Journal of Algebra 220, 612᎐628Ž. 1999 Article ID jabr.1999.7936, available online at http:rrwww.idealibrary.com on
Flat and Stably Flat Modules
U E. Alcock
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, View metadata, citation and similar papers at core.ac.uk brought to you by CORE E1 4NS, London, United Kingdom E-mail: [email protected] provided by Elsevier - Publisher Connector
Communicated by Peter M. Neumann
Received December 1, 1998
We study Kropholler’s generalisation of Lazard’s criterion for a module to be flat. The context is complete cohomology and modules of type Ž.FP ϱ. Let G be a ᑠ group in the class H1 of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let R be a strongly G-graded algebra. We provide a characterisation of the stably flat R-modules under the assumption that ᮊ R1 is coherent of finite global dimension. 1999 Academic Press Key Words: group; cohomology; graded ring; flat module.
1. INTRODUCTION
Given a commutative ring k and a group G, a strongly G-graded k-algebra is a k-algebra R together with a k-module decomposition s [ s g R g g G R ggsuch that RRhR ghfor all g, h G.If H is a subgroup s [ of G we denote by RHgg g H R the k-subalgebra of R supported on H. The simplest example of such an R is the group algebra kG. Other natural examples are twisted group algebras, skew group algebras, and even crossed products. A crossed product is a strongly G-graded k-algebra
in which each R g contains a unit. A twisted group algebra is a crossed product in which k is carried isomorphically onto R1. A skew group algebra is a crossed product R which admits a homomorphism from G to the group of units of R in which each g g G is carried to an element of
R g . It is natural to study strongly group-graded algebras as the proofs often become more streamlined, indeed this was why Dadewx 11 introduced the notion as a means of streamlining the study of the representations of a group H relative to a normal subgroup N of H.
* The author is supported by a grant from EPSRCŽ. Ref. 96003462 .
612 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. FLAT AND STABLY FLAT MODULES 613
For an arbitrary ring S a projective resolution of an S-module M is a resolution P# ¸ M by projective modules. We denote by ⍀ nM the nth kernel of M, that is,
⍀ n s ª M KerŽ.Pny1 Pny2 .
Although ⍀ is not a functor on the category of all modules it is a well defined functor on the stable category ModŽ.S having as its objects all
S-modules and morphism elements of HomSŽ.M, N for all S-modules M and N. Here HomSŽ.M, N denotes the quotient of Hom SŽ.M, N by the additive subgroup of homomorphisms which factor through projective modules. The projective dimension of a module M is the least integer n Žif it exists. such that ⍀ nM is projective. If no such integer exists M is said to wx have infinite projective dimension. Kropholler 19 has$ shown that a 0 s module M has finite projective dimension if and only if ExtSŽ.M, M 0. We denote the projective dimension of a module M by proj dim S M. Following Krophollerwx 20 , an S-module N is said to be stably flat if and only if $ 0 s ExtS Ž.M, N 0 for all S-modules M of type Ž.FP ϱϱ. Recall that a module of type Ž.FP is a module M possessing a resolution P# ¸ M by finitely generated projec- tive modules. Stably flat modules were introduced as a tool for studying the modules of type Ž.FP ϱ over a strongly G-graded k-algebra R, where the grading group belongs to the class LH ᑠ of locally hierarchically decomposable groups and the base ring R1 is coherent of finite global dimension. We provide the following characterisation of the stably flat ᑠ R-modules where the grading group lies in the subclass H1 of groups acting on a finite dimensional contractible CW-complex, in such a way that the setwise stabiliser of each cell is equal to the pointwise stabiliser and is finite: ᑠ THEOREM A. LetGbeanH1 -group and let R be a strongly G-graded k-algebra such that R1 is coherent of finite global dimension. IfNisan R-module, then the following are equi¨alent: Ž.i N is stably flat, Ž.ii N has finite projecti¨e dimension. This result generalises Theorem 6.6 ofwx 1 , however our proof avoids the need for complicated diagram chasing. One of the tools used in proving Theorem A is the following result concerning strongly group-graded alge- bras where the grading group is finite. 614 E. ALCOCK
THEOREM B. Let G be a finite group and let R be a strongly G-graded k-algebra such that R1 has finite global dimension. If N is a flat R-module ¨ ¨ which is projecti easanR1-module, then N is projecti e.
This is related to a question asked by Benson inwx 1 concerning the group algebra kG and we include in Section 5 results showing that, with mild constraints on R, a flat R-module which is projective as an R1-mod- ule is in fact projective. Bensonwx 2 has independently obtained these results in the case of the group algebra kG. Recently Benson and Good- earl have announced that if G is a finite group and S is any ring, not necessarily commutative, then every flat SG-module which is projective as a S-module is projective. This statement is still an open question in the case of a strongly G-graded k-algebra but it seems conceivable that the result will carry through to this, more general, setting. We observe that for any R-modules M and N the set of R -homomorphisms Hom Ž.M, N 1 R1 admits an action of the group algebra kG. This action does not appear to have been utilised before in the study of strongly group graded algebras and we include as an application a new proof of Maschke’s theorem for strongly group-graded k-algebras which avoids the use of Cohen᎐Mont- gomery dualityŽ seewx 25, 8. . Thus we are always assuming that the order of a finite group G is not invertible in R1.
2. STRONGLY GROUP GRADED ALGEBRAS
First, we recall an equivalent definition of a G-graded k-algebra and of a strongly G-graded k-algebra. These definitions were used by Cornick and Krophollerwx 9 and more details may be found there. Contrary to that claimed inwx 9 , this equivalent definition of a group-graded algebra was earlier observed by Bergmanwx 4 . Recall that kG has a natural bialgebra structure where the coalgebra map kG ª kG m kG is defined by the diagonal map g ¬ g m g. Here and throughout this paper an unembel- lished tensor denotes tensor over k.
DEFINITION 2.1. Let G be a group. A G-graded k-algebra is a k-alge- bra R together with a k-algebra map ␥ : R ª R m kG which makes R into a kG-comodule.
From now on R denotes a G-graded k-algebra and S will denote an arbitrary ring. The importance of Definition 2.1 is that one can make R act semi-diagonally on certain tensor products. In particular, let M be an R-module and V be a kG-module; then M m V is an R m kG-module. So FLAT AND STABLY FLAT MODULES 615
M m V is an R module via ␥,
Ž.m m ¨ r [ mr m ¨g
g for r R g . Let N be an R-module, then R acts semi-diagonally on the set of k-module homomorphisms Hom kŽ.V, N via
r Ž.¨ [ Ž. Ž.¨gy1 r
g ¨ g ᎐m for r R g and for all V. Thus we have two functors V and ᎐ Hom kŽ.V, from the category of R-modules to itself. Indeed these two are adjoint as shown by the following lemma.
LEMMA 2.2. Let M and N be R-modules and let V be a kG-module. There is a natural R-isomorphism:
m ( Hom RRŽ.M V, N Hom Ž.M, Hom k Ž.V, N .
COROLLARY 2.3. If V is a projecti¨ek-module then ᎐m V carries projec- ti¨eR-modules to projecti¨eR-modules. Cornick and Kropholler also provided the following equivalent defini- tion of strongly G-graded k-algebra:
DEFINITION 2.4. Let G be a group. Then a strongly G-graded k-algebra is a G-graded k-algebra R with the property that for all g g G, there exist g y g finitely many elements x1,..., xngR 1 and y1,..., yngR such that s Ýiiixy 1.
Let M and N be R-modules and consider the set of R1-homomor- phisms Hom Ž.M, N .If g Hom Ž.M, N , then we define RR11
g [ Ž.m Ý Žmxii . y i
g g for all m M and g G, where i, xii, and y are chosen as in Defini- tion 2.4.
LEMMA 2.5. Let G be a group and let R be a strongly G-graded k-algebra. If M and N are R-modules, then with the definition abo¨e Hom Ž.M, N R1 admits a well defined action of kG. Proof. We show first that our definition is well defined. Given that
g s Ž.m Ý Žmxii . y i 616 E. ALCOCK
for some choice of x ,..., x g R y1 and y ,..., y g R with Ý xys 1 1 ng 1 ngX X iii g y we may choose some other sequence of elements x1,..., xmgR 1 and X X g X X s y1,..., ymgR such that Ý iiix y 1. Then
g s Ž.m Ý Žmxii . y i s X X Ý Ž.mx jjiy xy i i, j s X X Ý Ž.mx jjiiy xy i, j s X X Ý Ž.mx jjy . j
Now suppose that r g R , g Hom Ž.M, N and choose x ,..., x g 1 R1 1 n y g s R g 1 , y1,..., yngR such that Ý iiixy 1. Then
g s Ž.mr Ý Žmrxii . y i s Ý Ž.mxjj y rx i y i i, j s Ý Ž.mxjjii y rx y i, j s Ý Ž.mxjj y r j s g Ž.mr
g and so we have that is an R1-homomorphism. Finally we check that we have an action of kG on Hom Ž.M, N . Let R1 X X g g y g g g, h G and choose x1,..., xngR 1 , y1,..., yngR , and x1,..., xm X X X X y g s s Rh 1 and y1,..., ymhR such that Ý iiixy 1 and Ý iiix y 1. Then
ggh s XX Ž.Ž.m Ý Žmxii . y i s X X Ý Ž.mxijxyy ji i, j s ghŽ.m ,
XX XX g y g s since the xxij RŽ gh. 1 , yyjiR ghand Ý i, jijjixxyy 1. FLAT AND STABLY FLAT MODULES 617
LEMMA 2.6. Let G be a group and let R be a strongly G-graded k-algebra. Let M and N be R-modules and g Hom Ž.M, N . Then R1 y1 g Ž.mr s gh Ž.mr g g for all r Rh, g, h G. X X X X g y g Proof. We may choose x1,..., xmhR 1 and y1,..., ymhR such X X s that Ýiiix y 1. Then
y1 g Ž.mr s gh h Ž.mr s ghy1 X X Ý Ž.mrxiiy i s ghy1 X X Ý Ž.mrxiiy i
y1 s gh Ž.mr as required. We provide here an alternative proof of Maschke’s theorem which avoids the use of Cohen᎐Montgomery dualitywx 7 as used by Passman w 25 x and Cornickwx 8 .
THEOREM 2.7. Let G be a finite group and let R be a strongly G-graded < g s 1r< Let H _G denote the right G-set of cosets Hg of G. Another applica- tion of Definition 2.4 is to show that if R is a strongly G-graded k-algebra we have a tensor identity. LEMMA 2.8Ž. Tensor identity . Let H be a subgroup of G. If M is an R-module then there is a natural isomorphism M m R ( M m kHwx_G R H ¨ m ¬ m g gi en by the map m r mr Hg for r R g . 3. STABLY FLAT MODULES Tatewx 6 developed a cohomology theory for finite groups which sub- sumed homology and cohomology into one theory. Vogel and Mislin independently discovered a generalised Tate cohomology theorywx 13, 23 which works for any group. A different, yet isomorphic, approach has been developed by Vogelwx 13 and by Benson and Carlson wx 3 . The definition adopted here is due to Bensonwx 1 and shall be called the complete cohomology. Since ⍀ is a functor on the stable category we have a natural homomor- phism ª ⍀ ⍀ HomSSŽ.M, N Hom ŽM, N . and via this we define the complete cohomology to be $ Exti Ž.M, N s lim Hom Ž⍀iqjjM, ⍀ N .. S ª S j With this definition of the complete cohomology it is sometimes useful to consider a variation on$ the usual stable category ModŽ.S . Following wx Kropholler 20 , we write ModŽ.S for the category$ with the same objects as 0 ModŽ.S but with morphisms the elements of ExtSŽ.M, N . DEFINITION 3.1. Let S be a ring. An S-module N is said to be stably flat if and only if $ 0 s ExtS Ž.M, N 0 for all S-modules M of type Ž.FP ϱ. Motivation and comparison with Lazard’s criterionwx 21 for a module to be flat can be found inwx 20 . The following is an immediate consequence of the definition, together with Lemma 4.2 ofwx 19 . FLAT AND STABLY FLAT MODULES 619 LEMMA 3.2. Let S be a ring. If N is a stably flat S-module of typeŽ. FP ϱ, then N has finite projecti¨e dimension. Let ᑲŽ.S denote the class of stably flat S-modules. Brownwx 5 proved i ᎐ that if M is of type Ž.FP ϱ then the functor Ext SŽ.M, commutes with direct limits for all i G 0. This, combined with a result of Krophollerwx 19 concerning the completion of a cohomological$ functor, shows that if M is i ᎐ a module of type Ž.FP ϱ then the functor ExtSŽ.M, commutes with direct limits for all i g ޚ. The proof of this result is essentially the same for ᑲŽ. filtered colimits. Thus S is closed under taking$ direct summands, direct i ᎐ sums, and filtered colimits. Since the functor ExtSŽ.M, induces long exact sequences which extend infinitely far to the left and right, ᑲŽ.S is extension closed. The following lemma is useful in the study of stably flat modules and may be proved in the same way as for ordinary cohomology. LEMMA 3.3Ž. Eckmann᎐Shapiro . Let T be a subring of S with S flat as a T-module. For M a T-module and N an S-module then $$ iim ( ExtSTŽ.M S, N ExtTSŽ.M,Hom Ž.S, N for all i g ޚ. COROLLARY 3.4. Let S, T, and N be as in the lemma. If N is a stably flat S-module then N is stably flat as a T-module. The following appears as Corollary 6.7 ofwx 20 . LEMMA 3.5. Suppose an exact sequence of modules ª ª иии ª ª ª 0 Nm N0 N 0 ¨ is gi en in which each Ni is stably flat. Then N is stably flat. COROLLARY 3.6. Let N be a filtered colimit of modules of finite weak dimension. Then N is stably flat. Proof. It suffices to prove that a module of finite weak dimension is stably flat. Let M be a module of type Ž.FP ϱ. First suppose that N is flat. wx Then, by Lazard’s theorem$ 21 , N is a direct limit of finitely generated 0 ᎐ free modules. Since ExtSŽ.M, commutes with direct limits and vanishes on projective modules it follows that N is stably flat. Now suppose that N is of finite weak dimension, so that there is an exact sequence ª ª иии ª ª ª 0 Nm N0 N 0 with each of the Ni flat and hence stably flat. It follows from Lemma 3.5 that N is stably flat as required. 620 E. ALCOCK Thus ᑲŽ.S contains all modules which may be expressed as a filtered colimit of modules of finite weak dimension and such modules are not necessarily of finite projective dimension. 4. COFIBRANT MODULES AND COMPLETE RESOLUTIONS Let G be a group and let R be a strongly G-graded k-algebra. Following Kropholler we write B [ BGŽ., k for the kG-module of func- tions from G to k which take only finitely many values. This module is a commutative k-algebra with ring operations defined by pointwise multipli- cation, and G acts on it as a group of algebra automorphisms. The reader is referred towx 10 for more details. Via the semidiagonal action of Section 2, M m B is an R-module. Krophollerwx 20 calls an R-module M cofibrant if and only if M m B is a projective R-module. It is also noted that if M is a cofibrant module then the natural map $ ª 0 HomR Ž.M, N ExtR Ž.M, N is an isomorphism for all R-modules N. When a module M is such that M m B has finite projective dimension Cornick and Krophollerwx 10 show that M has a complete resolution. We repeat here their definition of a complete resolution. For an arbitrary ring S, a complete resolution of an S-module M is an acyclic complex of projectives P# such that: ⅷ P# coincides with a projective resolution of M in sufficiently high dimensions ⅷ # Hom SŽ.P , Q is acyclic for every projective S-module Q. When complete resolutions exist they can be used to compute complete cohomology. s ª LEMMA 4.1. Let S be a ring and let M KerŽ.Py1 Py2 be the zeroth kernel of a complete resolution. If : M ª Q is a homomorphism to a ¨ u projecti e module Q, then factors through the inclusion M Py1. ª ␣ ¸ Proof. Let d00: P Py10factor through M via maps : P M and  u ␣ g ␣ s : M Py1. Then we have a map Hom SŽ.P01, Q such that d 0 g Hom SŽ.P1, Q . But we have the exact sequence U U U d d d иии ªy101ª ª ª иии Hom S Ž.Py1 , Q Hom S Ž.P0 , Q Hom S Ž.P1 , Q , g U s s ␣ so there exists Hom SŽ.Py100, Q such that d Ž. d . But ␣ s s ␣ ␣ s  then d0 and is surjective so as required. FLAT AND STABLY FLAT MODULES 621 PROPOSITION 4.2. Let G be a group and let R be a strongly G-graded k-algebra. Let M and N be R-modules with M m B of finite projecti¨e ª $dimension. If : M NisanR-homomorphism which goes to zero in 0 ¨ ExtRŽ.M, N , then factors through an R-module Y of finite projecti e dimension. Proof. In Theorem 3.5 ofwx 10 it is shown that if M m B has finite projective dimension then M has a complete resolution P#. Moreover the # X zeroth$ kernel of P is a cofibrant module M which is isomorphic to M in ModŽ.R . Thus we have $$ 00( XX( ExtRRŽ.M, M Ext ŽM , M .HomR ŽM , M .. ␣ g Ž X . So there exists$ an element HomR M , M which corresponds to the 0 X ª identity in ExtRŽ.M, M . Choose an R-homomorphism : M M repre- senting ␣ and consider the composition X M ª M ª N. $ This composition represents zero in ModŽ.R but $ 0 XX( ExtR Ž.M , N HomR Ž.M , N and so , factors through a projective module Q. By Lemma 4.1 we may X assume that M embeds in Q and so we have a commutative diagram X 6 MM 6 6 6 NQ in which is a monomorphism. Adding Q to M we have the commutative diagram with exact row M i y 6 6 X Ž.6 6 6 0 M M [ Q Y 0 Ž. , 0 6 6 N 6 where Y is the cokernel ofŽ.y and i is just the inclusion of M in M [ Q. It follows that factors through the cokernel Y. We have a long exact 622 E. ALCOCK sequence of complete cohomology induced from the short exact sequence in the diagram $$$␥ иии ª y10ª X ª 0[ ExtRRRŽ.Y, Y Ext ŽY, M .Ext ŽY, M Q . $ ª 0 ª иии ExtR Ž.Y, Y . $ X However since M is isomorphic to M in ModŽ.$R and Q is projective we ␥ 0 s must have that is an isomorphism and so ExtRŽ.Y, Y 0 and so by Lemma 4.2 ofwx 19 , Y has finite projective dimension as required. PROPOSITION 4.3. Let G be a group and let R be a strongly G-graded k-algebra. If M is a cofibrant R-module which has finite projecti¨e dimension, then M is projecti¨e. Proof. We have a k-split short exact sequence of kG-modules, k u B ¸ B, where k is mapped into B as the constant functions and is split by the k-module map B ª k given by evaluation at 1. B is the cokernel of this inclusion and so is projective as a k-module. Thus we may tensor this exact sequence repeatedly yielding Bmi u Bmi m B ¸ Bmiq1 . Tensoring with M and iterating we obtain the exact sequence of R-mod- ules: M u M m B ª M m B m B ª иии ª M m Bmny1 m B ¸ M m Bmn . Since M is cofibrant and B is projective as a k-module it follows by Corollary 2.3 that each of the intermediate terms in the above sequence is projective. Suppose now that M has projective dimension n. Then M m Bmn also has projective dimension n and so, by the extended version of Schanuel’s lemma, we have that M is projective. 5. FINITE GROUPS AND PROJECTIVITY OF FLAT MODULES In this section we let G be a finite group and let R be a strongly G-graded k-algebra. For finite groups the kG-module B is isomorphic to kG. Thus if N is an R-module which is projective as an R1-module it follows from the tensor identity that N is cofibrant. FLAT AND STABLY FLAT MODULES 623 PROPOSITION 5.1. Let G be a finite group and let R be a strongly G-graded k-algebra. If N is an R-module of finite projecti¨e dimension which is ¨ ¨ projecti easanR1-module, then N is projecti easanR-module. Proof. Follows immediately from the fact that N is cofibrant and Proposition 4.3. Recall that theŽ. right finitistic dimension of a ring S is the least integer n Ž.if it exists such that all Ž. right S-modules M of finite projective F dimension satisfy proj dim S M n. PROPOSITION 5.2Ž. Jensen . Let S be a ring with finite right finitistic dimension. Then any flat right S-module has finite projecti¨e dimension. Proof. Seewx 18, Proposition 6 . LEMMA 5.3. Let G be a finite group and let R be a strongly G-graded k-algebra such that R1 has finite global dimension. Then R has finite finitistic dimension bounded by the global dimension of R1. Proof. Suppose that M is an R-module of finite projective dimension and that R1 has global dimension d.If ⍀ d u ª ª иии ª ¸ M Pdy1 Pdy20P M is a truncated projective resolution of M of length d, then, by the ⍀ d extended version of Schanuel’s lemma, M is projective as an R1-module and is of finite projective dimension as an R-module so is projective as an R-module by Proposition 5.1. Thus M has projective dimension at most d as required. We are now in a position to prove Theorem B which we restate here: THEOREM B. Let G be a finite group and let R be a strongly G-graded k-algebra with R1 of finite global dimension. If N is a flat R-module which is ¨ ¨ projecti easanR1-module, then N is projecti easanR-module. Proof. Follows from Proposition 5.2, Lemma 5.3, and Proposition 5.1. Inwx 1 Benson asked whether there could exist a non-projective flat kG-module which is projective as a k-module. By Theorem B the answer is no if k has finite global dimension. It is possible to show that under certain other conditions a flat R-module which is projective as an R1-mod- ule is indeed projective and we include those results here. Benson and Goodearl have recently announced that for any ring S, not necessarily commutative, a flat SG-module which is projective as an S-module is projective as an SG-module. This statement is still an open question in the more general setting of strongly group-graded algebras, but it seems conceivable that the result will carry through. 624 E. ALCOCK First we are assuming that the order of G is not invertible in R1; otherwise every R-module would be projective byŽ. Maschke’s Theorem 2.7. PROPOSITION 5.4Ž. Osofsky . Let S be any ring and n g . Then an / ¨ q n-related flat module is of projecti e dimension at most n 1. Proof. See Osofskywx 24, Theorem 2.45 . THEOREM 5.5. Let G be a finite group, let R be a strongly G-graded g / k-algebra, and let n . If N is a flat n-related R-module which is ¨ ¨ projecti easanR1-module, then N is projecti easanR-module. Proof. Follows immediately from Proposition 5.4 and Proposition 5.1. PROPOSITION 5.6Ž. Gruson and Jensen . Let S be any ring of cardinality / g ¨ ¨ q n, n . Then e ery flat S-module has projecti e dimension at most n 1. Proof. Follows fromwx 16 and Proposition 10.5 ofwx 17 . THEOREM 5.7. Let G be a finite group and let R be a strongly G-graded / g ¨ k-algebra of cardinality n, n . If N is a flat R-module which is projecti e ¨ as an R1-module, then N is projecti easanR-module. Proof. Follows from Proposition 5.6 and Proposition 5.1. DEFINITION 5.8. Let A be an abelian category. A SerreŽ.´ epaisse subcategory C of A is a full subcategory C of A such that for all short exact sequences X Y M u M ¸ M X Y of A, M is an object in C if and only if M and M are objects of C. Given a Serre subcategory C of A one may form the new category ArC called the quotient category of A by C. See Gabrielwx 12 for more details. We give two equivalent definitions of theŽ. left Krull dimension of a ring S. There is some confusion in the literature between various dimensions defined in this area so our definitions may differ slightly from those found elsewhere. A good reference is Gordon and Robson’s bookwx 14 . DEFINITION 5.9. Let M be a left S-module. The Krull dimension of M, s written Kdim S M, is defined by transfinite recursion, as follows: If M 0, sy ␣ l ␣ s ␣ Kdim SSM 1; if is an ordinal and Kdim M , then Kdim SM provided there is no infinite descending chain s ) ) иии M M01M r l ␣ of submodules MiSsuch that Kdim Ž.Miy1 Mi . FLAT AND STABLY FLAT MODULES 625 If S is any ringŽ. with 1 let Mod Ž.S denote the category of all right S-modules. An equivalent categorical definition of Krull dimension was provided by Gordon and Robsonwx 15, p. 462 in terms of an ascending chain of Serre subcategories of A s ModŽSop .. We have altered the indexing slightly in order that our definition agrees under certain condi- tions with the dimension of a small abelian category; seewx 17, p. 267 . We define this chain by transfinite recursion as follows: s Ay10Ä40; A is the Serre subcategory of all left S-modules which are Artinian; if ␣ is a successor ordinal then A␣ is the Serre subcategory of all r ␣ left S-modules which are Artinian as objects of A A␣y1;if is a limit ordinal then A␣ is the Serre subcategory D - ␣A . PROPOSITION 5.10Ž. Gordon-Robson . Let M be a left S-module, ␣ an ordinal. Then the Krull dimension of M equals ␣ if and only if M g A␣ but f M A␣y1. Proof. See Proposition 1.5 ofwx 15 and note that our numbering has been altered. In either of the two definitions theŽ. left Krull dimension of S is defined to be the Krull dimension of S as a left S-module. Gruson and Jensenwx 17 defined the dimension of an arbitrary small abelian category and, noting that for a Noetherian ring this coincided with the Krull dimension, proved the following: PROPOSITION 5.11Ž. Gruson᎐Jensen . Let S be a left Noetherian ring with left Krull dimension d. Then any flat right S-module has projecti¨e dimension at most d. Proof. Seewx 17, Corollary 7.2 . THEOREM 5.12. Let G be a finite group and let R be a strongly G-graded k-algebra with R1 Noetherian of finite Krull dimension. If N is a flat R-module ¨ ¨ which is projecti easanR1-module, then N is projecti easanR-module. Proof. Since G is finite, R is finitely generated as a left R1-module, thus R isŽ. left Noetherian. McConnell and Robsonw 22, 6.5.3Ž.Ž. ii cx showed X X that for an arbitrary ring extension S of S such that S is finitely X generated as an S-module the Krull dimension of S is bounded by that of S. So theŽ. left Krull dimension of R is bounded by that of R1, hence finite. The result follows by Proposition 5.11 and Proposition 5.1. 6. THE PROOF OF THEOREM A ᑠ H1 denotes the class of all groups which act on a finite dimensional contractible CW-complex in such a way that the setwise stabiliser of each 626 E. ALCOCK cell is equal to the pointwise stabiliser and is finite. Theorem A ofwx 9 ᑠ states that if G is an H1 -group and R is a strongly G-graded k-algebra then every R-module has finite projective dimension if and only if it has finite projective dimension as an RH-module for each finite subgroup H of G. By Lemma 3.4 a stably flat R-module is stably flat as an RH-module for each finite subgroup H of G and so we may reduce the proof of Theorem A to the following: X THEOREM A. Let G be a finite group and let R be a strongly G-graded k-algebra such that R1 is coherent of finite global dimension. If N is a stably flat R-module then the following are equi¨alent: Ž.i N is stably flat, Ž.ii N has finite projecti¨e dimension. Moreo¨er, the projecti¨e dimension of such a module is bounded by the global dimension of R1. Proof. Ž.i « Žiii . . Since G is finite, R is coherent. Thus if N is finitely presented it has type Ž.FP ϱ and so is of finite projective dimension as an R-module by Lemma 3.2. So we may assume that N is infinitely presented and express N as a filtered colimit of finitely presented R-modules N s lim6 N. For each , since G is finite, N m B ( Nm kG ( N m R R1 has finite projective dimension$ and, as N is stably flat, the natural map ª 0 N N represents zero in ExtRŽ.N, N . Thus by Proposition 4.2 each such map factors through a module Y of finite projective dimension. Now R has finite finitistic dimensionŽ. Lemma 5.3 and so the projective dimension of the Y must be bounded by the global dimension of R1, n say. Let K be R ᎐ any left R-module. Then by the functoriality of Tornq1Ž., K and the fact R s that Tornq1Ž.Y, K 0 we have that the natural map R ª R Tornq1Ž.N , K Tornq1 Ž.N, K R ᎐ is zero for each . Since Tori Ž., K commutes with filtered colimits for all i we have an isomorphism R R lim Tor q Ž.N , K ( Tor q Ž.N, K ª n 1 n 1 constructed from the natural maps above which hence is zero. Thus R s ⍀ n Tornq1Ž.N, K 0 and so N has weak dimension at most n. Then N is ⍀ n flat and projective as an R1-module and so N is projective by Theorem B and N has projective dimension at most n as required. FLAT AND STABLY FLAT MODULES 627 Ž.ii « Ž.i is a straightforward application of the definition of complete cohomology. Theorem A follows immediately in the way described above. We note the following corollary which is in a similar vein to the results in Section 5. COROLLARY 6.1. Let G be a finite group and let R be a strongly G-graded k-algebra such that R1 is coherent of finite global dimension. If N is a stably ¨ ¨ flat R-module which is projecti easanR1-module, then N is projecti easan R-module. X Proof. Follows immediately from Theorem A and Proposition 5.1. X Theorem A also yields the following corollary which is in the wider setting of strongly group-graded algebras where the grading group is in the class LH ᑠ of locally hierarchically decomposable groups. COROLLARY 6.2. LetGbeanLH ᑠ-group and let R be a strongly G-graded k-algebra such that R1 is coherent of finite global dimension. Let N be an R-module. Then N is stably flat if and only if N has finite projecti¨e dimension as an RE-module for all finite elementary abelian subgroups E of G. X Proof. Follows immediately from Theorem A and Theorem B ofwx 20 . ACKNOWLEDGMENTS This work is part of Ph.D. research in progress. I would like to thank P. H. Kropholler and C. R. 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