Comodule categories and the geometry of the stack of formal groups N. Naumann Abstract We generalise recent results of M. Hovey and N. Strickland on comodule cate* *gories for Landweber exact algebras using the formalism of algebraic stacks. 1. Introduction Extensions of comodules over flat Hopf algebroids play an important role in alg* *ebraic topology as the E2-term of the Adams-Novikov spectral sequence based on a sufficiently w* *ell-behaved ring theory. It is "well-known" that the category of comodules is equivalent to the * *category of quasi- coherent sheaves of modules on an algebraic stack associated to the flat Hopf a* *lgebroid. The purpose of this note is to make this precise and to demonstrate that the switch of pers* *pective from flat Hopf algebroids to algebraic stacks is not purely formal. To this end, we generalise* * recent results of M. Hovey and N. Strickland ([HS ]) using only a minimum of the theory of formal gr* *oups and general facts about algebraic stacks. We hope to make clear that all these results are * *rather immediate consequences of the simple geometric structure of the stack of formal groups. We review the individual sections in more detail. In section 2 we give the rela* *tion between flat Hopf algebroids and algebraic stacks, following essentially [P ]. In section 3 * *we collect a number of technical results on algebraic stacks. The analogues for flat Hopf algebroids o* *f some of these results are known. In section 4 we apply this theory to the stack XFG of formal groups * *over Z(p)and isolate the relation of Landweber exactness (as considered in [HS ]) with the geometry * *of XFG (theorem 18). We then deduce the equivalences of comodule categories and change of rings theo* *rems generalising those of [HS ]. Acknowledgements I would like to thank E. Pribble for making a copy of [P ] available to me. 2.Algebraic stacks and flat Hopf algebroids In this section we review parts of [P ] giving the relation between flat Hopf a* *lgebroids and their categories of comodules and a certain class of stacks and their catagories of q* *uasi-coherent sheaves of modules. _______________________________________________________________________________* *__ 2000 Mathematics Subject Classification 55N22 Keywords: Comodules, flat Hopf algebroids, algebraic stacks N. Naumann 2.1 The 2-category of flat Hopf algebroids We refer to [R1 ], Appendix A for the notion of (flat) Hopf algebroid. To give * *a Hopf algebroid (A, ) is equivalent to giving (X0 := Spec(A), X1 := Spec( )) as a groupoid in affine * *schemes [LM-B ], 2.4.3 and we will formulate most results involving Hopf algebroids this way. Recall that this means that X0 and X1 are affine schemes and we are given morph* *isms s, t : X1 -! X0 (source and target), ffl : X0 -! X1 (identity), ffi : X1 sx,X0,tX1-! X1 (co* *mposition) and i : X1 -! X1 (inverse) verifying suitable identities. The corresponding maps of* * rings are denoted jL, jR (left- and right unit), ffl (augmentation), (comultiplication) and c (* *antipode). The 2-category of flat Hopf algebroids H is defined as follows. Objects are Hop* *f algebroids (X0, X1) such that s and t are flat (and thus faithfully flat because they allow ffl as * *a right inverse). A 1- morphism of flat Hopf algebroids from (X0, X1) to (Y0, Y1) is a pair of morphis* *ms of affine schemes fi : Xi -! Yi (i = 0, 1) commuting with all the structure. The composition of 1* *-morphisms is component wise. Given two 1-morphisms (f0, f1), (g0, g1) : (X0, X1) -! (Y0, Y1)* *, a 2-morphism c : (f0, f1) -! (g0, g1) is a morphism of affine schemes c : X0 -! Y1 such that* * sc = f0, tc = g0 and the diagram (g1,cs) x X1 _______//Y1s,Y0,tY1 (ct,f1)|| ffi| fflffl| fflffl|| Y1sx,Y0,tY1_ffi//_Y1 commutes. For (f0, f1) = (g0, g1) the identity 2-morphism is given by c := fflf* *0. Given two 2- 0 morphisms (f0, f1)c__//_(g0,_g1)c//_(h0,th1)heir composition is defined as 0 (c0,c) x ffi c O c : X0 ____//_Y1s,Y0,tY1//_Y1. One checks that the above definitions make H a 2-category which is in fact clea* *r because (except for the flatness of s and t) they are merely a functorial way of stating the ax* *ioms of a groupoid, a functor and a natural transformation. For technical reasons we will sometimes c* *onsider Hopf alge- broids for which s and t are not flat. 2.2 The 2-category of rigidified algebraic stacks Let S be an affine scheme. We denote by AffSthe category of affine S-schemes (w* *ith some cardinality bound as to make it small). We generally drop S from the notation. We endow Aff* *with with the fpqc`topology, i.e. a cover of X 2 Affis a finite family of flat morphisms Xi-!* * X in Affsuch that Xi- ! X is faithfully flat. We denote by Affalso the site thus defined. We wi* *ll consider stacks over Affand all notations and conventions concerning stacks will be those of [L* *M-B ] except that we work with the fpqc rather than the 'etale topology, c.f. [LM-B ], x9. Definition 1. A stack X (over the site Aff) is algebraic if its diagonal X is * *representable and affine and there is an affine scheme X0 and a faithfully flat morphism P : X0 -* *! X. Any 1-morphism of algebraic stacks from an algebraic space to an algebraic stac* *k is representable and affine, see the proof of [LM-B ], 3.13. In particular, it makes sense to sa* *y that P is faithfully flat. By definition, every algebraic stack is quasi-compact, hence so is any 1-morphi* *sm between algebraic stacks ([LM-B ], 4.16, 4.17). One can check that finite limits and colimits of * *algebraic stacks (taken 2 Comodules and algebraic stacks in the 2-category of fpqc-stacks, [LM-B ] 3.3) are again algebraic stacks. A morphism P as in definition 1 is called a presentation of X. As far as we are* * aware, this definition of "algebraic" is due to P. Goerss [G ] and is certainly motivated by the equivale* *nce given in subsection 2.3 below. We point out that the notion of "algebraic stack" well-establish in * *algebraic geometry ([LM-B ],4.1) is different from the above, for example the stack associated to * *(BP*, BP*BP ) (c.f. section 4) is algebraic in the above sense but not in the sense of algebraic ge* *ometry because its diagonal is not of finite type, [LM-B ] 4.2. Of course, in the following we wil* *l use the term "algebraic stack" in the sense defined above. The 2-category S of rigidified algebraic stacks is defined as follows. Objects * *are presentations P : X0 -! X as in definition 1. A 1-morphism from P : X0 -! X to Q : Y0 -! Y is a p* *air consisting of f0 : X0 -! Y0 (a morphism in Aff) and f : X -! Y (a 1-morphism of stacks) su* *ch that the diagram f0 X0 ____//_Y0 P || Q|| fflffl|fflffl| X __f_//_Y is 2-commutative. Composition of 1-morphisms is component wise. Given 1-morphis* *ms (f0, f), (g0, g) : (X0 -! X) -! (Y0 -! Y) a 2-morphism in S from (f0, f) to (g0, g) is by definiti* *on a 2-morphism from f to g in the 2-category of stacks, [LM-B ], 3. 2.3 The equivalence of H and S We now establish an equivalence of 2-categories between H and S. We define a fu* *nctor K : S -! H as follows. K( X0 _P__//_X) := (X0, X1 := X0 P,xX,PX0) has a canonical structure of groupoid ([LM-B ], 3.8), X1 is affine because X0 i* *s affine and P is representable and affine and the projections s, t : X1 ____//_//_X0are flat bec* *ause P is; so (X0, X1) is Q a flat Hopf algebroid. If (f0, f) : (X0 _P__//_X)-! (Y0 ____//_Y)is a 1-morphis* *m in S we define * * Q K((f0, f)) := (f0, f0x f0). If we have 1-morphisms (f0, f), (g0, g) : (X0 _P__/* */_X)-! (Y0____//_Y) in S and a 2-morphism (f0, f) -! (g0, g) then we have by definition a 2-morphis* *m f____//_g: X -! Y. In particular, we have X0 : Ob(XX0) -! Mor (YX0) = Hom Aff(X0, Y1) and* * we define K( ) := X0(idX0). One checks that K : S -! H is a 2-functor. We define a 2-functor G : H -! S as follows. On objects we put G((X0, X1)) := (* *X0 can-! [ X1 ____//_//_X0]), the stack associated with the groupoid (X0, X1) together w* *ith its canonical pre- sentation ([LM-B ], 3.4.3; identify the Xi with the flat sheaves they represent* * to consider them as "S-espaces", see also subsection 3.1). This is a rigidified algebraic stack: Sa* *ying that the diagonal of X is representable and affine means that for any algebraic space X and morphism* *s x1, x2 : X -! X the sheaf Isom_X(x1, x2) on X is representable by an affine X-scheme. This prob* *lem is local in the fpqc topology on X because affine morphisms satisfy effective descent in the fp* *qc topology [SGA1 ], expos'e VIII, theorem 2.1. So we can assume that the xilift to X0 and the asser* *tion follows because (s, t) : X1 -! X0 xS X0 is affine. A similar argument shows that P : X0 -! X * *is (representable 3 N. Naumann and) faithfully flat because s and t are faithfully flat. Given a 1-morphism (f0, f1) : (X0, X1) -! (Y0, Y1) in H there is a unique 1-mor* *phism f : X -! Y making X1 ____//_//_X0P//_X_ ____ f1|| f0|| f______ fflffl|/fflffl|fflffl___/Q Y1 ____//__Y0_//_Y 2-commutative ([LM-B ], proof of 4.18) and we define G((f0, f1)) := f. Given a 2-morphism c : X0 -! Y1 from the 1-morphism (f0, f1) : (X0, X1) -! (Y0,* * Y1) to the 1-morphism (g0, g1) : (X0, X1) -! (Y0, Y1) in H we have a diagram X1 ____//_//_X0P//_X_________ _________________________________* *______________ f1||g1||f0||g0||f_g_____________________________* *__________________________________________ |fflfflfflffl|fflffl|fflffl|________________~~_* *_______________//Q Y1_____//_Y0__//_Y and need to construct a 2-morphism = G(c) : f -! g (in the 2-category of stac* *ks). We will do this in some detail because we omit numerous similar arguments. Fix U 2 Aff, x 2 Ob(XU) and a representation of x (c.f. [LM-B ], proof of 3.2) 0 0 0 00 0 0 oe (U -! U, x : U -! X0, U := U xU U - ! X1), i.e. U0 -! U is a cover in Aff, x0 2 X0(U0) = Hom Aff(U0, X0) and oe is a desce* *nt datum for x0 with respect to the cover U0 -! U. Hence, denoting by ss1, ss2 : U00-! U0 and s* *s : U0 -! U the projections we have oe : ss*1x0-~! ss*2x0in XU00, i.e. x0ss1 = soe and x0ss2 = * *toe. Furthermore, oe satisfies a cocycle condition which we do not spell out. We have to construct a morphism x : f(x) -! g(x) inYU which we do by descent from U0 as follows. We have a morphism 0 0 OE0 * 0 ss*(f(x)) = f(ss*(x) = x ) = f0x -! ss (g(x)) = g0x in YU0 given by OE0:= cx0: U0 -! Y1. We also have a diagram ss*1(OE0) 0 ss*1(ss*(f(x))) = f0x0ss1//_ss*1(ss*(g(x))) = g0x ss1 oef|| oeg|| fflffl| ss*02(OE0) fflffl| 0 ss*2(ss*(f(x))) = f0x_ss2//_ss*2(ss*(g(x))) = g0x ss2 in YU00where oef and oeg are descent isomorphisms for f(x0) and g(x0) given by * *oef = f1oe and oeg = g1oe. We check that this diagram commutes by computing in Mor (YU00): 0 0 * * (*) oegO ss*1(OE ) = ffiY(g1oe, cx ss1) = ffiY(g1oe, csoe) = ffiY(g* *1, cs)oe = 0 * * * 0 = ffiY(ct, f1)oe = ffiY(ctoe, f1oe) = ffiY(cx ss2, f1oe) = ss2(* *OE ) O oef. Here ffiY is the composition of (Y0, Y1) and in (*) we used the commutative squ* *are in the definition of 2-morphisms in H. So OE0is compatible with descent data and thus descents to the desired x : f(x* *) -! g(x). We omit 4 Comodules and algebraic stacks the verification that x is independent of the chosen representation of x and n* *atural in x and U. We now check that K and G are inverse equivalences. We have G O K(X0 -P!X) = ( X0 _can//_[X0xXX0__////_X0]) and there is a unique 1* *-isomorphism P : [X0xXX0 ____//_//_X0]-! X with p O can = P ([LM-B ], 3.8). One checks tha* *t this defines an isomorphism of 2-functors GK -'!idS. Next we have K O G(X0, X1) = (X0, X0 Px,X,PX0), where (X0 -P! X) = G(X0, X1), a* *nd X1 ' X0 Px,X,PX0([LM-B ], 3.4.3) and one checks that this defines an isomorphism of* * 2-functors idH-'! KG. The forgetful functor from rigidified algebraic stacks to algebraic stacks is n* *ot full but we have the following. Proposition 2. If (X0, X1) and (Y0, Y1) are flat Hopf algebroids with associate* *d rigidified algebraic stacks P : X0 -! X and Q : X0 -! Y and X and Y are 1-isomorphic as stacks then * *there is a chain of 1-morphisms of flat Hopf algebroids from (X0, X1) to (Y0, Y1) such tha* *t every morphism in this chain induces a 1-isomorphism on the associated algebraic stacks. This result implies theorem 6.5 of [HS ]: As we will see in section 4, the assu* *mptions of loc. cit. imply that the flat Hopf algebroids (B, B) and (B0, B0) considered there have the s* *ame open substack of the stack of formal groups as their associated stack. So they are connected * *by a chain of weak equivalences by proposition 2 (see also remark 6 for the notion of weak equival* *ence). Proof.Let f : X -! Y be a 1-isomorphism of stacks and form the cartesian diagram X01_f1_//Y1 || |||| || || fflffl|fflffl|fflffl|fflffl|0 X0__f0_//Y0 | |Q P0| | fflffl|fflffl| X __f__//Y. To be precise, the upper square is cartesian for either both source or both tar* *get morphisms. Then (f0, f1) is a 1-isomorphism of flat Hopf algebroids. Next, Z := X000x X0 is * *an affine scheme P ,X,P because X00is and P is representable and affine. The obvious 1-morphism Z -! X * *is representable, affine and faithfully flat because P and P 0are. Writing W := Z xX Z ' X01 x* *X X1 we have that X ' [ W ____//_//_Z] by (the flat version of) [LM-B ], 4.3.2. There are ob* *vious 1-morphisms of flat Hopf algebroids (Z, W ) -! (X00, X01) and (Z, W ) -! (X0, X1) covering idX(in p* *articular inducing an isomorphism on stacks) and we get the sought for chain as (Y0, Y1) - (X00, * *X01)_- (Z, W ) -! (X0, X1). * * |__| 5 N. Naumann 2.4 Comodules and quasi-coherent sheaves of modules For basic results concerning the category Mod qcoh(OX) of quasi-coherent sheave* *s of modules on an algebraic stack X we refer the reader to [LM-B ], 13. Fix a rigidified algebraic stack X0 -P!X corresponding as in subsection 2.3 to * *the flat Hopf algebroid (X0 = Spec(A), X1 = Spec( )) with structure morphisms s and t. As P is affine i* *t is in particular quasi-compact, hence fpqc, and thus of effective cohomological descent for quas* *i-coherent modules, [LM-B ],13.5.5,i). In particular, P *induces an equivalence P *: Modqcoh(OX) -'!{F 2 Modqcoh(OX0) + descent data}, c.f. [BLR ], Chapter 6 for similar examples of descent. A descent datum on F 2 * *Mod qcoh(OX0) is an isomorphism ff : s*F -! t*F in Mod qcoh(OX1) satisfying a cocycle conditi* *on. Giving ff is equivalent to giving either its adjoint _l: F -! s*t*F or the adjoint of ff-1, * *_r : F -! t*s*F . Writing M for the A-module corresponding to F , ff corresponds to an isomorphis* *m j M -! * * L,A jR,AM of -modules and _r and _l correspond respectively to morphisms M -! * * A M and M -! M A of A-modules. One checks that this is a 1-1 correspondence betw* *een descent data on F and left- (respectively right-) -comodule structures on M. For exampl* *e, the cocycle condition of ff corresponds to the coassociativity of the coaction. In the foll* *owing we will work with left- -comodules only and call them simply -comodules. The above construc* *tion provides an identification of the category of -comodules and Mod qcoh(OX) which can also b* *e proved using the Baer-Beck theorem, [P ], 3.22. The identification of Mod qcoh(OX) with -comodules allows to (re)understand a * *number of results on -comodules from the "geometric" point of view and we now give a short list * *of such applications which we will use later. The adjunction (P *, P*) : Modqcoh(OX) -! Modqcoh(OX0) corresponds to the forge* *tful functor from -comodules to A-modules, respectively to the functor "induced/extended comodul* *e". The structure sheaf OX corresponds to the trivial -comodule A, hence taking the primitives o* *f a -comodule (i.e. the functor Hom (A, .) from -comodules to abelian groups) corresponds t* *o Hom OX(OX, .) = H0(X, .) and Ext*(A, .) corresponds to quasi-coherent cohomology H*(X, .). By [LM-B ], 14.2.7 there is a 1-1 correspondence between closed substacks Z X* * and quasi-coherent ideal sheaves I OX such that OZ ' OX=I and these I correspond to -subcomodul* *es I A, i.e. invariant ideals. In this situation, the diagram Spec( =I ) ____//_Spec( ) || || || || fflffl|fflffl|fflffl|fflffl| Spec(A=I)_____//Spec(A) | | | | fflffl| fflffl| Z___________//_X is cartesian. Note that the Hopf algebroid (A=I, =I ) is induced from (A, ) b* *y the map A -! A=I because A=I A A A=I ' =(jLI + jRI) = =I because I is invariant. i If U ,! X is a quasi-compact open immersion of stacks then the stack U is algeb* *raic as one easily checks. In general, an open substack of an algebraic stack need not be algebrai* *c (c.f. the introduction of section 4). We conclude this subsection by giving a finiteness result for quasi-coherent sh* *eaves of modules. Let 6 Comodules and algebraic stacks X be an algebraic stack. We say that F 2 Modqcoh(OX) if finitely generated if t* *here is a presentation P : X0 = Spec(A) -! X such that the A-module corresponding to P *F is finitely * *generated. If F is finitely generated then for any presentation P : X00= Spec(A0) -! X the A0-modu* *le corresponding to P 0*F is finitely generated as one sees using [Bou ], I, x3, proposition 11. Proposition 3. Let (A, ) be a flat Hopf algebroid, M a -comodule and M0 M a* * finitely generated A-submodule. Then M0 is contained in a -subcomodule of M which is fi* *nitely generated as an A-module. * * __ Proof.[W ], proposition 5.7. * * |__| Note that in this result, "finitely generated" cannot be strengthened to "coher* *ent" as is shown by the example of the simple BP*-comodule BP*=(v0, . .).which is not coherent as a* * BP*-module. Proposition 4. Let X be an algebraic stack. Then any F 2 Modqcoh(OX) is the fil* *tering union of its finitely generated quasi-coherent subsheaves. * * __ Proof.Choose a presentation of X and apply proposition 3 to the resulting flat * *Hopf algebroid. |__| Compare to [LM-B ], 15.4. 3. Properties of morphisms In this section we relate properties of 1-morphisms (f0, f1) of flat Hopf algeb* *roids to properties of the induced morphism f : X -! Y of algebraic stacks and the adjoint pair (f*, f*) :* * Mod qcoh(OX) -! Mod qcoh(OY ) of functors. 3.1 The epi/monic factorisation By a flat sheaf we will mean a set valued sheaf on the site Aff. The topology o* *f Affis subcanonical, i.e. every representable presheaf is a sheaf. We can thus identify the category* * underlying Affwith a full subcategory of the category of flat sheaves. Every 1-morphism f : X -! Y of stacks factors canonically X -! X0- ! Y into an * *epimorphism followed by a monomorphism, [LM-B ], 3.7. The stack X0is determined up to uniqu* *e 1-isomorphism and is called the image of f. For a 1-morphism (f0, f1) : (X0, X1) -! (Y0, Y1) of flat Hopf algebroids we den* *ote (1) ff := tss2 : X0 fx Y1 -! Y0 and 0,Y0,s fi := (s, f1, t) : X1 -! X0 fx Y1 x X0 . 0,Y0,st,Y0,f0 The 1-morphism f : X -! Y induced by (f0, f1) on algebraic stacks is an epimorp* *hism if and only if ff is an epimorphism of flat sheaves; this is clear from the definition of e* *pimorphism of stacks, [LM-B ], 3.6; f is a monomorphism if and only if fi is an isomorphism, as is ea* *sily checked. 7 N. Naumann 0 Writing X1 := X0 fx Y1 x X0 , (f0, f1) factors as 0,Y0,st,Y0,f0 f01:=fi 0ss2 X1 ____//_X1__//_Y1 || || || || ss1|ss3| || 0 fflffl|fflffl|fflffl|fflffl|fflffl|fflffl|f0:=* *idX0f0 X0 ____//_X0__//_Y0 and the factorisation of f induced by this is the epi/monic factorisation. Note* * that even when (X0, X1) and (Y0, Y1) are flat Hopf algebroids, (X0, X01) does not have to be f* *lat. 3.2 Flatness and isomorphisms The proof of the next result will be given at the end of this subsection. The e* *quivalence of ii) and iii) is equivalent to theorem 6.2 of [HS ] but we will obtain refinements of it* * below (proposition 11 and proposition 12). Theorem 5. Let (f0, f1) : (X0, X1) -! (Y0, Y1) be a 1-morphism of flat Hopf alg* *ebroids with associated morphisms ff and fi as in (1) and inducing f : X -! Y on algebraic s* *tacks. Then the following are equivalent: i) f is a 1-isomorphism of stacks. ii) f* : Modqcoh(OX) -! Modqcoh(OY ) is an equivalence. iii) ff is faithfully flat and fi is an isomorphism. Remark 6. This result shows that the weak equivalences of [H ], 1.1.4 are exact* *ly those 1-morphisms of flat Hopf algebroids which induce 1-isomorphisms on the associated algebraic* * stacks. It is possible for Modqcoh(OX) and Modqcoh(OY ) to be equivalent without X and * *Y being isomorphic which answers conjecture 6.3 of [HS ] to the negative. In [R ], 1.4. one finds * *two non-isomorphic finite groups G1 and G2 with identical character table. This implies that their catego* *ries of representa- tions over C are equivalent as abelian categories. The constant groups schemes * *over C defined by G1 and G2 define algebraic stacks X1 and X2 (their classifying stacks, [LM-B ] * *2.4.2). The category of representations over C of Giis equivalent to Mod qcoh(OXi), hence one gets t* *he desired example because X1 6' X2 by Tannaka theory ([D ]). Even though this invalidates the above mentioned conjecture as it is stated, it* * leaves room for some speculation: The example of [R ] was meant to illustrate that in Tannakian theo* *ry one cannot ignore the additional structure on the representation categories furnished by the tens* *or product. Taking this structure into account, Tannakian theory may be subsumed by saying that th* *ere is an equiva- lence of 2-categories between a category of (rather special) algebraic stacks (* *namely gerbes bound by affine group schemes over fields of characteristic zero) and the category of* * Tannakian categories. Even though this has been generalised lately ([W ]) there is no such result for* * algebraic stacks as general as those considered here. It is still conceivable that partial results * *may be interesting: Using the notation from section 4, the classification of hereditary torsion the* *ories of BP*-comodules ([HS ], theorem B) amounts to saying that the hereditary torsion theories insid* *e Mod qcoh(OXFG) are exactly given as ker(j*n) for the open immersions jn : Un ,! XFG (0 6 n < 1). I* *t is easy to see that the Un exhaust all quasi-compact open substacks of XFG and a suitable "Tan* *nakian" corre- spondence (between quasi-compact open substacks and hereditary torsion theories* *) would allow to recover [HS ], theorem B from this simple geometric fact. See [Lu], theorem 5.1* *1 for a result in this direction. 8 Comodules and algebraic stacks We next give two results about the flatness of morphisms. Proposition 7. Let (f0, f1) : (X0, X1) -! (Y0, Y1) be a 1-morphism of flat Hopf* * algebroids, P : X0 -! X and Q : Y0 -! Y the associated rigidified algebraic stacks and f : * *X -! Y the induced 1-morphism of algebraic stacks. Then the following are equivalent: i) f is (faithfully) flat. ii) f* : Modqcoh(OY ) -! Modqcoh(OX) is exact (and faithful). iii) ff := tss2 : X0 fx Y1 -! Y0 is (faithfully) flat. 0,Y0,s iv) the composition X0 -P!X -f!Y is (faithfully) flat. Proof.The equivalence of i) and ii) is by definition, the one of i) and iv) is * *because P is fpqc and being (faithfully) flat is a local property for the fpqc topology. Abbreviating* * Z := X0 f x Y1 we * * 0,Y0,s have a cartesian diagram Z _____ff____//Y077________ ______ ss1|| _f0______|Q|_____ |fflfflP______fflffl|____f X0 ____//_X___//_Y which, as Q is fpqc, shows that iv) and iii) are equivalent. We check that this* * diagram is in fact cartesian by computing: X0 fxP,Y,QY0= X0Qfx0,Y,QY0' ' X0 fx Y0 x Y0 ' X0 x Y1 = Z, 0,Y0,idQ,Y,Q f0,Y0,s * * __ and under this isomorphism the projection onto the second factor corresponds to* * ff. |__| Proposition 8. Let (Y0, Y1) be a flat Hopf algebroid, f0 : X0 -! Y0 a morphism * *in Aff and (f0, f1) : (X0, X1 := X0 f x Y1 x X0 ) -! (Y0, Y1) the 1-morphism of Hopf al* *gebroids from 0,Y0,st,Y0,f0 the induced Hopf algebroid and Q : Y0 -! Y the rigidified algebraic stack assoc* *iated to (Y0, Y1). Then the following are equivalent: i) the composition X0 f0-!Y0 -Q!Y is (faithfully) flat. ii) ff := tss2 : X0 fx Y1 -! Y0 is (faithfully) flat. 0,Y0,s If either of this maps is flat, then (X0, X1) is a flat Hopf algebroid. The last assertion of this proposition does not admit a converse: For (Y0, Y1) * *= (Spec(BP*), Spec(BP*BP )) and X0 := Spec(BP*=In) -! Y0, the induced Hopf algebroid is flat but X0 -! Y is* * not (c.f. section 4). Proof.The proof of the equivalence of i) and ii) is the same as in proposition * *7, using that Q is fpqc. Again denoting Z := X0 fx Y1 one checks that the diagram 0,Y0,s ZO__ff_//Y0OOO | | | |f0 | t | X1 ____//_X0 is cartesian, hence the final assertion of the proposition follows because flat* *ness_is stable under base change. * * |__| 9 N. Naumann We will need the next result in section 4. Proposition 9. Let (Y0, Y1) be a flat Hopf algebroid, f0 : X0 -! Y0 a morphism * *in Affsuch that the composition X0 f0-!Y0 -Q!Y is faithfully flat, where Q : Y0 -! Y is the rig* *idified algebraic stack associated to (Y0, Y1). Let (f0, f1) : (X0, X1) -! (Y0, Y1) be the canoni* *cal 1-morphism with (X0, X1) the Hopf algebroid induced from (Y0, Y1) by f0. Then (X0, X1) is a fla* *t Hopf algebroid and (f0, f1) induces a 1-isomorphism on the associated algebraic stacks. Proof.The 1-morphism f induced on the associated algebraic stacks is a monomorp* *hism by con- struction. Proposition 8 shows that (X0, X1) is a flat Hopf algebroid and that * *f is an epimorphism,_ hence a 1-isomorphism by [LM-B ], 3.7.1. * * |__| We now start to take the module categories into consideration. Given f : X -! Y in Affwe have an adjunction _f : idModqcoh(OY)-! f*f*. We recognise the epimorphisms of representable flat sheaves as follows. Proposition 10. Let f : X -! Y be a morphism in Aff. Then the following are equ* *ivalent: i) f is an epimorphism of flat sheaves. ii) There is some OE : Z -! X in Affsuch that fOE is faithfully flat. If i) and ii) hold, then _f is injective. If f is flat, the conditions i) and ii) are equivalent to f being faithfully fl* *at. For an example of such an f which is not flat, c.f. [Bou ], I, x3, ex. 5. Proof.That i) implies ii) is seen by lifting idY2 Y (Y ) after a suitable faith* *fully flat cover Z -! Y to some OE 2 X(Z). To see that ii) implies i), fix some U 2 Affand u 2 Y (U) and form the cartesia* *n diagram OE f ZO_____//X____//YOOO |v| u|| | | W __________//_U. Then W -! U is faithfully flat and u lifts to v 2 Z(W ) and hence to OEv 2 X(W * *). To see the assertion about flat f, note first that a faithfully flat map is tri* *vially an epimorphism of flat sheaves. Secondly, if f is flat and an epimorphism of flat sheaves, then there * *is some OE : Z -! X as in ii) and the composition fOE is surjective (on the topological spaces unde* *rlying these affine schemes), hence so is f, i.e. f is faithfully flat. The injectivity of _f is a * *special_case of [Bou ], I, x3, proposition 8 i). * * |__| We have a similar result for epimorphisms of algebraic stacks. Proposition 11. Let (f0, f1) : (X0, X1) -! (Y0, Y1) be a 1-morphism of flat Hop* *f algebroids inducing f : X -! Y on associated algebraic stacks and write ff := tss2 : X0 fx* * Y1 -! Y0. Then 0* *,Y0,s the following are equivalent: i) f is an epimorphism. ii) ff is an epimorphism of flat sheaves. iii) There is some OE : Z -! X0 fx Y1 in Affsuch that ffOE is faithfully flat. 0,Y0,s If these conditions hold then idModqcoh(OY)-! f*f* is injective. 10 Comodules and algebraic stacks Proof.The equivalence of i) and ii) is "mise pour memoire", the one of ii) and * *iii) has been proved in proposition 10. Assume that these conditions hold and let g : X0 -! X be any* * morphism of algebraic stacks. Assume that idModqcoh(OY),! (fg)*(fg)*. Then we have that the* * composition idModqcoh(OY)-! f*f* -! f*g*g*f* = (fg)*(fg)* is injective and hence so is idMo* *dqcoh(OY)-! f*f*. Taking g := P : X0 -! X the canonical presentation we see that we can ass* *ume that X = X0, in particular f : X0 -! Y is representable and affine (and an epimorphism). Now* * let Q : Y0 -! Y be the canonical presentation and form the cartesian diagram g0 (2) Z0 ____//_Y0 P|| Q|| fflffl|ffflffl| X0 ____//_Y. As Q is fpqc we have idModqcoh(OY),! f*f* if and only if Q* ,! Q*f*f* ' g0,*P ** *f* ' g0,*g*0Q* (we used flat base change, [LM-B ] 13.1.9) and this will follow from idModqcoh(OY0)* *,! g0,*g*0because Q is flat. As f is representable and affine, Z0 is an affine scheme hence, by proposition * *10, we are done_because g0 is an epimorphism of flat sheaves, [LM-B ], 3.8.1. * * |__| There is an analogous result for monomorphisms of algebraic stacks. Proposition 12. Let (f0, f1) : (X0, X1) -! (Y0, Y1) be a 1-morphism of flat Hop* *f algebroids, P : X0 -! X the rigidified algebraic stack associated to (X0, X1), f : X -! Y t* *he associated 1- morphism of algebraic stacks, : f*f* -! idModqcoh(OX)the adjunction and fi = * *(s, f1, t) : X1 -! X0f0x,Y0,sY1xt,Y0,f0X0. Then the following are equivalent: i) f is a monomorphism. ii) fi is an isomorphism. iii) P*OX0 is an isomorphism. If f is representable then these conditions are equivalent to: iiia) is an isomorphism. iiib) f* is fully faithful. Remark 13. This result may be compared to the first assertion of theorem 2.5 of* * [HS ]. There it is proved that is an isomorphism if f is a flat monomorphism. We will determine the essential image of f* below. I do not know whether every monomorphism of algebraic stacks is representable, * *c.f. [LM-B ], 8.1.3. Proof.We already know that i) and ii) are equivalent. Consider the diagram f X0______P_//_X______//Y ________________________________________OOOOOO| ______________________________________|ss|| * * 0 0___________________________f_____________________* *______ss1|1|f| _~~______________~~______|x|| ss : Z_0_//_Xf,Y,fXss//_X P 2 in which the squares made of straight arrows are cartesian. As fP is representa* *ble and affine, we have fP = Spec_(f*P*OX0) (c.f. [LM-B ], 14.2) and ss = Spec_(f*f*P*OX0). We kno* *w that i) is equivalent to the diagonal of f, f, being an isomorphism [LM-B ], 2.3.1. As f* * is a section of ss1 this is equivalent to ss1 being an isomorphism. As P is an epimorphism, this is* * equivalent to ss01 being an isomorphism by [LM-B ], 3.8.1. Of course, ss01admits 0 := (idX0, fP * *) as a section so, 11 N. Naumann 0 0 finally, i) is equivalent to being an isomorphism. One checks that = Spec_(* * P*OX0) and this proves the equivalence of i) and iii). Now assume that f is representable and a monomorphism. We will show that iiia) * *holds. Consider the cartesian diagram f0 Z ____//_Y0 P || Q|| fflffl|fflffl|f X ____//_Y. We have 0* * 0* 0 * P *f*f* ' f Q f* ' f f*P . As P *reflects isomorphism, iiia) will hold if the adjunction f0*f0*-! idModqco* *h(OZ)is an isomor- phism. As f is representable, this can be checked at the stalks of z 2 Z, and w* *e can replace f0 by the induced morphism Spec(OZ,z) -! Spec(OY0,y) (y := f0(z)) which is a monom* *orphism. In particular, we have reduced the proof of iiia) to the case of affine schemes* *, i.e. the following assertion: If OE : A -! B is a ring homomorphism such that Spec(OE) is a monomo* *rphism (i.e. the ring homomorphism corresponding to the diagonal B A B -! B, b1 b2 7! b1b2 is* * an isomor- phism) then, for any B-module M, the canonical homomorphism of B-modules M A B* * -! M is an isomorphism. This is however easy: M A B ' (M B B) A B ' M B (B A B) ' M B B ' M, and we leave it to the reader to check that the composition of these isomorphis* *ms is the natural map M A B -! M. Finally, the proof that iiia) and iiib) are equivalent is a formal manipulation* * with adjunctions_which we leave to the reader, and trivially iiia) implies iii). * * |__| We promised to identify the essential image of f*. Proposition 14. In the situation of proposition 12 assume that f is representab* *le and a monomor- phism, let Q : Y0 -! Y be the rigidified algebraic stack associated to (Y0, Y1)* * and form the cartesian diagram g0 (3) Z0____//_Y0 P || Q|| fflffl|fflffl|f X ____//_Y. Then Z0 is an algebraic space and a given F 2 Mod qcoh(OY ) is in the essential* * image of f* if and only if Q*F is in the essential image of g0,*. Consequently, f* induces an * *equivalence between Mod qcoh(OX) and the full subcategory of Mod qcoh(OY ) consisting of such F. Proof.Z0 is an algebraic space because f is representable. We know that f* is f* *ully faithful by proposition 12, iiib) and need to show that the above description of its essent* *ial image is correct. If F ' f*G then Q*F ' Q*f*G ' g0,*P *G so Q*F lies in the essential image of g0* *,*. To see the 12 Comodules and algebraic stacks converse, extend (3) to a cartesian diagram g1 Z1____//_Y1 || || || || fflffl|fflffl|fflffl|fflffl|g0 Z0____//_Y0 P || Q|| fflffl|fflffl|f X ____//_Y. Note that X ' [ Z1____//_//_Z0], hence (Z0, Z1) is a flat groupoid (in algebrai* *c spaces) representing X. Now let there be given F 2 Mod qcoh(OY ) and G 2 Mod qcoh(OZ0) with Q*F ' g0,*G* *. We define oe to make the following diagram commutative: s*Q*F _can~//_t*Q*F ~|| ~ || fflffl| fflffl| s*g0,*G t*g0,*G ~|| ~ || fflffl| fflffl| g1,*s*G_~oe//_g1,*t*G. As f is representable and a monomorphism, so is g1 and thus g*1g1,*-~!idModqcoh* *(OZ1)and g1,*is fully faithful by proposition 12,iiia), iiib). We define o to make the followin* *g diagram commutative: g*1(oe)* * g*1g1,*s*G~_//_g1g1,*t G ~ || ~ || fflffl|o fflffl| s*G ________//_t*G. Then o satisfies the cocycle condition because it does so after applying the fa* *ithful functor g1,*. So o is a descent datum on G, and G descents to G 2 Mod qcoh(OX) with P *G ' G * *and we have Q*f*G ' g0,*P *G ' Q*F, hence f*G ' F, i.e. F lies in the essential image of f** * as_was to be shown. * * |__| To conclude this subsection we give the proof of theorem 5 the notations and as* *sumptions of which we now resume. Proof.If iii) holds then f is an epimorphism and a monomorphism (by proposition* * 11, iii) ) i) and proposition 12, ii) ) i)) hence i) holds by [LM-B ], 3.7.1. The proof that * *i) implies ii) is left to the reader and we assume that ii) holds. Since (f*, f*) is an adjoint pair o* *f functors, f* is a quasi-inverse for f* and : f*f* -! idModqcoh(OX)is an isomorphism so fi is an* * isomorphism by 12, iii) ) ii). As f* is in particular exact and faithful, ff is faithfully flat by* * proposition_7, ii) ) iii) and iii) holds. * * |__| 13 N. Naumann 4.Landweber exactness and change of rings Let p be a prime number. In this section we will consider the algebraic stack a* *ssociated to the flat Hopf algebroid (BP*, BP*BP ) where BP denotes Brown-Peterson homology at p. We will work over S := Spec(Z(p)), i.e. Aff will be the category of Z(p)-algebr* *as with its fpqc topology. We refer the reader to [R1 ], Chapter 4 for basic facts about BP , e.* *g. BP* = Z(p)[v1, . .]. where the vi denote either the Hazewinkel- or the Araki-generators, it does not* * matter but the reader is free to make a definite choice at this point if she feels like doing * *so. (V := Spec(BP*), W := Spec(BP*BP )) is a flat Hopf algebroid and we denote by P* * : V -! XFG the corresponding rigidified algebraic stack. We point out that it is not a pri* *ori clear what XFG is: For U = Spec(R) 2 Aff, XU should of course be the groupoid of one dimensional, * *commutative formal groups (not group laws) over the Z(p)-algebra R and checking this amount* *s to understanding fpqc descent for formal groups. More than enough material to do this should be * *contained in [S] but we do not claim to have checked the details. Of course, we will not use the* * above description of XFG but always consider it as the stack associated to (V, W ). For n > 1 the ideal In := (v0, . .,.vn-1) BP* is an invariant prime ideal whe* *re we agree that v0 := p, I0 := (0) and I1 := (v0, v1, . .).. As explained in subsection 2.4, corresponding to these invariant ideals there i* *s a sequence of closed substacks XFG = Z0 Z1 . . .Z1 . The stack Zn should be the stack of formal groups all of whose geometric fibres* * have height at least n. We denote by Un := XFG - Zn (0 6 n 6 1) the open substack complementary to Z* *n and have an ascending chain ; = U0 U1 . . .U1 XFG . For 0 6 n < 1, In if finitely generated, hence the open immersion Un XFG is q* *uasi-compact and Un is an algebraic stack. However, U1 is not algebraic: If it was, it coul* *d be covered by an affine (hence quasi-compact) scheme and the open covering U1 = [n>0,n6=1Un woul* *d allow a finite subcover, which it does not. 4.1 Flatness and the Landweber condition Fix some 0 6 n < 1. The stack Zn is associated to the flat Hopf algebroid (Vn, * *Wn) where Vn := Spec(BP*=In) and Wn := Spec(BP*BP=InBP*BP ) (the flatness of this Hopf al* *gebroid is established by direct inspection) and we have a cartesian diagram " (4) Wn O____//_W = W0 || || || || fflffl|fflffl|fflffl|fflffl|Oi"n Vn_____//_V = V0 Qn || Q|| fflffl|O "fflffl| Zn______//_XFG in which the horizontal arrows are closed immersions. We have an ascending chain of open substacks ; = Zn \ Un Zn \ Un+1 . . .Zn \ U1 Zn 14 Comodules and algebraic stacks and Zn\Un+1 should be the stack of formal groups all of whose geometric fibres * *have height exactly n. Let X0 -OE!Vn be a morphism in Aff corresponding to BP*=In -! R := (X0, OX0). * *Slightly generalising definition 4.1 of [HS ] we define the height of OE as ht(OE) := max{N > 0|R=IN R 6= 0} which may be 1 and we agree to put ht(OE) := -1 in case R = 0, i.e. X0 = ;. Rec* *all that a geometric point of X0 is a morphism -ff!X0 in Affwhere = Spec(K) is the spectrum of a* *n algebraically in closed field K. The composition -ff!X0 -OE!Vn ,!V specifies a p-typical forma* *l group law over K and ht(inOEff) is the height of this formal group law. The relation between h* *t(OE) and the height of formal group laws is the following. Proposition 15. In the above situation we have ht(OE) = max{ht(inffOE)|ff : -! X0 a geometric point}, with the convention that max ; = -1. This proposition means that ht(OE) is the maximum height in a geometric fibre o* *f the formal group law over X0 parametrised by inOE. Proof.Clearly, ht(_OE) 6 ht(OE) for any morphism _ : Y -! X0 in Aff. For any 0 * *6 N0 6 ht(OE) we have that IN0R 6= R so there is a maximal ideal of R containing IN0R and a geom* *etric point ff of * * __ X0 supported at this maximal ideal will satisfy ht(inffOE) > N0. * * |__| Another geometric interpretation of ht(OE) is given by considering the composit* *ion f : X0 -OE!Vn Qn-! Zn. Proposition 16. In this situation we have ht(OE) + 1 = min{N > 0|f factors throughZn \ UN ,! Zn} with the convention that min; = 1 and 1 + 1 = 1. Proof.For any 1 > N > n we have a cartesian square j (5) VnN ______//_Vn | | | |Qn fflffl|i fflffl| Zn \ UN ____//_Zn SN-1 -1 where VnN = Vn - Spec(BP*=IN ) = i=n Spec((BP*=In)[vi ]) hence f factors throu* *gh i if and only if OE : X0 -! Vn factors through j. As j is an open immersion, this is equ* *ivalent to |OE|(|X0|) |VnN| |Vn| where | . | denotes the topological space underlying a scheme. But* * this condition can_ be checked using geometric points and the rest is easy, using proposition 15. * * |__| Recall from [HS ], 2.1 that, if (A, ) is a flat Hopf algebroid, an A-algebra f* * : A -! B is said to be Landweber exact over (A, ) if the functor M 7! M A B from -comodules to B-mo* *dules is exact. For (X0 := Spec(A), X1 := Spec( )), OE := Spec(f) : Y0 := Spec(B) -! X0 and P :* * X0 -! X the rigidified algebraic stack associated to (X0, X1) this exactness is equivalent * *to the flatness of the composition Y0 -OE!X0 -P!X (see also proposition 8). In case X = Zn this flatne* *ss has the following 15 N. Naumann decisive consequence which paraphrases the fact that the image of a flat morphi* *sm is stable under generalisation. Proposition 17. Assume that n > 0 and that OE : ; 6= X0 -! Vn is Landweber exac* *t of height N := ht(OE) (hence n 6 N 6 1). Then for any n 6 j 6 N there is a geometric poin* *t ff : -! X0 such that ht(inOEff) = j. Proof.Let OE correspond to BP*=In -! R. We first note that vn, vn+1, . .2.R is * *a regular sequence: Assume to the contrary that there is some i > n such that K := ker(R=Ii-1R -.vi* *!R=Ii-1R) 6= 0. .vi We have an injective homomorphism BP*=Ii-1,! BP*=Ii-1of (BP*=In, BP*BP=In)-como* *dules which by flatness (i.e. Landweber exactness) pulls back to give the contradicti* *on K = 0. Now fix n 6 j 6 N. Then vj 2 R=Ij-1R 6= 0 is not a zero divisor and thus there * *is a minimal prime ideal of R=Ij-1R not containing vj. A geometric point supported at this p* *rime_ideal solves the problem. * * |__| The main result of this subsection is the following. Theorem 18. Assume that n > 0 and that ; 6= X0 -! Vn is Landweber exact of heig* *ht N (hence n 6 N 6 1). Let (X0, X1) be the Hopf algebroid induced from (V, W ) along the c* *omposition in X0 -OE!Vn ,!V . Then (X0, X1) is a flat Hopf algebroid and its associated algeb* *raic stack is given as [ X1 ____//_//_X0] ' Zn \ UN+1 ifN 6= 1 and [ X1 ____//_//_X0] ' Zn ifN = 1. Proof.Note that (X0, X1) is also induced from the flat Hopf algebroid (Vn, Wn) * *along OE and thus is flat using the final statement of proposition 8 and the Landweber exactness of * *OE. We first assume that N 6= 1. Then by proposition 16 the composition X0 -OE!Vn -! Zn factors as X0 -_* *!Zn\UN+1 -i! Zn and _ is flat because i is an open immersion and X0 -! Zn is flat by assumpt* *ion. By proposition 9 we will be done if we can show that _ is in fact faithfully flat. For this we* * consider the presentation Zn \ UN+1 ' [ WnN+1 ____//_//_VnN+1] given by the cartesian diagram WnN+1 ______//Wn || || || || fflffl|fflffl|fflffl|fflffl| VnN+1 ______//_Vn | | | |Qn fflffl| fflffl| Zn \ UN+1_____//Zn and note that _ lifts to ae : X0 -! VnN+1 and induces ff := tss2 : X0 x WnN+* *1 -! VnN+1 which ae,VnN+1,s is flat and we need it to be faithfully flat (to apply proposition 7, iii) ) iv* *) and conclude that _ is faithfully flat), i.e. surjective (on the topological spaces underlying the sch* *emes involved). This surjectivity can be checked on geometric points and for any such geometric* * point -~!VnN+1 we have that j := ht( -~! VnN+1 -! Vn) satisfies n 6 j 6 N. By proposition 17 * *there is a geometric point 0-! X0 with ht( 0-! X0 -! Vn) = j and we can assume that =* * 0because the corresponding fields have the same characteristic (namely 0 if j = 0 and p * *otherwise). As any 16 Comodules and algebraic stacks two formal group laws over an algebraically closed field having the same height* * are isomorphic we find some oe : -! WnN+1fitting into a commutative diagram X0ae,xVnN+1,sWnN+1ff//_VnN+1 OO ooo77o ( ,oe)||oo~oooo | oooo . As ~ was arbitrary this shows that ff is surjective. We leave the obvious modif* *ications_for the case N = 1 to the reader. * * |__| Remark 19. We will see in the next subsection that this result implies many of * *the recent results of M. Hovey and N. Strickland ([HS ]) so it may be worthwhile to point out that* * we have not used any of the fundamental results of P. Landweber ([L]) except for the invariance * *of the In BP*. Besides the language of stacks we only used basic facts about (p-typical) forma* *l group laws, and one may wonder if the results of [L] may be recovered from this point of view. We can get the classification of finitely generated radical ideals I BP* as f* *ollows: I corresponds to some closed substack Z XFG with complement U XFG which is algebraic beca* *use I is finitely generated. Composing a presentation X0 -! U with the inclusion U XFG* * we have a flat 1-morphism f : X0 -! XFG hence by theorem 18 (with n = 0) the image of f equals* * some Un, so U = Un. As Z is reduced (because I is radical) we conclude Z = Zn, i.e. I = In. The fact that any non-zero BP*-comodule has a non-zero primitive means that any* * non-zero quasi- coherent OXFG-module F has H0(XFG , F) 6= 0, certainly a striking result. We be* *lieve that this is due to the faithful Gm -action on XFG (corresponding to the grading) and it mig* *ht be interesting to generalise this result from the example XFG to general algebraic stacks with Gm* * -action. Finally, the result that any BP*-comodule is the union of its finitely generate* *d subcomodules gen- eralises, c.f. proposition 4. We conclude this subsection by proving the expected characterisation of Landweb* *er exactness. Proposition 20. Let n > 0 and OE : X0 -! Vn be a morphism in Affcorresponding t* *o BP*=In -! R. Then the following are equivalent: i) OE is Landweber exact. ii) The composition X0 -OE!Vn -! Zn is flat. iii) The sequence vn, vn+1, . .2.R is regular. Proof.The equivalence of i) and ii) was explained in the paragraph preceding pr* *oposition 17 and the implication i) ) iii) has been established during the proof of proposition * *17. The proof that iii) implies ii) is an immediate generalisation of the proof of the exact funct* *or theorem [L]. To show that TorOZn1(OX0, F) = 0 for all F 2 Mod qcoh(OZn) we can assume that F is fini* *tely generated by proposition 4. By proposition 21 below, then, F corresponds to a finitely gener* *ated BP*-comodule M such that InM = 0. Hence every subquotient of the Landweber filtration of M i* *s (isomorphic_to a shift of) BP*=Im for some m > n and the result follows. * * |__| 4.2 Equivalence of comodule categories and change of rings In this subsection we will spell out some consequences of the above results in * *the language of comodules but need some elementary preliminaries first. Let A be a ring, I = (f1, . .,.fn) A (n > 1) a finitely generated ideal and M* * an A-module. We 17 N. Naumann have a canonical map M M ixi xjj Mfi-! Mfifj, (xi)i7! __- __ i i 0 the category Mod qcoh(OZn) is equivalent to the f* *ull subcategory of BP*-comodules M such that InM = 0. For any 0 6 n 6 N < 1 the category Mod qcoh(OZn\UN+1) is equivalent to the full* * subcategory of BP*-comodules M such that InM = 0 and M is IN+1=In-local as a BP*=In-module. Proof.Fix 0 6 n < 1. The 1-morphism Zn ,! XFG is representable and a closed imm* *ersion (in particular a monomorphism) because its base change along V -! XFG is a clos* *ed immersion and being a closed immersion is fpqc-local on the base, [EGA IV2 ], 2.7.1, xii)* *. Proposition 14 identifies Modqcoh(OZn) with the full subcategory of Modqcoh(OXFG) consisting o* *f those F such that Q*F ' in,*G for some G 2 Mod qcoh(OVn) ( with notations as in (4)). Identifying* *, as in subsection 2.4, Mod qcoh(OXFG) with the category of BP*-comodules, F corresponds to some B* *P*-comodule M and Q*F corresponds to the BP*-module underlying M. So the condition of propo* *sition 14 is that the BP*-module M is in the essential image of in,*, i.e. M is an BP*=In-mo* *dule, i.e. InM = 0. Now fix 0 6 n 6 N < 1. We apply proposition 14 to i : Zn \ UN+1 -! XFG which is* * representable and a quasi-compact immersion (in particular a monomorphism) because it sits in* * a cartesian diagram j VnN+1________//V | | | Q| |fflffli fflffl| Zn \ UN+1____//_XFG , c.f. (5), in which j is a quasi-compact immersion and one uses [EGA IV2 ], 2.7.* *1, xi) as above. Arguing as above, we are left with identifying the essential image of j* which,* * as explained at the beginning of this subsection, corresponds to the BP*-modules M such that InM = * *0 and_M_is IN+1=In-local as a BP*=In-module. * * |__| Corollary 22. Let n > 0 and let BP*=In -! R 6= 0 be Landweber exact of height N* * (hence n 6 N 6 1). Then (R, ) := (R, R BP* BP*BP BP* R) is a flat Hopf algebroid an* *d its category of comodules is equivalent to the full subcategory of BP*-comodules M such that* * InM = 0 and M is IN+1=In-local as a BP*=In-module. (The last condition is to be ignored in ca* *se N = 1) Proof.By theorem 18, (R, ) is a flat Hopf algebroid with associated algebraic * *stack Zn \ UN+1 (resp. Zn if N = 1). So the category of (R, )-comodules is equivalent to Modqc* *oh(OZn\UN+1)_(resp. Mod qcoh(OZn)). Now use proposition 21. * * |__| 18 Comodules and algebraic stacks The case n = 0 corresponds to the situation treated in [HS ] where (translated * *into the present terminology) Mod qcoh(OUN+1) is identified as a localisation of Mod qcoh(OXFG).* * This can be done because f : UN+1 -! XFG is flat, hence f* exact. To relate more generally Mod q* *coh(OZn\UN+1) to Mod qcoh(OXFG) it seems more appropriate to identify the former as a full su* *bcategory of the latter as we did above. However, using proposition 1.4 of loc. cit. and proposi* *tion 12 one sees that Mod qcoh(OZn\UN+1) is equivalent to the localisation of Modqcoh(OXFG) with resp* *ect to all morphisms ff such that f*(ff) is an isomorphism where f : Zn \ UN+1 -! XFG is the immersi* *on. As f is not flat for n > 1 this condition seems less tractable than the one in corollary 22. Of course, equivalences of comodule categories give rise to change of rings the* *orems and we refer to [HS ] for numerous examples (in the case n = 0) and only point out the followin* *g (c.f. [R2 ], theorem B.8.8 for the notation and a special case): If n > 1 and M is a BP*-comodule su* *ch that InM = 0 and vn acts invertibly on M then Ext*BP*BP(BP*, M) ' Ext* (n)(Fp[vn, v-1n], M BP* Fp[vn, v-1n]). In fact, this is clear from the case n = N of corollary 22 applied to the obvio* *us map BP*=In -! Fp[vn, v-1n] which is Landweber exact of height n by proposition 20. To make a final point, in [HS ] we also find many of the fundamental results of* * [L] generalised to Landweber exact algebras (whose induced Hopf algebroids are presentations of ou* *r UN+1). One may generalise these further to the present case (i.e. to Zn \ UN+1 for n > 1) but * *again we leave the fun of doing this to the reader and only point out an example: In the situation of * *corollary 22 every non-zero (R, )-comodule has a non-zero primitive. To prove this, consider the comodule as a quasi-coherent sheaf F on Zn \ UN+1 a* *nd use that the primitives we are looking at are H0(Zn \ UN+1, F) ' H0(XFG , f*F) 6= 0 because * *f* is faithful and using the result of P. Landweber about XFG recalled in remark 19. References BLR S. Bosch, W. L"utkebohmert, M. Raynaud, N'eron models, Ergebnisse der Math* *ematik und ihrer Gren- zgebiete (3), 21, Springer-Verlag, Berlin, 1990. Bou N. Bourbaki, Alg`ebre Commutative, Hermann, Paris, 1961. D P. Deligne, Cat'egories tannakiennes, The Grothendieck Festschrift, Vol. I* *I, 111-195, Progr. Math., 87, Birkh"auser Boston, Boston, MA, 1990. G P. Goerss, (Pre-)sheaves of ring spectra over the moduli stack of formal g* *roup laws, Axiomatic, enriched and motivic homotopy theory, 101-131, NATO Sci. Ser. II Math. Phys. Chem.,* * 131, Kluwer Acad. Publ., Dordrecht, 2004. EGA IV2 A. Grothendieck, 'El'ements de g'eom'etrie alg'ebrique IV, Secon* *de partie, Publications Math'ematiques de l'IH'ES, 24 (1965) 5-231. Ha R. Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Spr* *inger-Verlag, Berlin-New York, 1967. H M. Hovey, Homotopy theory of comodules over a Hopf algebroid, Homotopy the* *ory: relations with algebraic geometry, group cohomology, and algebraic K-theory, 261-304, Con* *temp. Math., 346, Amer. Math. Soc., Providence, RI, 2004. HS M. Hovey, N. Strickland, Comodules and Landweber exact homology theories, * *math.AT/0301232. L P. Landweber, Homological properties of comodules over MU*(MU) and BP*(BP)* *, Amer. J. Math. 98 (1976), no. 3, 591-610. LM-B G. Laumon, L. Moret-Bailly, Champs alg'ebriques, Ergebnisse der Mathemati* *k und ihrer Grenzgebiete, 3. Folge, 39, Springer-Verlag, Berlin, 2000. Lu J. Lurie, Tannaka Duality for Geometric Stacks, math.AG/0412266. 19 Comodules and algebraic stacks P E. Pribble, Algebraic stacks for stable homotopy theory and the algebraic * *chromatic convergence theorem, PhD thesis. R D. Ramakrishnan, Pure motives and automorphic forms, Motives (Seattle, WA,* * 1991), 411-446, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994. R1 D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure * *and Applied Mathemat- ics, 121. Academic Press, Inc., Orlando, FL, 1986. R2 D. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals o* *f Mathematics Studies, 128, Princeton University Press, Princeton, NJ, 1992. S N. Strickland, Formal schemes and formal groups, Homotopy invariant algebr* *aic structures (Baltimore, MD, 1998), 263-352, Contemp. Math., 239, Amer. Math. Soc., Providence, RI,* * 1999. SGA1 A. Grothendieck, Rev^etements 'etales et groupe fondamental, S'eminaire d* *e G'eom'etrie Alg'ebrique du Bois Marie 1960-1961 (SGA 1), Lecture Notes in Mathematics, Vol. 224, Spri* *nger-Verlag, Berlin-New York, 1971. W T. Wedhorn, On Tannakian duality over valuation rings, available * *at http://www.math.uni- bonn.de/people/wedhorn/prepr.html. N. Naumann niko.naumann@mathematik.uni-regensburg.de NWF I- Mathematik, Universit"at Regensburg, Universit"atsstrasse 31, 93053 Rege* *nsburg 20