_ff. Combining these two equations gives _Nfføx(_Mff)-1 = _Nfiøx0(_Mfi)-1, so oey is independent of the choice of ff. In particular, if y = x, we can tak* *e ff to be the identity map of x. The cocycle condition implies that _Mffand _Nffare identity maps, and so oe x = øx. We now show that oe commutes with the structure maps of M and N on the essential image of . Suppose that (R, y) -f!(S, y0) is a map of points of Y0, * *and that y is in the essential image of . Choose a morphism ff from x to y for so* *me point (R, x) of X0. Let ff0 = Y1(f)(ff), so that ff0 is a morphism from x0 to * *y0, where x0= X0(f)(x). Since ø is a map of sheaves, we get the commutative square below. M x --øx--!N x ? ? `M (f, x)?y ?y`N (f, x) M x0 ----!ø N x0 x0 We would like to know that the square below is commutative. My --oey--!Ny ? ? `M (f,y)?y ?y`N (f,y) My0 ----!oe> Ny0. hy We claim that is an isomorphism from the top square to the bottom square, and so the bottom square must be commutative. Indeed, in the upper left corner this isomorphism is _Mff, in the upper right corner it is _Nff, in the lower left co* *rner it is _Mff0, and in the lower right corner it is _Nff0. All the required diagrams * *commute to make this a map of squares. This uses the fact that _M and _N are maps of sheaves and the well-definedness of oe. We now check that oe commutes with _, on the essential image of . Suppose we have a morphism fi :y -! y0in (Y0(R), Y1(R)), and that y is in the essential im* *age of . Let ff be a morphism from x to y for some point (R, x) of X0. Consider t* *he following diagram. M _Mfi M x -_ff---!My----! My0 ? ? ? øx?y oey?y ?yoey0 N x ----! Ny ----! Ny0 _Nff _Nfi By definition of oe, the left-hand square is commutative. The cocycle condition implies that _fiO _ff= _fiff, so the definition of oe also implies that the out* *side MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 13 square commutes. Since the horizontal maps are isomorphisms, the right-hand square must also be commutative. We now extend the definition of oe to an arbitrary point (R, y) of Y0. The sh* *eaf- theoretic essential image of is all of Y0, we can choose a cover {R fi-!Si} o* *f R in the topology T such that yi= Y0(fi)(y) is in the essential image of for all i* *. Let yjk denote the image of y in Y0(Sj R Sk). We then have a commutative diagram Q Q My ----! Myi - ---! Myjk Q oey?? ??Q yi y oeyjk Q Q Ny ----! Nyi - ---! Nyjk where the right-hand horizontal maps are the difference of the two restriction * *maps. Thus each row expresses its left-hand entry as a kernel. The diagram commutes since oe is a map of sheaves on the essential image of . Thus, there is a uniq* *ue map oey: My -!Ny making the diagram commute. We now check that oey is independent of the choice of cover. It suffices to s* *how that oey is unchanged if we replace the cover {R -!Si} by a refinement {R -!Tj}, since any two covers have a common refinement. If we denote the map coming from the refinement by oe0y, then we would have to have oe0yi= oeyi, since some of t* *he Tj form a cover of Siand oe is a map of sheaves on the essential image of . Then * *the sheaf condition forces oe0y= oey as well. In particular, if y is already in the* * essential image of , then we can take the identity cover to find that the new definition* * of oe is an extension of our old definition. We now show that oe is a map of sheaves over Y0. Suppose we have a map (R, y) -f!(S, y0) of points of Y0. Choose a cover {R -gi!Ti} of R such that yi = Y0(gi)(y) is in the essential image of for all i. Then there is an induced c* *over {S -hi!Ui= S R Ti} of S. The map f induces corresponding maps fi: (Ti, yi) -! (Ui, y0i), where y0i= Y0(hi)(y0). Since oe is a map of sheaves on the essential* * image of , we have the commutative diagram below. oeyi Myi ----! Nyi ?? ? y ?y My0i ----!oeNy0i y0i The sheaf condition and the definition of oe then show that the diagram My --oey--!Ny ?? ? y ?y My0 ----!oeNy0 y0 is commutative, and so oe is a map of sheaves over Y0. The proof that oe commutes with _, and so is a map of sheaves over (Y0,_Y1), * *is similar. |__| Finally, we show that * is essentially surjective. 14 MARK HOVEY Proposition 3.5.Suppose : (X0, X1) -! (Y0, Y1) is an internal equivalence of presheaves of groupoids on AffT . Then *: ShT(Y0,Y1)-!ShT(X0,X1)is essentially surjective. Proof.Suppose that N is a sheaf over (X0, X1). We must construct a sheaf M over (Y0, Y1) and an isomorphism *M -! N of sheaves. We first construct My for y in the essential image of , and show that it has the desired properties there.* * For every point (R, y) in the essential image of , choose a point (R, x(y)) of X0 * *and a morphism ff(y) from x(y) to y. Note that this only requires choosing over a s* *et, since Aff is a small category. Define My = Nx(y). We now construct the restriction of the structure map `M to the essential im* *age of . Suppose that we have a map (R, y) -f!(S, y0) between points of Y0, where (R, y) is in the essential image of . Let ff0= Y1(f)(ff(y)), so that ff0is a m* *orphism from x0to y0, where x0= X0(f)(x(y)). Then ff(y0)-1ff0is a morphism from x0 to x(y0). Since is full and faithful, there is a unique morphism fl of X1(S)* * from x0to x(y0) such that fl = ff(y0)-1ff0, We then define `M (f, y): My -! My0to be the composite N (f,x(y)) _Nfl My = Nx(y)`-------!Nx0--! Nx(y0)= My0. We must check the functoriality conditions for `M (restricted to the essential* * image of ). First of all, if f is the identity map, then fl will be the identity mo* *rphism of y. Since is faithful, it follows that fl is the identity morphism of x(y).* * The cocycle condition forces _Nflto be the identity map, and so `M (1, y) is the id* *entity as required. If g :(S, y0) -!(T, y00) is another map of points of Y0, a diagram* * chase involving the cocycle condition for _N and the fact that _N is a map of sheav* *es shows that `M (gf, y) is the composition `M (g, y0)`M (f, y). We now show that M is a sheaf on the essential image of . Indeed, suppose (R, y) is a point in the essential image of , and {R -! Si} is a cover of R in* * T . We must check that Y Y My -! Myi' Myjk is an equalizer diagram. We have an equalizer diagram Y Y My = Nx(y)-! Nx(y)i' Nx(y)jk since N is a sheaf. We construct an isomorphism from the bottom diagram to the top, from which it follows that the top is also an equalizer diagram. The morphism ff(y): x(y) -! y induces a morphism ff(y)i: x(y)i -! yi. We also have the morphism ff(yi): x(yi) -! yi. The composition (ff(yi))-1 O ff(y)i = * *fl for a unique fl :x(y)i -! x(yi), since isQfull and faithful.Q Then _fl:Nx(y)i* *-! Nx(yi)= MyidefinesQthe desiredQisomorphismQ Nx(y)i-! Myi. One constructs the isomorphism Nx(y)jk-! Nx(yjk)= Myjkin the same manner, using the morphisms ff(y)jk: x(y)jk-! yjk and ff(yjk). The proof that the diagram below Nx(yi)= Myi ----! Nx(yij)= Myij x? x ? ?? Nx(y)i ----! Nx(y)ij is commutative is a computation using the fact that _N is a map of sheaves, the cocycle condition, and the fact that is faithful. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 15 We now construct the restriction of the map _M to the essential image of . Suppose fi is a morphism from y to y0, where y is in the essential image of . * *Then ff(y0)-1fiff(y) is a morphism from x(y) to x(y0). Since is full and faithf* *ul, there is a unique morphism fl from x(y) to x(y0) such that fl = ff(y0)-1fiff(y* *). Hence we can define _Mfi= _Nfl. We leave to the reader the diagram chase showing that _ is a map of sheaves. We now construct the desired isomorphism of sheaves ø : *M -! N. (Since *M is determined by the restriction of M to the image of , we can do this even though we have not completed the definition of M). Suppose (R, x) is a point of X0. Then ff( x) is a morphism from (x( x)) to x. Since is full and faithful, there is a unique morphism fi from x( x) to x such that fi = ff( x). We define øx = _Nfi:M x = Nx( x)-! Nx. Obviously øx is an isomorphism, but we must check it is compatible with the str* *uc- ture maps. We leave these checks to the reader; both are diagram chases. We have now defined a sheaf M on the essential image of , and to complete the proof we need only extend it to a sheaf on all of (Y0, Y1). For each point * *(R, y) of Y0, choose a cover C(y) = {R fi-!Si} such that yi= Y0(fi)(y) is in the essen* *tial image of for all i, making sure to choose the identity cover when y is alread* *y in the essential image of . Once again, we can do this since Aff is a small categ* *ory. We then define My as we must if we are going to get a sheaf, as the equalizer of the two maps of R-modules Y Y Myi' Myjk. i jk This definition of My will of course depend on the choice of cover C(y). Supp* *ose D = {R -!Tm } is some other cover such that ym is in the essential image of f* *or all m. We claim that there is a canonical equalizer diagram Y Y My -! Mym ' Mynp. To see this, let MDy denote the pullback of the two arrows Y Y Mym ' Mynp. m np We claim that there is a canonical isomorphism MDy-! My. It suffices to check t* *his when D is a refinement of C(y), since any two covers have a common refinement. In this case, there is a diagram Y Y My -! Mym ' Mynp, m np where the first map is induced by first mapping to Myi, and then using the stru* *cture maps of M restricted to the essential image of to map further to Mym. It suff* *ices to prove that this diagram is an equalizer. It is easy to check that My maps in* *to the equalizer. If t 2 My maps to 0 in each Mym, then, using the fact that M restric* *ted to the essential image of is a sheaf, we find that t mapsQto 0 in each Myi. By definition of My, then, t = 0. Similarly, suppose (tm ) 2 Mym is in the equal* *izer. Again using the fact that M restrictedQto the essential image of is a sheaf, * *we construct an element (ti) 2 Myi. The images of ti and tj in Myijcoincide, sin* *ce they coincide after restriction to the induced cover. Thus we get an element t * *2 My 16 MARK HOVEY restricting to the ti. It follows that t restricts to the tm as well, and so My* * is the desired equalizer. Now we can construct the structure maps of M. Suppose (R, y) -! (S, z) is a map of points of Y0. The cover C(y) = {R -!Si} of R induces a cover D = {S -! S R Si} of S, and the restriction zi of z is in the essential image of for a* *ll i, since yi is so. Thus we get a map from Y Y Myi' Myjk to Y Y Mzi' Mzjk, and so an induced map My -!MDzon the equalizers. After composing this with the canonical isomorphism MDz-! Mz, we get the desired structure map ` :My -!Mz. Since we chose the identity cover when y was already in the essential image of * * , this extends the definition we have already given in that case. We leave it to * *the reader to check the functoriality of `. We now show that M is a sheaf. Suppose (R, y) is a point of Y0 and {(R, y) -! (Tm , ym )} is a cover of R. Let C(y) = {(R, y) -! (Si, yi)} be the given cove* *r of R, so that each yi is in the essential image of . Then {Si -! Tm R Si} is a cover of Si, and each ymi is the essential image of since each yi is. Simila* *rly, {Tm -! Tm R Si} is a cover of Tm . Thus we get the commutative diagram below. Q Q My ----! m Mym ----! npMynp ?? ? ? y ?y ?y Q Q Q iMyi? ----! mi?Mymi ----! npiMynpi? ?y ?y ?y Q Q Q jkMyjk ----! mjk Mymjk ----! npjkMynpjk The subscripts m, n, and p all refer to the Tm , and the subscripts i, j and k * *all refer to the Si. So, for example, ynpiis the image of y in Y0(Tn R Tp R Si). The rig* *ht- hand horizontal arrows are all the differences of the two restriction maps. Th* *is means that the second and third rows express their left-hand entries as kernels, since M restricted to the essential image of is a sheaf. Similarly, the bott* *om vertical arrows are also differences of the two restriction maps. It follows th* *at each column expresses its top entry as a kernel, since the definition of M does not * *depend on which cover we choose, up to isomorphism. A diagram chase then shows that the top row expresses My as a kernel, which means that M is a sheaf. We now construct the isomorphism _ :dom *M -! codom*M. Suppose ff: y -! z is a morphism in Y1(R). Let {R -!Si} be the given cover of (R, y), so that ea* *ch yi is in the essential image of . It follows that zi is also in the essential * *image of for all i. Let ffi: yi-! zi denote the image of ff in Y1(Si), and similarly l* *et ffjk denote the image of ff in Y1(Sj R Sk). Then we have a commutative diagram Q Q My ----! Myi ----! Myjk Q_ff?? ??Q iy y _ffjk Q Q Mz ----! Mzi ----! Mzjk. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 17 Here the right-hand horizontal arrows are differences of restriction maps, as u* *sual. The top row is an equalizer by definition, and we have proved that the bottom r* *ow is also an equalizer diagram. Hence there is a unique map _ff:My -!Mz, necessarily an isomorphism, making the diagram commute. The facts that _ satisfies the cocycle condition and is a map of sheaves are the usual sheaf-theoretic diagram_ chases, and we leave them to the reader. |__| 4.Quasi-coherent sheaves The object of this section is to prove Theorem C, showing that if : (X0, X1)* * -! (Y0, Y1) is an internal equivalence of presheaves of groupoids in the flat topo* *logy, then *: Shqc(Y0,Y1)-!Shqc(X0,X1)is an equivalence of categories of quasi-coher* *ent sheaves. This theorem can be viewed as a manifestation of faithfully flat desce* *nt; we have seen already that *: ShT(Y0,Y1)-!ShT(X0,X1)is an equivalence of categorie* *s, and we use faithfully flat descent to conclude that quasi-coherent sheaves are * *a full subcategory of sheaves in the flat topology. Recall that a cover of R in the flat, or fpqc, topology is a finiteQcollectio* *n of maps {R -!Si} such that each Siis flat over R, and the product Siis faithfully flat over R. This also defines the flat topology on Aff*. We use faithfully flat descent in the form of the following well-known lemma. Lemma 4.1. Suppose {R -! Si} is a cover of R in the flat topology on Aff, and M is an R-module. Then the diagram Y Y M -! Si R M ' Sj R Sk R M i jk is an equalizer in the category of R-modules. Q Of course, the two maps in the equalizerQtake s m 2 Si M to (1 si m) 2 jiSj R Si R M and to si 1 m 2 ikSi R Sk R M. As usual, this lemma also works in the graded case, with the same proof. Q Proof.Let S = iSi. Since the product is finite, it suffices to show that M -! S R M ' S R S R M is an equalizer for all R-modules M. Since S is faithfully flat, it suffices to* * show that S R M -! S R S R M ' S R S R S R M is an equalizer for all M. But, before tensoring with M, this sequence is just * *the beginning of the bar resolution of S as an R-algebra; since the bar resolution * *is_ contractible, this diagram remains an equalizer after tensoring with M. * *|__| Lemma 4.1 leads immediately to the following proposition, which is also true * *in the graded case. Proposition 4.2.Suppose M is a quasi-coherent sheaf over a presheaf of groupoids (X0, X1) on Aff. Then M is a sheaf in the flat topology. Proof.Suppose (R, y) is a point of X0, and {(R, y) -!(Si, yi)} is a cover in th* *e flat topology. We must show that the diagram Y Y Ey = (My -! Myi' Myjk) 18 MARK HOVEY is an equalizer diagram. But, since M is quasi-coherent, Ey is isomorphic to the diagram Y Y My -! Si R My ' Sj R Sk R My, which is an equalizer diagram by Lemma 4.1. |___| We will also need a lemma about purity of equalizer diagrams. Definition 4.3.Suppose E is an equalizer diagram of the form A -!B ' C in the category of R-modules for some commutative ring R. We say that E is pure if S R E is still an equalizer diagram for all commutative R-algebras S. One can also define purity using arbitrary R-modules S. We prefer this defini* *tion because it is the concept we need, but in fact the two definitions are equivale* *nt. Either definition also works in the graded case with the obvious changes. Lemma 4.4. Suppose E is an equalizer diagram of R-modules for some commuta- tive ring R. Suppose {Si} isLa set of flat commutative R-algebra such that Si R* * E is pure for all i and S = iSi is faithfully flat over R. Then E is pure. Proof.Suppose T is an arbitrary R-algebra. Then (T R Si) Si(Si R E) is an equalizer diagram since Si R E is pure, but (T R Si) Si(Si R E) ~=(T R Si) T (T R E). Thus (T R S) T (T R E) is also an equalizer diagram, being a direct sum of equalizer diagrams. Since T R S is faithfully flat over T , it follows that_T * * R E is an equalizer diagram. |__| We can now prove that quasi-coherent sheaves are homotopy invariant in the fl* *at topology. The following theorem is Theorem C of the introduction. Theorem 4.5. Suppose : (X0, X1) -!(Y0, Y1) is an internal equivalence of pre- sheaves of groupoids on AffT, where T is the flat topology. Then *: Shqc(Y0,Y1* *)-! Shqc(X0,X1)is an equivalence of categories. This theorem is also true in the graded case, with the same proof. Proof.Since *: ShT(Y0,Y1)-!ShT(X0,X1)is an equivalence of categories, and quas* *i- coherent sheaves are a full subcategory of sheaves in the flat topology by Prop* *osi- tion 4.2, we find immediately that *: Shqc(Y0,Y1)-!Shqc(X0,X1)is full and fait* *hful. It remains to show that it is essentially surjective. Suppose N is a quasi-coherent sheaf over (X0, X1). Because *: ShTY0,Y1-! ShT(X0,X1)is an equivalence of categories, there is a sheaf M in the flat topol* *ogy, over (Y0, Y1), such that *M ~=N. We will show that M is in fact quasi-coherent, so * *that * is essentially surjective on quasi-coherent sheaves. To do so, we must show * *that, M (f) if (R, y) f-!(S, y0) is a map of points of Y0, then the adjoint S R My æ----!M* *y0 of the structure map of M is an isomorphism. First suppose that y is in the essential image of . Then there is an x 2 X0(* *R) and a map ff: x -!y. Let x0= f(x) 2 X0(S), so that f(ff) = X1(f)(ff): x0-! MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 19 z. Then we have the commutative diagram below. N(f) S R Nx -æ---! Nx0 ? ? ~=?y ?y~= M (f) S R M x -æ---! M x0 ? ? 1 _ff?y ?y_fff S R My ----! My0 æM (f) The top square of this diagram commutes because *M ~=N as sheaves, and the bottom square commutes because _ is a map of sheaves. The vertical maps are isomorphisms, and the top horizontal map is an isomorphism since N is quasi- coherent. Hence the bottom horizontal map is an isomorphism as well. In fact, if y is in the essential image of and {R -! Si} is a cover of R in* * the flat topology, we claim that the equalizer diagram Y Y (4.6) E = Ey = (My -! Myi' Myjk) is pure. Indeed, suppose S is an R-algebra, so we have f :(R, y) -! (S, y0). Th* *en {S -! S R Si} is a cover of S in the flat topology. It follows from what we ha* *ve just done (and the fact that covers in the flat topology are finite), that the * *diagram S R Ey is isomorphic to Ey0, and so is still an equalizer diagram. Now suppose y is an arbitrary point of Y0. Since the sheaf-theoretic essenti* *al image of is all of Y0, we can choose a cover {R -!Si} such that each yi is in* * the essential image of . There is an induced cover {S -! S R Si} of S, and maps fi: (Si, yi) -!(S R Si, y0i), so each y0iis also in the essential image of . * *We then get the commutative diagram below, which is a map from the diagram S R Ey to Ez. Q 1 d Q S R My ----! S R Myi ----! S R Myjk ? ? ? æf?y Q æ(fi)?y ?yQæ(fjk) Q Q Mz ----! Mzi ----! Mzjk. d Here the map d is the difference between the two restriction maps, so the bottom row expresses Mz as a kernel. We have already seen that the maps æ(fi) and æ(fjk) are isomorphisms, so if we knew that S R Ey were an equalizer diagram, we would be able to conclude that æ(f) is an isomorphism, and therefore that M is quasi-coherent. In particular, if S is flat over R, we conclude that the diagram S R Ey is isomorphic to the equalizer diagram Ey0. In case y0 is in the essential image o* *f , we have proved thatQEy0is pure. In particular, Si R E is a pure equalizer diagr* *am for all i. Since Si is faithfully flat over R, it follows from Lemma 4.4 that* * the equalizer diagram E is pure. Thus, for any S, S R E is an equalizer diagram,_a* *nd so M is quasi-coherent. |__| 20 MARK HOVEY 5. Hopf algebroids In this section, we prove Theorem D of the introduction, characterizing those maps of Hopf algebroids which induce internal equivalences in the flat topology* * of the corresponding presheaves of groupoids. Suppose f = (f0, f1): (A, ) -!(B, ) is a map of Hopf algebroids. See [Rav86* * , Definition A1.1.7] for an explicit definition of this, though of course f is eq* *uivalent to a map = f* :(SpecB, Spec ) -! (SpecA, Spec ) of sheaves of groupoids on Aff. A map of Hopf algebroids induces a map B A A B -jL-f1-jR----! jLf0 A f1jL f1jR A jRf0 ~-! , where ~ denotes multiplication. Note that ~ makes sense since f1jL = jLf0 and f1jR = jR f0. By abuse of notation, we denote this map simply by jL f1 jR . Our goal is to characterize those f for which f* is a weak equivalence. We be* *gin by determining when f* is faithful. Proposition 5.1.Suppose f = (f0, f1): (A, ) -! (B, ) is a map of Hopf al- gebroids. Then f* :(SpecB, Spec ) -! (SpecA, Spec ) is faithful if and only if jL f1 jR :B A A B -! is an epimorphism in Rings. Recall that an epimorphism in Rings need not be surjective; the map from the integers to the rational numbers is a ring epimorphism. Also note that the obvi* *ous generalization of this proposition holds for graded Hopf algebroids. Proof.Given ff, fi : -!R, ff O (jL f1 jR ) = fi O (jL f1 jR ) if and only if ff and fi have the same domain and codomain when thought of as__ morphisms of (SpecB, Spec )(R) and f*ff = f*fi. The proposition follows. |__| We now determine when f* is full. Proposition 5.2.Suppose f = (f0, f1): (A, ) -! (B, ) is a map of Hopf al- gebroids. Then f* :(SpecB, Spec ) -! (SpecA, Spec ) is full if and only if jL f1 jR :B A A B -! is a split monomorphism of rings. Once again, the obvious generalization of this proposition is true in the gra* *ded case. Proof.The map f* is full if and only if every morphism fi :f*x -!f*y 2 (SpecA, Spec )(R) is equal to f*ff for some morphism ff: x -!y of (SpecB, Spec )(R). Said another way, f* is full if and only if every ring homomorphism x fi y :B A A B -! R can be extended through jL f1 jR to a ring homomorphism -! R. This is __ equivalent to jL f1 jR being a split monomorphism. |__| Corollary 5.3.Suppose f = (f0, f1): (A, ) -! (B, ) is a map of Hopf alge- broids. Then f* :(SpecB, Spec ) -! (SpecA, Spec ) is fully faithful if and only if jL f1 jR :B A A B -! is an isomorphism. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 21 Proof.Any map g :R -! S of rings that is both a split monomorphism and a ring epimorphism is an isomorphism. Indeed, Rings (g, T ): Rings (S, T ) -! Rings(R, T ) is monic since g is a ring epimorphism and epic since g is a split_ monomorphism, so is an isomorphism for all T . |__| Finally, we need to determine the sheaf-theoretic essential image of f* is al* *l of SpecA. For this we need the map f0 jR :A -!B A defined as the composite A ~=A A A f0-jR---!B A . Proposition 5.4.Suppose f = (f0, f1): (A, ) -! (B, ) is a map of Hopf alge- broids. Then the sheaf-theoretic essential image of f* :(SpecB, Spec ) -!(SpecA, Spec ) is all of SpecA if and only if there is a ring map g :B A -! C such that g(f0 jR ) exhibits C as a faithfully flat extension of A. This proposition is also true in the graded case, with the same proof. Proof.We first determine when y :A -!R is in the essential image of f*. For this to happen we need an object x: B -! R and a morphism ff: -! R from f*x to y. A morphism ff from f*x to anywhere is equivalent to the composite B A x-ff--!Rxf0 A ffjLR ~-!R, which we also denote, by abuse of notation, by x ff. The codomain of ff is the composite (x ff)(f0 jR ): A -! R. Altogether then, y is in the essential im* *age of f* if and only if there is a map h: B A such that h(f0 jR ) = y. Now, suppose the sheaf-theoretic essential image of f* is all of SpecA. Then there must be a cover {A -hi!Si} such that the image of the identity map of A, namely hi, is in the essential image of f* for all i. By the preceding paragrap* *h, this is true if and only if there exist maps gi: B A -!Sisuch that gi(f0 jR ) = * *hi. Let C be the product of the Si and let g :B A -! C be the product of the gi. Then g(f0 jR ) is the product of the hi, which displays C as a faithfully* * flat extension of A since {A hi-!Si} is a cover of A. Conversely, suppose there is a ring map g :B A -!C such that h = g(f0 jR ) exhibits C as a faithfully flat extension of A. Suppose y :A -! R is an arbitra* *ry point of (SpecA, Spec )(R). Then R ~=A A R h-1-!C A R is a cover of R. One can easily check that the image of y in (SpecA, Spec )(C A* * R) is the composite A h-!C ~=C A A 1-y-!C A R. Since h = g(f0 jR ), the image of y is in the essential image of f*, and so_y* *_is in the sheaf-theoretic essential image of f*. |__| Note that the proof of Proposition 5.4 can be easily modified to prove the kn* *own result that f* is essentially surjective if and only if f0 jR :A -!B A is a* * split monomorphism. Altogether then, we have the following theorem, which is Theorem D of the introduction. 22 MARK HOVEY Theorem 5.5. Suppose f = (f0, f1): (A, ) -!(B, ) is a map of Hopf algebroids. Then f* :(SpecB, Spec ) -!(SpecA, Spec ) is an internal equivalence in the flat topology if and only if jL f1 jR :B A A B -! is an isomorphism and there is a ring map g :B A -! C such that g(f0 jR ) exhibits C as a faithfully flat extension of A. This characterization of internal equivalences shows in particular that is * *deter- mined by (A, ) and f0. In fact, if (A, ) is any Hopf algebroid, and f :A -!B * *is a ring homomorphism, there is a unique (up to isomorphism) Hopf algebroid (B, f) and map of Hopf algebroids (f, f1) such that the map jL f1 jR is an isomor- phism. To show existence, we take f = B A A B and define the structure maps as follows: jL: B ~=B A A A A 1-jL-f---!B A A B; jR: B ~=A A A A B -f-jR-1--!B A A B; ffl:B A A B -1-ffl-1-!B A A A B ~=B A B -~!B; c: B A A B -1-c-1-!B A jR jL A B -ø!B A jL jR A B; : B A A B -1---1-!B A A A B ~=B A A A A A B -1-1-f-1-1---!B ~ A A B A A B = (B A A B) B (B A A B). We leave it to the reader to check that this does define a Hopf algebroid. We d* *efine f1: -! f to be the composite ~=A A A A f-1-f--!B A A B. We leave it to the reader to check that this defines a map of Hopf algebroids, * *and also to check our uniqueness claims. We therefore have the following corollary. Corollary 5.6.Suppose f = (f0, f1): (A, ) -! (B, ) is a map of Hopf alge- broids. Then f* :(SpecB, Spec ) -! (SpecA, Spec ) is an internal equivalence in the flat topology if and only if (B, ) is isomorphic over (A, ) to (B, f0* *) and there is a ring map g :B A -!C such that g(f0 jR ) exhibits C as a faithful* *ly flat extension of A. The conditions in Corollary 5.6 have appeared before, in [HS99 , Theorem 3.3] and in [Hop95 ]. Of course, in the situation of Corollary 5.6, Theorem 4.5 giv* *es us an equivalence of categories between (A, )-comodules and (B, f)-comodules. This equivalence of categories takes an (A, )-comodule M to B A M. 6.Formal groups In this section, we apply Corollary 5.6 and the theory of formal group laws to prove Theorem E. We also recover the change of rings theorems of Miller- Ravenel [MR77 ] and Hovey-Sadofsky [HS99 ]. This section requires familiarity with formal group laws and how they are use* *d in algebraic topology. A good source for this material is [Rav86 ], especially App* *endix 2 for formal group laws and Chapter 4 for their use in algebraic topology. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 23 Fix a prime p for use throughout this section. Recall that (BP*, BP*BP ) is the universal Hopf algebroid for p-typical formal group laws. Here BP* = Z(p)[v1, v2, . .]., and BP*BP = BP*[t1, t2, . .].; see [Rav86 , Section 4.1]. T* *he fact that (BP*, BP*BP ) is universal means that a p-typical formal group law over a ring R is equivalent to a ring homomorphism BP* -! R, and a strict isomor- phism of p-typical formal group laws over R is equivalent to a ring homomorphism BP*BP -! R. In case R is graded, let us call a p-typical formal group law over R homogeneous if its classifying map BP* -!R preserves the grading. (An example of a non-homogeneous formal group law is the formal group law over Fp whose classifying map takes vi to 0 for i 6= n and vn to 1). Recall also the invariant ideal In = (p, v1, . .,.vn-1). The element vn is a * *primi- tive modulo In. This means that there is a Hopf algebroid (A, ) = (v-1nBP*=In, v-1nBP*BP=In). Definition 6.1.A p-typical formal group law over a ring R is said to have strict height n if its classifying map factors through v-1nBP*=In. Our application of Theorem 4.5 is then the following theorem, which is Theo- rem E of the introduction. Theorem 6.2. Fix a prime p and an integer n > 0. Let (A, ) denote the Hopf algebroid (v-1nBP*=In, v-1nBP*BP=In). Suppose B is a graded ring equipped with a homogeneous p-typical formal group law of strict height n, classified by f :A -* *!B. Then the functor that takes an (A, )-comodule M to B A M defines an equivalence of categories from graded (A, )-comodules to graded (B, f)-comodules. Proof.Let D = A Fp[vn,v-1n]B. Let x: A -! D denote the ring homomorphism defined by x(a) = a 1, and let y :B -! D denote the ring homomorphism defined by y(b) = 1 b. Then x and the composite yf induce two formal group laws F and G over D, both p-typical and of strict height n. Furthermore, x(vn) = yf(vn). A r* *esult of Lazard, as modified by Strickland [HS99 , Theorem 3.4], then implies that th* *ere is a faithfully flat graded ring extension h: D -! C and a strict isomorphism from* * h*G to h*F . This strict isomorphism is represented by a graded ring homomorphism ff: -! C. Let g :B -! C be the composite hy. Since the domain of ff is h*G, ffjL = gf :A -!C. This means that there is a well-defined map g ff: B A g-ff--!Cgf A ffjLC ~-!C. Furthermore, (g ff) O (f jR ) represents the codomain of ff, so is hx. We k* *now already that h is a faithfully flat ring extension, and we claim that x is also* * a faithfully flat ring extension. Indeed, since Fp[vn, v-1n] is a graded field, B* * is a free Fp[vn, v-1n]-module, and so x makes D into a free A-module. Corollary 5.6_and Theorem 4.5 complete the proof. |__| In particular, we can take B = E(m)*=In, where m n and E(m) is the Landweber exact Johnson-Wilson homology theory introduced in [JW75 ]. This leads to the following corollary. Corollary 6.3.Let p be a prime and m n > 0 be integers. Then the functor that takes M to E(m)* BP* M defines an equivalence of categories (v-1nBP*=In, v-1nBP*BP=In)-comodules -!(v-1nE(m)*=In, v-1nE(m)*E(m)=In)-comodules. 24 MARK HOVEY Using the method of [MR77 ], we then get the following change of rings theore* *m, which is Theorem F of the introduction. Theorem 6.4. Let p be a prime and m n > 0 be integers. Suppose M and N are BP*BP -comodules such that vn acts isomorphically on N. If either M is finitely presented, or if N = v-1nN0 where N0 is finitely presented and In-nilpotent, th* *en Ext**BP*BP(M, N) ~=Ext**E(m)*E(m)(E(m)* BP* M, E(m)* BP* N). Note that, when M = BP*, this is the Hovey-Sadofsky change of rings theo- rem [HS99 , Theorem 3.1]. When m = n and M = BP*, we get the Miller-Ravenel change of rings theorem [MR77 , Theorem 3.10]. Proof.By Lemma 3.11 of [MR77 ], N is the direct limit of comodules v-1nN0, where N0 is finitely presented and In-nilpotent. Since we are assuming either that M * *is finitely presented or that N = v-1nN0, in either case we may as well take N = v* *-1nN0. Then Lemma 3.12 of [MR77 ] reduces us to the case N = v-1nBP*=In. In this case, one can check using the cobar resolution (as in [MR77 , Proposition 1.3]) that * *we have canonical isomorphisms Ext**BP*BP(M, N) ~=Ext**v-1nBP*BP=In(v-1nM=In, N) and Ext**E(m)*E(m)(E(m)* BP* M, E(m)* BP* N) ~= Ext**v-1nE(m)*E(m)=In(E(m)* BP* v-1nM=In, E(m)* BP* N) Now Corollary 6.3 implies that Ext**v-1nBP*BP=In(v-1nM=In, N) ~= Ext**v-1nE(m)*E(m)=In(E(m)* BP* v-1nM=In, E(m)* BP* N). This completes the proof. |___| References [DG70]Michel Demazure and Pierre Gabriel, Groupes alg'ebriques. 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[JT91]Andr'e Joyal and Myles Tierney, Strong stacks and classifying spaces, Cat* *egory theory (Como, 1990), Springer, Berlin, 1991, pp. 213-236. [JW75]D. C. Johnson and W. S. Wilson, BP-operations and Morava's extraordinary * *K-theories, Math. Zeit. 144 (1975), 55-75. [Mor85]J. Morava, Noetherian localizations of categories of cobordism comodules* *, Ann. of Math. (2) 121 (1985), 1-39. MORITA THEORY FOR HOPF ALGEBROIDS AND PRESHEAVES OF GROUPOIDS 25 [MR77]H. R. Miller and D. C. Ravenel, Morava stabilizer algebras and the locali* *zation of Novikov's E2-term, Duke Math. J. 44 (1977), 433-447. [Rav86]D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, * *Academic Press, 1986. [Str99]Neil P. Strickland, Formal schemes and formal groups, Homotopy invariant* * algebraic structures (Baltimore, MD, 1998), Amer. Math. Soc., Providence, RI, 1999,* * pp. 263-352. Department of Mathematics, Wesleyan University, Middletown, CT 06459 E-mail address: hovey@member.ams.org