GROTHENDIECK GROUPS OF INVARIANT RINGS: FILTRATIONS
KENNETH A. BROWN and MARTIN LORENZ
[Received 31 March 1992—Revised 12 August 1992]
ABSTRACT
We investigate the Grothendieck group Ga(R) of finitely generated modules over the ring of invariants R = SG of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G0(R) that is defined in terms of (Gabriel-Rentschler) Krull dimension, properties of G0(R) are derived to a large extent from the connections between the sets of prime ideals of 5 and R. A crucial ingredient is an equivalence relation ~ on Spec/? that was introduced by Montgomery [25]. For example, we show that
rank Go(/?) ^ rank G0(S)G + £ (#Q - 1), a where Q runs over the —-equivalence classes in Spec R and (-)c denotes G-coinvariants. The torsion subgroup of G0(R) is also considered. We apply our results to group actions on the Weyl algebra in positive characteristics, the quantum plane, and the localized quantum plane for roots of unity.
Introduction
This article is a continuation of the work in [5] on the structure of the G Grothendieck group Ga(R), where R = S is the ring of invariants of a finite group G acting on a right Noetherian ring 5. While the latter paper focussed on the use of the skew group ring S * G and the G-fixed point functor in computing G0(R), our main technique here will be a certain filtration {Fj(R)} of G0(R) and the restriction map from 5-modules to R-modules. In the first section, we define the aforementioned filtration {Fj(R)} of GQ(R) for any right Noetherian ring R, very much in the spirit of the commutative theory developed in [8] and [9] but using a dual approach. In fact, since localization is not generally available in a non-commutative setting, our definition of {Fj(R)} will be based on the notion of an exact dimension function for finitely generated /^-modules rather than on codimension (height of primes). This leads to minor notational clashes with [8] and [9], but the transition between the two notations is easily made in the most standard situations (cf. 1.2, Remarks). We also consider 'higher class groups' W;(R), essentially generalizing the definition of [8] to a non-commutative setting. The interest of these class groups mainly stems from the following two facts:
Wj(R) canonically maps onto the ith slice Fj(R) = Fi(R)/Fi_x(R) of the above filtration; if R is a fully bounded Noetherian (FBN) ring then Wj(R) is accessible via an explicit presentation. The group W^R) can also be viewed as the £?_,-term of a spectral sequence converging (under suitable assumptions) to G*(R).
The second author was supported in part by NSF Grant DMS-9005597. 1991 Mathematics Subject Classification: primary 16W20, 16E20, 16D90; secondary 19A49. Proc. London Math. Soc. (3) 67 (1993) 516-546. GROTHENDIECK GROUPS OF INVARIANT RINGS 517
A primary objective in § 1 is to investigate the transfer of the above structural data in a Morita context (A, B, X, Y,f, g) which fails to be an equivalence. Here, A and B are right Noetherian rings, AXB and BYA are bimodules, and /: X<8>BY—>A and g: Y<£)AX^>B are bimodule maps satisfying the usual associativity conditions. If / and g are both surjective, then the Morita context yields an equivalence of the module categories of A and B, and so G0(A) = GQ(B). We will study the case where / and g are not necessarily both surjective but the size of their cokernels can still be controlled in terms of the underlying dimension functions. We find that, under suitable hypotheses, Fj(A) = Fj(B) and Wj(A) = Wj(B) provided / is larger than the dimensions of the cokernels of /and g (1.6, Proposition). In the important special case where one of the maps, say/, is surjective, F({A), F,{A), and Wj(A) are all images of the corresponding data for B (1.6, Theorem). The material in § 1 is developed with particular emphasis on two types of rings and dimension functions: (a) Noetherian polynomial identity (PI) algebras with Gelfand-Kirillov dimen- sion, and (b) FBN rings with (Gabriel-Rentschler) Krull dimension. Affine Noetherian Pi-algebras are subsumed under both (a) and (b), because in this case Gelfand-Kirillov dimension and Krull dimension coincide (and are integer valued) for each finitely generated module. Section 2 contains the main results of this article. We apply the above G machinery to study G0(R) where R = S is the ring of invariants of the action of a finite group G on a right Noetherian ring 5. For the sake of explicitness, we assume in this section that the underlying dimension function d(-) is (Gabriel- Rentschler) Krull dimension and our main emphasis will be on the case where S is an FBN ring. Furthermore, in view of our results on Morita contexts in § 1, we will concentrate on the case where the trace map tr: S—>R is surjective. Besides the standard situation where the order of G is invertible in 5, this covers the case where S is simple Artinian and G acts by outer automorphism on S [26, Theorems 2.5 and 2.7], amongst others. It turns out that, in the case where S is an FBN ring, many properties of the restriction map from G0(S) to GQ(R) can be derived from the connections between the sets Spec 5 and Spec R of prime ideals of 5 and R. In 2.4 we recall, in slightly generalized form, the basic facts about Montgomery's equivalence relation ~ on Spec R [25]. This equivalence relation is always trivial for commutative rings, and in some non-commutative cases as well (cf. for example 3.2), but in general it represents a complicating factor, compared with the commutative setting. For example, we show in 2.6, Theorem 1 that
rank G0(R) *s rank GQ(S)G + 2 (#Q - 1), where Q runs over the —-equivalence classes in Speci? and (-)c denotes G-coinvariants. The estimate actually is an equality in some important examples (cf. § 3). In particular, if ~ is trivial then rank GQ(R) =s rank GQ(S)G. More precisely, in this case, we show in 2.6, Theorem 2(i) that
the cokernel of the restriction map Res£: G0(S)—> G0(R) is torsion. Furthermore, a bound for the exponent can be given in terms of the ramification of primes between R and S. This result, and some of its consequences and 518 KENNETH A. BROWN AND MARTIN LORENZ refinements that are described in 2.6, Theorem 2, generalize corresponding results established in [6] for commutative rings. The latter paper also considers, in a commutative setting, so-called Galois actions (and generalizations thereof) and induction from R to S, topics which are not touched upon in the present article. The final § 3 is devoted to the discussion of a number of examples: the Weyl algebra in positive characteristic, the quantum plane for a root of unity, and the localized quantum plane for a root of unity. (The characteristic 0, respectively non-root of unity, cases are more straightforward.) Our main emphasis in this section is on the ranks of the Grothendieck groups in question. In a forthcoming article, we will consider further specific examples in greater detail.
1. Filtration of Grothendieck groups 1.1. Module categories Let R be a right Noetherian ring with an exact dimension function d = dR that is defined on the category mob-/? of all finitely generated right /?-modules. We assume that d has values in some totally ordered set Q which has a smallest element denoted —°°. The precise requirements on d which will be relevant for us are as follows (cf. [3]): (i) d(M) = -oo if and only if M = 0; (ii) for each short exact sequence 0—»AT—»Af—»M"—»0 in mob-/?, one has d(M) = max{d(M'), d(M")} ('exactness').
Often Q can be taken to be {-oo}uZ&() but, for the most part, the special properties of the range of d will be immaterial here. S s For each i e Q, we let M, = Mt{R) denote the full subcategory of mob-/? consisting of all finitely generated /?-modules M such that d(M)^i. Clearly, s SK,-2 iTCy if i^y. Weput
j and 2«|._
s So aKf-_c2K|.c iK,.+ . In case Q = {-«>} UZ&() one has Wli+ = Wli+x and Tlt_ = s S Mj-t. Exactness of d implies that M,- is a Serre subcategory of mob-/?, that is, if 0—»M'—>M—>M"-*0 is an exact sequence in mob-/? then M belongs to sMj if and only if both M' and M" do. Similar results hold for Mi+ and 2ft,-_. Thus one s can form the (abelian) quotient categories Wlj/Wlj, SJ^/Sft,-., !&tf,+/ JJ?/_ etc. (see, s for example, [10, Chapter 15]). Note that Wlj/Mlj; s9ft. if/ = -oo and Mt = mob-/? if i
EXAMPLES. (1) If /? is a commutative Noetherian ring of finite (Krull) dimension n then we can take d(M) to be the usual (Krull) dimension of M (see, for example, [4, Chapter 8, § 1, no. 4 and no. 5]; Bourbaki's notation for 9}J, in this case is ^«,). The category iP?(, has objects the /?-modules of finite length. If R s is a domain, Mn-x consists of all finitely generated torsion modules over /?. In the special case where /? is an affine algebra over some field k such that dim R/p = n holds for all minimal primes p of /?, our categories 9J£, are identical with the s categories Mn-j of [8]. This follows from the formula dim /?/p + ht p = n which is valid for all primes p of R (cf. [23,14.H]). (2) More generally, let /? be right Noetherian with finite Krull dimension n in GROTHENDIECK GROUPS OF INVARIANT RINGS 519 the sense of Gabriel-Rentschler. Then, letting d(M) denote the (Gabriel- S Rentschler) Krull dimension of M, we note that MO still consists of the R-modules of finite length, and if R is prime, then ^)ln-i consists of all finitely generated torsion modules over R (cf. [24, Proposition 6.3.11(iii)]). (3) Now let R be a Noetherian Pi-algebra over some field k and let d(-) denote Gelfand-Kirillov dimension over k. Then the foregoing applies, since d is known to be exact for Noetherian Pi-algebras. If, in addition, R is affine over k, then
00 In all these examples, we can take Q = {- } U Z&0-
1.2. Filtration of G*{R) Keeping the notation and hypotheses of 1.1, we now consider the (Quillen) K-groups G*(R) = K^mob-R). The embedding Hft,-c mob-/? gives rise to a homomorphism K*(SR;)-» G*(/?) whose image will be denoted by O'G*:
Thus, with similar definitions for
(1) {0} = <*>-°°G* c ... c O'-G, c
This is the content of the Theorem below which is a consequence of the following s s observation. If h^i^j then WljIWj is a Serre subcategory of Mh/ Mj with quotient category equivalent to SJ^/Sft,-. Thus we have a localization sequence [30, Theorem 5]
(2) ... (This is also valid with h, i, or j equal to /+ or /—, provided the indices occur in the appropriate order.)
THEOREM. Assume that Q = {-°°} UZg0. Then there is a converging spectral sequence l Gp+q{R). Here we interpret K* and G* as 0 for * <0. So the spectral sequence is concentrated in the range 0 =sp =s n = d(R), p + q 2= 0.
Proof. Note that our assumption on Q implies that n = d(R)<<*>. S Furthermore, 3Rp = 0 for p < 0 and MP = mob-/? for p^n. Thus the filtration (1) of G*(/?) is bounded: {0} = Q^G^R) c O°G*(/?) c ... c O"G*(/?) c ... c ^" 520 KENNETH A. BROWN AND MARTIN LORENZ
Viewing the localization sequence (2) in the form
...^Kl(mp/mp.l)^K0(mp.1)^KQ(mp)^KQ(mp/mp.l)^o as an exact couple via Dpq = Kp+q(Mp) and Epq = Kp+q($RpiyRp_x), we see that the spectral sequence follows as in [32, proof of Theorem 11.13], for example.
For the most part, we will concentrate on G0(R). Thus, for simplicity, we put
Fi = Fi(R) = VG0(R), and similarly for F,_ and Fi+. Furthermore, we let
Note that, in terms of the localization sequence (2), Ft =
W, = W,(R) = ^(^(1(/1,-)) <= K^JW,-) (i < n), where 0/:=0/+i/i/_. Since 0n,,-,,•_ factors through 0,+,/,,-, we have canonical epimorphisms (3) w,: W^F,.
If Q = {—°°} UZ»n then jrn_, is in fact an isomorphism, because 9K,+ = ^Jln for i = n-\.
REMARKS. Versions of the groups V^ were introduced in [8] as possible candidates for higher class groups of R. Indeed, let R be an integrally closed commutative domain which is a localization of an affine algebra over some field k, and let d(-) denote the usual Krull dimension function. Then, putting n = d(R), as above, we have that Wn_l(R) is the usual class group C(R) of R (cf. [8]). We remark that our Wj(R) corresponds to Ww_,-(/?) in [8] (see 1.1, Example 1).
In terms of the above spectral sequence (and with the assumption that Q = {-00} U Z^o), Wj(R) has the following description:
1 This follows from the isomorphism £•?_, = £,-_,/d (E-+1_,), together with the exact sequence (from (2))
1.3. Fully bounded Noetherian rings Our goal in this section is to describe the Zs'-page of the spectral sequence in 1.2, Theorem in the special case where R is a right FBN ring and d(-) is (Gabriel-Rentschler) Krull dimension. In particular, for finite /, we will obtain an explicit description of the terms in the following presentation of V^ which is an immediate consequence of (2) (and can also be found in [8]):
(4) /c,« jm,) -iw.w.) -^u wi —> o. GROTHENDIECK GROUPS OF INVARIANT RINGS 521
The proposition below can essentially be viewed as a non-commutative extension of [8, Propositions 2.4 and 2.5]. The following well-known facts about finitely generated right modules M over right FBN rings will be used in the proof:
(a) there exist submodules 0 = MocM1c.,, cMs = M such that, for each /, pj = annA?(A/y/A/y_,) is a prime ideal of R and A/y/A/y_i is isomorphic to a uniform right ideal of Rip, (cf. [12, Theorem 8.6]);
(b) d(M) = d\m(R/annR(M)), where dim(-) denotes the classical (ordinal- valued) Krull dimension of the ring in question (cf. [24, 6.4.8 and 6.4.12]). We put Xj = Xj(R) = {p: p is a prime ideal of R with dim(/?/p) = i}. Recall that the Goldie skew field of a prime Noetherian ring R is the endomorphism ring, D say, of the unique simple module of the classical simple Artinian ring of quotients Q(R). Thus Q{R) is a matrix ring over D.
PROPOSITION. Suppose R is right FBN and d(-) is (Gabriel-Rentschler) Krull dimension. For each p eXh let Dp denote the Goldie skew field of Rip. Then:
peAf, In particular, the following hold. (i) A^o(3K,/3K,_)=0peA',Z(p), the free abelian group on Xt. An explicit basis for Kofflli/ffii-) is given by {[t/p],-: peXj}, where Uv is a fixed uniform right ideal of Rip and [£/p], denotes the element of K0(Plil'3RiJ) corresponding to the module Uv. s (ii) Kl(Wil Mi-)=^peXi D*I[D*, Dp], where D* is the multiplicative group ofDv.
Proof. We will give a direct proof for Ko which will also establish the crucial facts about the category 9K,/^_ that are needed for the general case. The result will then follow from [30, § 5, Corollary 1] (or [2, Theorem III.3.4] for * = 0 or 1). Suppose that M e 3ft,. For a given prime ideal p e Xh define /P(A/) to be the number of subquotients My/A/,., in a fixed series for M as in (a) above such that MjlMj-x is isomorphic to a uniform right ideal of Rip. To see that this definition does not depend on the particular chosen series for M, note that any two series have a common refinement of the required form. Furthermore, if 0=£ V c U are /^-modules so that U is uniform with d(U)<
s Since each /p(-) is clearly additive on short exact sequences in Mh /(•) is likewise. 522 KENNETH A. BROWN AND MARTIN LORENZ
Moreover, l(M) = 0 holds for M e SM/ if and only if M e »?,_. Therefore, by (2), / yields a homomorphism
/: K0(2K,/aK,_)-> 0 Z
. An inverse, a, for / is obtained by putting
o(M) = [U9], (peXt), where Uv is a fixed uniform right ideal of R/p and [(/p], denotes the element of KuQMj/WljJ) corresponding to the module Uv. By [24, Lemma 3.3.4], every uniform right ideal of R/p embeds into Uv, and we have seen above that the cokernel has dimension less than /. Thus the isomorphism type of Up in Sfy/Sft,-. only depends on p e Xt. Clearly, 1° o = Id. Surjectivity of o follows from the fact that if M e 3K,- has a series as in (a) then, in /C0(iUy^,_), one has
The above argument shows that /(•) is a length function on Sftf/SK,-.. Hence the foregoing establishes the following facts: every object of sM,/sMi_ has finite length, and a full set of representatives for the isomorphism classes of simple objects of Mi/Mi- is given by the above [t/J; for p e X,. By [30, §5, Corollary 1], the general assertion about K^ipli/Wl,--) is a consequence of these facts, provided we can show that
But (cf. [10,15.3])
Enda,w_(*/P) = lim YlomR{V, Uv/W), where the limit is over those V, W c (/„ with Uv/V e Sft,_ and W e aW,-_. Hence W = 0 and
En&m.m._(U») = lim Hom«(V, (7P), where V runs over the non-zero (hence essential) submodules of LJV. But one easily checks that
lim Hornby, Uv) = EndQ{Rfv)(Uv <8>R/p Q(R/p)) = Dp, which completes the proof of the proposition.
COROLLARY 1. Let R and d(-) be as in the proposition and suppose that n = dim(fi) < ». Then
»eXn where Up is a fixed uniform right ideal of R/p. Furthermore, Fn_x is the kernel of the homomorphism
p: ()()(()) 0 »eXmin projection
0 G0 \>eXn GROTHENDIECK GROUPS OF INVARIANT RINGS 523 where N is the nilpotent radical of R, Xmin is the set of minimal primes, and Q(-) denotes the classical ring of quotients. (If N is prime then p is Goldie's reduced rank function.)
Proof. By the proposition (and its proof), G0(R)/Fn_l = AToC^/^.,) is free abelian with basis the images of {[Up]: peXn} which implies the above decomposition of G0(R). The map p sends {[Up]: peXn) onto the canonical basis of 0p6A-,, Go(Q(R/p)) (Uv is mapped onto the simple module for Q(R/p)) and, using fact (a) about FBN rings stated above, one can easily see that Fn_x is mapped to {0}. This proves the assertion about p.
COROLLARY 2. Let R and d(-) be as in the proposition and suppose that n = dim(R) < <». Then there is a converging spectral sequence
El«= © Kp+q(D,)^pGp+q(R). \>eXp Proof. This is immediate from 1.2, Theorem and the proposition. 1.4. Change of rings: transfer across a bimodule Let A and B be right Noetherian rings, and let AXB be an .4-jB-bimodule which is finitely generated as fi-module. Thus tensoring with X yields a functor %=(-)®AXB: mob-A^mob-B. Suppose further that dA and dB are exact dimension functions on mob-y4 and mob-5, both with values in Q. In order to ensure that #f behaves well with respect to the sub- and quotient categories introduced in § 2, we will consider the following conditions:
(NBU) for each finitely generated ,4-module M one has dB(c£(M)) =s dA(M); and s (F), the functor Tor?(•, X): mob-A^ mob-B sends mob-A to Mt(B). Here (NBU) stands for No Blowing Up and (F), for Flat, as in [34] and [9], where similar conditions are considered. The following lemma shows that, in the situations that we will be most interested in, (NBU) is a consequence of fairly harmless finiteness conditions on X. (Exactness of dA and dB is irrelevant for this lemma.)
LEMMA. Assume that the bimodule AXB is finitely generated on both sides. Then (NBU) is satisfied in each of the following cases:
(a) A and B are algebras over a field k, and dA and dB are Gelfand-Kirillov dimension over k;
(b) A and B are FBN-rings, and dA and dB are (Gabriel- Rentschler) Krull dimension. Proof, (a) See [24, Proposition 6.3.14(ii)]. (b) For a given finitely generated ,4-module M, put 7 = ann/i(Af), Y = X/IX, and / = annfl(Y). Then, by [24, Proposition 6.4.12], dA(M) = dA(A/I) and, using Kdim for Gabriel-Rentschler Krull dimension of left modules, we see that [24, Proposition 6.4.13] implies that Kdim(AY) ^dB(Y) = dB(B/J). Since M<8>A X = Y is a finitely generated module over B/J, we further have dB(B/J)^ 524 KENNETH A. BROWN AND MARTIN LORENZ dB{M®AX). Similarly, Kdim(AY)^Kdim(AA/1) = dA(A/1), where the latter equality follows from [24, Corollary 6.4.10]. Therefore, dB(M®A X)^dA(M).
REMARK. By [35], (NBU) is not satisfied for (Gabriel-Rentschler) Krull dimension of arbitrary two-sided Noetherian rings.
The importance of (NBU) and (F), for our purposes comes from the following proposition.
PROPOSITION, (i) // (NBU) is satisfied then the functor %? = (•) ®A XB sends each to the corresponding Ti^B). Thus % yields functors %, f. l
(ii) //, in addition, (F), is satisfied then the functors c£itj are exact for all i^j^ t. (iii) Assume that (NBU) and (F), are satisfied. Then #f yields homomorphisms £,-,: K^W/mjiA))^K*(M-(B)/Wj(B)) (i&j&t) and a commutative dia- gram of localization sequences (2) for h^i^j^t:
K*(Wh(B)Wj(B))
In particular, %?yields homomorphisms f,: Fj(A)^> Fj(B) (i>t) and §,: Wt{A) Wj(B) (n> i> t), where the latter fit into a commutative diagram (cf. (4))
K1$Jli(A)/'mi-(A)) ^l W,(A) —^ 0
Kt(mi+(B)/W,(B)) -^gf K^UBVMUB)) ^^ W,(B) —> 0
Proof, (i) The first assertion is clear. Composing the functor $?, restricted to i s 3Ri(A), with the canonical functor 2Jli(B)-+ JRi(B)/'2Rj(B) we obtain a functor Tij: sMi(A)-^sMi(B)/Wlj(B) which has sMj(A) in its kernel. Thus #f passes down to a functor %itj: Mi(A)/'mj(A)-+Wli(B)WJiB). (ii) The functor <%itj is exact if and only if Thj is exact (cf. [10, Corollaries 15.9 and 15.10]). Since T,j is right exact, being a composite of a right exact functor and an exact functor, proving exactness amounts to showing that, for any monomorphism/: M'<-*M in %Rj(A), the morphism Ttjf is a monomorphism in 9)li(B)/Mj(B). By [10, Lemma 15.5], the latter condition is equivalent with < Ker(f®Aldx)e mJ(B). But Ker(/®i4Id^) is a factor of Tor?(M/AT, X) € WlAB) c sMj{B), and hence it also belongs to Wlj(B). (iii) We have the following commutative diagram of exact functors whenever
[xhJ Mh(B)/Mj(B) GROTHENDIECK GROUPS OF INVARIANT RINGS 525
The above commutative diagram of localization sequences now follows from functoriality of the localization sequence. Note that the foregoing applies withs Mj replaced by ^,_, as long as / > t. Thus, with h=n = max{dA(A), dB{B)}, we obtain a commutative diagram of Grothen- dieck groups
i Since Ft =
1.5. Change of rings: restriction, inflation, induction In this section, we briefly discuss some important special cases of the material in the previous section. Let oc: A —> B be a homomorphism of rings and assume that either
(a) A and B are Noetherian Pi-algebras over a field k, and dA and dB are Gelfand-Kirillov dimension over k, or
(b) A and B are FBN-rings, and dA and dfl are (Gabriel-Rentschler) Krull dimension.
Restriction and inflation. Assume that B is finitely generated as right A -module via oc. Then take BXA = BBA (so we are interchanging the roles of A and B in 1.4). Now 1.4, Lemma guarantees that X satisfies (NBU). Since (F), is trivially satisfied, for all t, the proposition implies that $f = (•) <8>B BA yields exact functors s i yRi(B)/ MJ(B)^>yjli(A)/ mj(A) for all i^j. Restriction is the case where oc is a monomorphism and inflation the case where oc is an epimorphism.
Induction. Assume that B is finitely generated as left A -module via a and that dB(Torf(M, B)) as t holds for all finitely generated v4-modules M. In other words, AXB = ABB satisfies (F),. Then, as above, the induction functor (•) ^ BB yields exact functors Wli(A)Wj(A)^>'mi(B)/'mj(B) for all i**j& t.
In each case, one obtains morphisms of localization sequences and maps between the appropriate Wj and Fh as in the proposition.
1.6. Morita contexts Let {A, B, X, Y,f, g) be a Morita context. So A and B are rings, AXB and BYA are bimodules, and /: X<8)BY-+A and g: Y<8>AX—>B are bimodule maps which satisfy the usual associativity conditions (see, for example, [24,1.1.6]). When A and B are algebras over some commutative field k, it will be understood that the left and right A>actions on all bimodules over A and B coincide. As above, we assume that dA and dB are exact dimension functions on mob--<4 and mob-#. Furthermore, we will assume throughout that A and B are right 526 KENNETH A. BROWN AND MARTIN LORENZ
Noetherian and that XB and YA are finitely generated. Thus we have functors %= and
REMARK 1. In the important special case where / is surjective, X and Y are finitely generated and projective over B (cf. [2, Theorem II.3.4]). If, in addition, B is right Noetherian then, by [5, Lemma 1.1], YA is finitely generated and A is right Noetherian. (Similarly, if B is left Noetherian then so is A and AX is finitely generated.) Thus our finiteness assumptions are satisfied in this case.
Throughout, we put A=A/lmf and B = B/lmg. We will study the situation where a bound on the dimensions of A and B is given, say dA(A)^d and dB(B)^d. Thus, in some sense, 6 measures the extent to which the given Morita context fails to provide a Morita equivalence. We remark that, in this situation, the kernels of/and g also have dimensions bounded by S. This is a consequence of the following lemma.
LEMMA. For a finitely generated right A-module V, consider the map
Id,,®/: (
The kernel and the cokernel of IdK ®f are both finitely generated right modules over A, and hence their dimensions are bounded by dA(A). In particular,
dA{V)>dA(A) => If % and <& both satisfy (NBU), then
dA(V)>dA(A) ^
Proof. First, putting I = lmf<=A, we have Im(Idy ®/) = W, and hence Coker(Idv (8)/) = V/VI which is clearly a finitely generated right module over A. As to Ker(Idv/ <8>/), consider elements a= £,- vt,®xt <8> y,;€ Ker(IdK <8>f) and /3 = TtjX'j <8>yj eX<8)BY. Then, using the associativity conditions on / and g, one computes
« • f(fi) = 2 Vi® Xi ® {y, • f{x] ® y'j)) = S«,-® x, ® (g{yi ® x}) • y'j) i.j i,j = 2 «/ ® (x, -giyi®x'j)) ®y'j = ^ Vi ® if{x, ®y,) -xj) ®y) ij i
= 0, where the last equality follows because £,v,.fix,:0V,) = 0. This shows that / annihilates Ker(Idv ®f) which is therefore a right /i-module. Finite generation is clear from our Noetherian assumptions. The first assertion about dA now follows from exactness of dA, applied to the exact sequence
0^ Ker(Idv, ®/)-* V®A X®B Y-+ V- GROTHENDIECK GROUPS OF INVARIANT RINGS 527 whose end terms have dimensions at most dA(A). Finally, the last assertion also follows, because (NBU) implies that dA{V)^dB{%{V))^dA{
PROPOSITION. Assume that $? and ty both satisfy (NBU) and put
d = m Then $? and % both satisfy (F),.,, and the functors %ti and %4 of 1.4, Proposition yield exact inverse equivalences between Mi(A)/sMj(A) and sM,(B)/sMj(B), for all i^j^d. In particular, we obtain isomorphisms §,: WM^+WtB) and £, ^ for all i > 6. Proof. For any finitely generated v4-module V, consider the exact sequence » V®A X®B Y^> V®A A = V-> VIV S Since Ker(Idv, 0/) and VIV • Imf both belong to M6(A), by the lemma, we s conclude that V and V<8>A X®B Y are isomorphic in xnob-A/ M6(A) (cf. [10, Lemma 15.6]). This shows that %j°S^ij is naturally equivalent to the identity on s i JEfii(A)/ 3Rj(A), for all i s»/ ^ d. The argument for %ltio oyLj is identical, and hence the categories sMj(A)/sMj(A) and sMi(B)/sMj(B) are equivalent for all i z*j ^ d via s s Sf and °y, and likewise the categories mob-A/ M6(A) and mo'0-B/ M6(B). Finally, since any functor defining an equivalence between two abelian categories is necessarily exact (cf. [11, Proposition 13]), we conclude that #? and % both satisfy (F)6 (cf. the proof of 1.4, Proposition (ii)). As the remaining assertions are clear from 1.4, Proposition, the proof is complete. REMARK 2. If dA(A) > 6 holds in the situation of the above proposition (which is uninteresting otherwise) then the lemma implies that dA(A) = dB(3?(A)) ^ dB(B). Thus dB(B) > d also, and we conclude that dA(A) = dB{B). We now specialize to the case where / is surjective, so A = 0. Note that B inherits a dimension function from B by restricting dB to mob-Z? viewed as the full subcategory of mob-Z? consisting of those M in mob-Z? with M • Img = 0. With this in mind, we can also consider sMj(B) etc. THEOREM. Assume that, in the given Morita context, f is surjective. Assume further that S6 and % both satisfy (NBU). Then, for all i s=y, there is a long exact sequence 528 KENNETH A. BROWN AND MARTIN LORENZ < where r is given by inflation and o by the functor 3/ = {-)®BYA. In particular, <3/ yields epimorphisms , and rjr. /5 for all i, and 77, and rj, are isomorphisms for i>dB(B). Proof. By Remark 1, BY is projective, and hence <2/ satisfies (F), for all i. If we put d = dB(B), the proposition implies that #? satisfies (¥)8. Therefore, by 1.4, Proposition, #? and <& give rise to functors %u: Tli{A)imj{A)^Wi{B)l'mi{B) and %: mi(B)mj(B)-»mi(A)/mA) such that %tj is exact for all i&j and %u is exact for /ss/ss S. Since/is in fact an isomorphism [2, Theorem II.3.4], one has natural equivalences for all i^j. Letting & denote the kernel of %j, that is, the full subcategory of l mi(B)Wj(B) consisting of those M eob(ijli(B)/ Mj(B)) = ob(Wi(B)) with %j(M) = 0 or, explicitly, M®B Y EWJ{A), we obtain an equivalence of categories (cf. [10, Proposition 15.18]). We claim that inflation mob-Z?-» mob-fi yields an equivalence of categories s s First note that Mi(B)/ MJ(B) can be identified with the subcategory, ^ say, of $Jli(B)/Wlj(B) consistingof those M with M • Img eWj(B). Indeed, inflation clearly maps ^1^8)^10) to % and (-) s shows that, in Mt(B)/^)lj(B), M is isomorphic to M%BB, and hence the two functors are inverse to each other. Next, if M belongs to 9, so M <8>B Y e Wft^A), then M<8>s Y®A X € Mj(B) and, since the latter module maps onto M Img, we have M € (S. Conversely, if M • Img e sMj(B) then, using the equality Img • Y = g(Y®A X) - Y = Y -f{X®BY) = Y, we obtain M®BY = M®B(lmg - Y) = (M • lmg) ®BYeMj(A). This establishes the asserted equivalence of categories Wlj(B)/sMj(B) = 9. The long exact sequence in the theorem is now a consequence of Quillen's localization theorem [30, Theorem 5 on p. 113]. In particular, with j = —00, we obtain that KQ(Tlj(B)) maps onto KQ(WHi{A)), and this fact in turn immediately yields the epimorphisms r)] and »),. Similarly, with j = i—, we obtain a GROTHENDIECK GROUPS OF INVARIANT RINGS 529 commutative diagram (cf. 1.4, Proposition) W^B) —> 0 onto Wj(A) > 0 whence 77, is surjective. The fact that 77, and fj, are isomorphisms for i > 6 follows from the proposition or, more directly, from the fact that '3Ri(B)/%Jli-(B) = 0 for COROLLARY. Assume that, in the given Morita context, f is surjective. Assume further that either (a) A and B are Noetherian Pi-algebras over a field k, and dA and dB are Gelfand-Kirillov dimension over k, or (b) A and B are FBN-rings, and dA and dB are (Gabriel-Rentschler) Krull dimension. Then all conclusions of the theorem hold. Proof. By Remark 1, surjectivity of/and the fact that B is Noetherian imply that the bimodules X and Y are finitely generated on either side. Thus, by 1.4, Lemma, (NBU) is satisfied by 2f and by <2/. 2. Rings of invariants of finite group actions 2.1. Notation The following notation will be kept fixed throughout this section: S will be a right Noetherian ring; G will be a finite group acting by automorphisms of S, written s*-+s8; R = SG will denote the ring of C-invariants in 5; and T = S*G will be the skew group ring of G over 5. Multiplication in T is based on the rule sg = gss for s e S, g e G. Furthermore, we will use the trace map tr: S—>R, defined by tr(5) = 2 ss (se S), geG and the map /: 5 <8>R S—» T that is defined by f(s<8>s') = sts' (s,s'eS) where t = 2 g e T. geG The trace map is an /?-/?-bimodule map. Viewing S as 7?-r-bimodule and as !T-/?-bimodule via the obvious isomorphisms S = tS = tT and S = St= Tt, we see that the map/becomes an 5-5-bimodule map. In this way one obtains the Morita context (R, T,RST,TSR, tr, /) of Chase, Harrison, the Rosenberg [7]. The cokernel of/will be denoted by Tand the image by /, so I = StS = TtT and T=T/I. 530 KENNETH A. BROWN AND MARTIN LORENZ For any ideal X of 5, we put G(X) = GZ(X)/GT(X), where GZ(X) = {g eG: Xs = X) is the decomposition group (or stabilizer) of X and Gr(X) = {g eG: ss-seX for all seS} is the inertia group of X. Thus G{X) acts faithfully on SIX. Finally, for the sake of explicitness, the dimension function d is always understood to be (Gabriel-Rentschler) Krull dimension in this section. We put <5 = ^(7"), the right Krull dimension of t. 2.2. Surjectivity of the trace map We shall assume throughout § 2 that the trace map is surjective, say (5) tr(*) = l (xeS). Putting e = xt = x 2 g, geG we have te = t. Thus e = e2 and TS = Tt = Te. Therefore, rS is projective, and similar remarks apply to ST. For s e S one has (6) ese = e tr(sjt). In particular, if r e R then ere = er and so eTe = eSe = eR. This yields the following description of R as a corner of T: 0: R >eTe = eR, (f)(r) = ere = er. Since T is right Noetherian, by assumption on S, Te is Noetherian as right e7e-module, and hence SR is Noetherian. All this is particularly well-known in the standard situation where |G|~' eS in which case one usually takes x = \G\~l. Similarly, [20, Theorem 3.3 (i)-(iii)] can be easily extended to the present slightly more general setting, without assuming that S is right Noetherian, by using the above idempotent e. (Part (iv) also generalizes, with the same Maschke averaging argument, as long as x can be chosen in the centre of S.) For example, the following lemma extends [20, Theorem 3.3 (ii)]. LEMMA 1. If Vs is an S-module of finite length n then VR has length at most \G\-n. nas Proof. First, V<8>5 T = ®g6c V ®g length \G\ • n as 5-module, and hence its length is at most \G\ • n as T-module. Also, viewing Te as right /^-module via the above isomorphism (f), one has Te = St = SR, and so V<8)sTe = VR. Finally, if 2 WA is a module over some ring A and a = a is an idempotent of A, then it is easy to see that the length of WaaAa is bounded above by the length of WA. LEMMA 2. If one of S, T, and R is right fully bounded, all three are. GROTHENDIECK GROUPS OF INVARIANT RINGS 531 Proof. First, T is right FBN if and only if 5 is (without surjectivity assumption on the trace), because S c T is a finite normalizing extension of rings and so [34, Theorem 21] applies. Since corners of right FBN rings are easily seen to be right FBN, it follows that R is right FBN if T is. Alternatively, one can use [36, Corollary 3.2] for this. Finally, if R is right FBN then [17, Proposition 4.9] implies that 5 is likewise, because SR is finitely generated. Surjectivity of the trace map passes to homomorphic images SIX of S as Z follows. Put H = G {X) and let tvH denote the corresponding trace map for SIX Sl and for S. Then, writing G-{JgjH (disjoint union) and y = Ylx , one has trH(y + X) = 1 and so trw maps SIX onto (SIX)". Note also that (7) H H since if s + X e {SIX) then ixH{ys) e S and s + X = trH{ys) + X. 2.3. Transfer from T to R: the G-fixed point functor The following result is 1.6, Theorem and Corollary specialized to our present setting. Recall that the module categories 30?, are now formed with respect to (Gabriel-Rentschler) Krull dimension. THEOREM. Assume that S is FBN. Then, for all i^j, there is a long exact sequence where x is given by inflation mob-f1—> mob-T and o by the G-fixed point functor (•)c: mob-r-»mob-/?. In particular, (-)G yields epimorphisms , and fjr. Ft(T)-»Fi(R) for all i, and rjj and r/, are isomorphisms for i> 6 = d(t). Proof. The only point requiring some explanation is the assertion that the c functor (•) ®TSR in 1.6, Theorem is equivalent to the G-fixed point functor (-) . But, with e = e2 as in 2.2, we have, for any T-module V, and the theorem follows. The long exact sequence in the theorem is a refinement, for FBN rings, of the sequence (8) ...^^^ of [5,1.3]. 532 KENNETH A. BROWN AND MARTIN LORENZ 2.4. Equivalence of prime ideals in R We recall the definition and some basic facts concerning Montgomery's equivalence relation for primes in R [25], extended slightly to the case where the trace map is surjective. This material is valid without the Noetherianness assumption on 5. Let e = e2 e T be as in 2.2 and identify R with eTe via (p. Using the well-known bijection (cf. [15] or [21, Lemma 4.5]), W: Spec e7Vr-»Specer = {H3e Spec 7| e $ ^} a H-» largest ideal $ of T with es#e = a, we obtain a bijection Proof First note that, for any ideal X of S, X°T is an ideal of T with eXl)Te = e(X C\R) = Since $ and its inverse preserve inclusions, we conclude that p is minimal over PDR <=> 3>(p) is minimal over P°T. By [21, Theorem 1.3(i)], the latter condition is equivalent with 0>(p) n 5 = Pl\ This proves the first assertion. The second follows from the fact that, for every prime ideal s$ of T, one has s$ H 5 = P{) for some prime P of S [21, Lemmas 1.1 and 3.1(i)], and if P and Q are prime ideals of S then Pl) = Q° holds if and only if P and Q are G-conjugate. In the situation of the above lemma, one says that P lies over p. Montgomery's equivalence relation ~ on Spec/? can now be defined as follows: p~q <^> LEMMA 2. Assume that S is FBN. Let p be a prime ideal of R and let P be a prime ideal of S with pD P D R. Then p is minimal over P D R if and only if d(R/p) = d(S/P). In particular, d is constant on —classes in Spec R. Proof. By 2.2, R is FBN as well and RS and SR are finitely generated. It follows from [12, Lemma 12.11] that p is minimal over P C\ R if and only if d(R/p) = d(R/PC\R). Finally, [12, Corollary 12.15] implies that d(R/P n R) = d(S/P). Since the G-action on Spec 5 respects each Xt(S) = {P eSpecS] d(S/P) = i}, the above bijection Spec S/G*-> Spec R/— for FBN rings breaks up into bijections 2.5. Quotient-rings and relative degrees The following lemma is well-known when \G\~l e S [26, Theorem 5.3] or, more generally, when the element x in (5) can be chosen central in 5 [13, Proposition A.3]. Instead, we assume that 5 is FBN. We also give a somewhat crude upper bound for the number of primes in a given equivalence class. (For general 5, a similar bound can be given in terms of so-called X-inner automorphisms.) LEMMA 1. Assume that S is FBN and let P be a prime ideal of S. (i) The ring R/PC\R has an Artinian classical ring of quotients Q(R/P D R) and the inclusion R/PHR^S/P extends to an embedding Q(R/PC\R)^ Q(S/P). Under this embedding, Q(S/P) = S/P • Q(R/P D R) and Q(R/P n R) is identified with the fixed ring Q(S/P)G(P\ (ii) An upper bound for the number of primes p of R that are minimal over PC\R is given by the order of the group Ginn(P) = {g e G(P): g yields an inner automorphism of Q(S/P)}. Proof, (i) The assertions that Q(R/POR) exists and is an Artinian subring of Q(S/P) follow from [12, Theorem 12.12], and the claimed equality Q(S/P) = S/P • Q(RIPf\R) is an easy consequence (cf. [3,7.3]). It remains to show that Q(R/P C\R) = Q(S/P)G(P\ To this end, replace 5 by S/P° (which we may do in view of (7) in 2.2), and thus reduce to the case where P n R = 0 and P° = 0. By the foregoing, regular elements of R are regular modulo P, and hence also modulo all conjugates of P and modulo P°. Thus Q(R) embeds into the semisimple Artinian ring (2(5). Since the minimal primes of 5 are exactly the ideals P8i where g, runs through a transversal for GZ(P) in G, Q{S) is the direct Si product of the rings Qt = Q(S/P ). The action of G on Q(S) permutes the Q{ transitively. It follows that projection of Q(S) onto Qx = Q(S/P), say, identifies Q(S)G with Q{SIP)G(P\ On the other hand, since Q(S) = S • Q(R), one has Q(S)G = Q(R) which completes the proof of (i). (ii) The minimal primes of R over PDR correspond to the prime ideals of Q(S/P)G(p) = Q(R/PnR). Via the above correspondence O, the primes of Q(S/P)G(P) correspond to a subset of the prime ideals of the skew group ring Q(S/P)*G(P) and, by [21, Theorem 2.5], the primes of Q(S/P)*G(P) in turn correspond in a one-to-one fashion with the G(P)-primes of a certain twisted group algebra C'[Ginn(P)], where C is the centre of Q(S/P), a commutative field. 534 KENNETH A. BROWN AND MARTIN LORENZ Clearly, the number of G(/>)-primes of C'[Ginn(P)] is bounded by dimcC'[Ginn(P)] = |Ginn(P)|. Now let P be a prime ideal of S and let /(•) denote composition length. We define l fP = p(s/p where p(S/P) = 1(Q(S/P)Q(S/P)) is the reduced rank of S/P and U is the unique simple <2(S/P)-module. Since G(P) has a surjective trace on SIP, and hence on Q(S/P), we conclude from 2.2, Lemma 1 that fP is an integer less than or equal to \G(P)\. In Lemma 2 below, we show that often fP does in fact divide \G(P)\. If 5 is FBN x then, by Lemma l,fP = p(S/P)~ • l(Q(S/P)Q(R/PnR)). Since fP is clearly invariant under replacing P by a G-conjugate, we can define, for any p e Spec R, fv =fp> where P is any prime ideal of S lying over p. LEMMA 2. Let P be a prime ideal of S and assume that the skew group ring associated with the action of G{P) on Q(S/P) is simple. Then fP = \G(P)\/l(VQ(S/P)Q{S/P)), where V is a simple right ideal of Q(S/P)G(P). In particular, this happens if G(P) acts by outer automorphisms on Q(S/P). Proof. Put A(P) = Q(S/P)*G(P), the skew group ring associated with the action of G(P) on Q(S/P), and B{P) = Q{S/P)G{P\ If A(P) is simple then the map/in the Chase-Harrison-Rosenberg Morita context, with Q(S/P) in the role of S, is surjective, and since the G(P)-trace maps onto Q(S/P)G{P) as well, it follows that A(P) and B(P) are Morita equivalent. Therefore, B(P) is simple Artinian and so B(P) = Vb, where V is a simple right ideal of B(P). It follows that b Q(S/P) = (VQ(S/P)) and l(Q(S/P)Q(S/n) = b • l(VQ(S/P)Q{S/P)). Furthermore, exactly as in the proof of [26, Theorem 2.7], one shows that Q(S/P) is free of rank \G(P)\ over B(P). Therefore, l(Q(S/P)B{P)) = \G(P)\ • l(B(P)B(P)) = \G(P)\ • b. This proves the formula for fP. The last assertion follows from [26, Theorem 2.7]. Note that Lemma 2 implies that fp = \G(P)\ whenever S/P is a domain and G(P) acts by outer automorphisms on Q(S/P), because Q(S/P) is a division ring in this case. This applies in particular when S/P is commutative, since in this case any faithful action is outer. 2.6. Transfer from S to R: restriction We will assume throughout this subsection that S is FBN. By 2.2, Lemma 2, it follows that R is FBN as well. s Restriction yields exact functors a)l;(5)/^(5)-> JW,-(/?)/a)iy(/?) for all i^j (cf. §1.5). Concentrating on the case where j = i—, we will now describe the GROTHENDIECK GROUPS OF INVARIANT RINGS 535 corresponding map (denoted £,,_ in 1.4, Proposition 2(iii)) The group G acts on the categories 3ft,(S) and 3Kl-(5)/9K,-(S) by conjugating modules, and hence G also acts on the corresponding Grothendieck groups. For j = i-, we know by 1.3, Proposition that K0(2Jl/(S)/3ft/_(S)) is isomorphic to the free abelian group with basis Xt{S) = {Pe Spec5| d(S/P) = i}. In terms of this s description, the action of G on Kl)(^Jli(S)/ Mi-(S)) is simply given by the permutation action of G on the basis Xj(S). Therefore, denoting the G- coinvariants of tfo(3ft,(S)/2tt,_(S)) by ^0(9W/(5)/2«/_(5))c, we have KoWCS)/2MS))cs 0 UP)- (P)eXi(S)IG Similarly, denoting the factor of Kni^Ri{R)IW,-{R)) that is obtained by identify- ing —equivalent primes in Xt{R) by /C0(3K,(/?)/3W,_(/?))~, we have, by definition, With this, we have the following proposition. PROPOSITION. Assume that S is FBN. Then, for each i, restriction from S to R yields an exact sequence Z//pZ->0. Proof By 1.3, Proposition, a Z-basis of /Co(^/(5)/^,_(5)) is given by the elements [UP]h where UP is a fixed uniform right ideal of SIP and P e X,(S). Let (PnR)eKlj$Jli(R)/Wli-(R))~ denote the basis element corresponding to the —equivalence class in Xt(R) consisting of the primes of R that are minimal over PC\R (cf. (2.4, Lemma 2). We will show that, under the map that is given by restriction, we have Since (PnR) = (QnR) holds for P, (26^,(5) if and only if P and Q are G-conjugate (2.4, Lemma 1), this will prove the proposition. Recall from the proof of 1.3, Proposition that UUP\R)(p), where UP\R denotes UP viewed as an /^-module and lx,{UP\R) is defined as follows. Fix a chain 1/PU = 1/,2(/,-, 3... =>*/„ = 0 such that each LJilU,_x is isomorphic to a uniform right ideal of R/q, where q, is some prime ideal of R. Then lp(UP\R) = #{/: q/ = p}. Thus, if /p((/P|n)#0 for pe*,•(/?) then pDannR(l/P) = PD/?, and since d(R/PHR) = i (cf. [12, 536 KENNETH A. BROWN AND MARTIN LORENZ Corollary 12.15]) we conclude that p is minimal over P D R. Conversely, all these primes occur among {c\t: 1 =s / ^ /}, since annR(UP) = PC\R. Hence Pi([Upl) = ( 2 lx,(UP\R))(PnR) = #{l: q, minimal over PnR}(PHR). \>=>pr\R Put A = Q(R/P fl R), which exists by 2.5, Lemma 1. We obtain a chain UP®RA = U,®RA 3 */,-i ®*>1 3... 3 f/,,®!,^ = 0, where LJ,(&RAIII,-X®RA = {U,IUI-X)®RA is a simple right ^4-module if q, is minimal over P D R and zero otherwise. Therefore, letting /(•) denote composi- tion length, we can rewrite the above formula as follows: Finally, using the embedding AcB = Q(S/P) of 2.5, Lemma 1, we see that UP®RA = UPB is a simple right ideal of B, since B = SIP. A. Thus l{Up®RAA)=fP, by (10), which completes the proof. As a first application of the foregoing, we give an upper bound for the rank of G()(R). We continue to use (-)G to denote G-coinvariants. THEOREM 1. Assume that S is FBN. Then rank G()(R) «s rank GO(S)C + 2 (#Q - 1), Q where Q runs over the —equivalence classes in Spec R. This result is of course vacuous if infinitely many —equivalence classes in Spec/? have more than one element. For commutative rings, however, all equivalence classes have just one element, and in 2.7 we will discuss some non-commutative situations where the sum is finite and the estimate is in fact an equality. Here we just mention that in the case when 5 (or equivalently R) is Artinian, we have rank G()(R) = #Spec R and rank GO(S)C = #Spec S/G = #Spec/?/~. Thus equality holds in this case. Proof of Theorem 1. Note that the action of G on G()(5) respects the filtration {Fj(S)}, and hence G acts on each Ft(S). Moreover, rank G{)(S)C = £,rank^(S)c and similarly for rank G0(R). Thus it suffices to show that, for all /, rank F^R) ^ rank ^(5)c + 2 (#Q - 1), where Q runs over the ~-equivalence classes in Xj(R). To this end, define ^(/?)_ to_be the pushout of ^(/?)^^0(^|.(/?)/2K|._(/?))->> Writings Mj = sJft,/Stf,_, we obtain a commutative diagram GROTHENDIECK GROUPS OF INVARIANT RINGS 537 where the vertical maps are surjective and the composite of the two maps in the top row becomes surjective upon tensoring with Q, by the proposition. Thus the same is true of the composite in the bottom row, and hence rank F,(S)G^ rank Ft(R)~. Letting xv e Fj(R) denote the image of the basis element of /C0(XftJ(/?)/3ft,_(/?)) corresponding to the prime p e Xi(R), we can obtain the group F^R)^. from Fj(R) by factoring out the subgroup generated by the elements xv — xq for p ~ q in Xi(R). Thus rank F^R)^ ^ rank F^R) - 2 (#Q - 1), where Q runs over the —equivalence classes in Xj(R). This completes the proof. We now discuss torsion in GQ(R). For this, we will concentrate on situations where the equivalence relation ~ is trivial, that is, all classes have just one element (as is always the case when S is commutative). We put /rl.c.m.{/p: p €*,(*)} = l.c.m.{/P: PeXt(S)}. LEMMA. Assume that S is FBN. If the equivalence relation ~ is trivial on X/(R) then fi annihilates the cokernels of the restriction maps Wj(S)—»Wj(R) and Fj(S)—>Fj(R). This applies, in particular, if G(P) acts by outer automorphisms on Q(S/P) for each P e X^S). In this case, fi divides l.c.m.{|C(P)|: P e X^S)} (and hence also \G\). Proof. Since tfo(3ft,(/?)/Hft,_(/?))-- = A:o(^,(/?)/^,_(/?)), the cokernel of the restriction map A:0W(5)/3K/_(5))^ KOWM/2R/-W) >S annihilated by fi, by the proposition. The first assertion thus follows from the commutative diagram W,iS) ^X Ft(S) I I I *O(2R,(/O/2M/O) where the vertical maps are the restriction maps and the horizontal maps are all surjective (cf. 1.4, Proposition and (3) in 1.2). The assertions about outer actions follow from 2.5, Lemmas l(ii) and 2. As an application of the foregoing we offer the following result. Parts (i) and (iii) extend [5, Proposition 2.3 and Theorem 2.4(i)] to a non-commutative setting. Our assumption on the triviality of Montgomery's equivalence relation is satisfied, of course, when 5 is commutative and, more generally, when G(P) acts by outer automorphisms on Q(S/P) for each prime ideal P of 5, by the above lemma. A non-commutative example will be described in 3.2. THEOREM 2. Assume that S is FBN of finite Krull dimension n = d(S) and that Montgomery's equivalence relation on Spec/? is trivial. 538 KENNETH A. BROWN AND MARTIN LORENZ (i) The cokernel of the restriction map Res£: G()(S) —»G{)(R) is annihilated by nuj, (ii) //F,,_,(5) is torsion then Fn-i(R) is the torsion subgroup of G()(R). (iii) Assume that G0(S) is cyclic. Then where U is a uniform right ideal of R/N (with N the nilpotent radical of R), and Fn_,(/?) is annihilated by Yl'llJ f. Proof. Restriction sends the terms of the filtration {0} = F.^S) c F()(S) c ... c Fn_t(S) <= Fn(S) = C0(S) of Go(5) to the corresponding terms Fj(R) for G0(R) and, by the above lemma, we have (11) /, • F,(R) <= ResJtf-(S)) + /<_,(/?) for all /. Part (i) follows from this. Also, if Fj(S) is a torsion group then so is Fj(R), and so part (ii) follows from the fact that GJFn_x is torsion free, by 1.3, Corollary 1. As to (iii), note that the same corollary and our assumption on G0(S) imply that F,,_,(5) = 0 and that the nilpotent radical of S is prime. Hence, by our assumption on Spec/?, the nilpotent radical of R is prime as well. Part (iii) now follows from 1.3, Corollary 1 together with (11) above. 2.7. Application to certain fixed-point-free actions In this subsection, we describe a situation where the estimate in 2.6, Theorem 1 is in fact an equality. The corresponding result, the proposition below, applies in particular to certain filtered algebras. In 2.8 and 2.9, we will discuss some explicit examples. Throughout this subsection we will keep the following notation. The ring S will be FBN and C will denote the centre of 5. We put D= H C(g), geG\{\) where C(g) is the ideal of C that is generated by the set {c — c8: ceC}. (Of course, it suffices to let c run over a set of generators of C here.) Note that D is a C-invariant ideal of C and, consequently, DT is an ideal of T. We let denote the canonical map. Thus 5 = S/DS, R = R/R D DS = Sc, and f = 5* G is a skew group ring of G over 5. Finally, recall that 6 = d(t). LEMMA, (i) If S is a finite module over C, then Equality holds if S = C is commutative. (ii) D|C|-'c/nC. (iii) Assume that P is a prime ideal of S so that P OR has more than one minimal covering prime in R. Then P D C 3 D. GROTHENDIECK GROUPS OF INVARIANT RINGS 539 Proof, (i) Our assumption on 5 implies that T is a finite normalizing extension of C/I D C. Therefore, by [24, Corollary 10.1.11], we have d = d(C/I n C). Since C/Ifl C is a factor of C/CtCD C, we have d(C/IflC)^d(C/CtCn C), with equality if 5 = C. Furthermore, by [5, Lemma 2.2], II c(g)cacncc n cfe). G\{l) Thus d(C/OCr\C) = max{d(C/C(g)): l^geG}, which proves (i). (ii) This is a consequence of the first of the above inclusions. (iii) Note that 2.5, Lemma l(ii) implies that some geG\{l} yields an inner automorphism of Q(S/P), and hence we have C(g) s P. This proves (iii). PROPOSITION. Assume that 5 = S/DS is Artinian. Then rank G()(T) - rank G0(f) ^ rank G0(R) - rank Gn(R) =s= rank G()(S)C - rank G0(S)G. Furthermore, if we assume rank G0(T) is finite, the first inequality is an equality precisely if the inflation map G0(T)^> G0(T) is injective. Proof. The only non-zero contributions in the sum over Q in 2.6, Theorem 1 come from —classes in SpecR with more than one element. By part (iii) of the lemma above, these classes are all accounted for by the —classes in Spec R. Since (9) implies that # Spec/?/~ = # Spec S/G, the estimate in Theorem 1 becomes (12) rank G()(R) ^ rank G0(S)G + # Spec R - # Spec S/G. Furthermore, since 5 and R are Artinian, we have (13) # Spec R = rank G0(R) and (14) # Spec S/G = rank G0(S)G. Therefore, (12) can be rewritten as rank G0(R) - rank G0(R) ^ rank G0(S)G - rank G0(S)G. In order to prove the first inequality of the proposition, note that the prime ideal correspondence in 2.4 implies that rank GQ(R) = #Spec R = #Spec7 f = rank GQ(f) - rank G0(f), where we have put f = T/I_+ DT = t/I. Part (ii) of the lemma above implies that thecanonical map 71-»"fhas a nilpotent kernel. Therefore, the inflation map a: G0(f)^>G0(T) is an isomorphism, and we get (15) rank G0(R) = rank G0{f) - rank G0(f). Furthermore, the exact sequence (from (8) in 2.3) G0(f) -^Go(r) -^U GoW-* 0 implies that (16) rank G0(R) ^ rank G0(T) - rank GQ(T). 540 KENNETH A. BROWN AND MARTIN LORENZ Assuming the ranks are finite, we see that equality occurs if and only if the inflation map T0: GO(T)^>GO(T) is a monomorphism, because G0(f) is a finitely generated free abelian group. The desired inequality now follows by subtracting (15) from (16), and the proposition is proved. REMARKS. (1) An alternative proof of the first inequality of the proposition which is independent of the assumption that 5 be Artinian is as follows. Consider the commutative diagram G0(f) -^ G0(f) -^ Go(fl) —> 0 I" l*> [Y G0(f) -^ G0(T) -£ G(){R) —> 0 where the rows come from the exact sequence (8) in 2.3 and the vertical maps are the inflation maps. Since the map a is an isomorphism, it follows that Ker j3 maps onto Ker y and Coker /? embeds into Coker y. Consequently, rank G0(R) - rank G0(R) = rank Coker y - rank Ker y s= rank Coker /3 - rank Ker j8 = rank G0(T) - rank G0(f), which proves the first inequality of the proposition. (2) Situations where S = S/DS is Artinian often arise in connection with fixed-point-free actions of G. Specifically, assume that 5 is an algebra over a field k and G acts by A>algebra automorphisms on 5. Furthermore, assume that 5 is a finite module over its centre C and C = k[V] for some G-invariant A>subspace V such that Vl~g = V holds for all 1 *g e G, where Vl~s = {v - vs: v e V). (If V is finite-dimensional, then the latter condition is equivalent with G acting fixed- point-freely on V.) Then V c C{g), and hence D = VC has codimension 1 in C and S = S/VS is finite-dimensional over k. Similar remarks apply when C = k[A] for some G-stable group of units A in C such that Al~s = {a(a~l)g: a eA} has finite index in A for all l^geG. In this case, D does at least have finite codimension in C. If A is free abelian of finite rank, the above condition is equivalent with the action of G on A being fixed-point-free. COROLLARY. Suppose that the following assumptions are satisfied. (a) 5 = U/s=o Sj is filtered, with 50 = k a central subfield of S, and gr 5 is right Noetherian of finite global dimension. The action of the group G respects the filtration {5,} and is trivial on So = k. Furthermore, char k does not divide \G\ and k is a splitting field for G. (b) 5/rad 5 is a central simple k-algebra, where 5 = S/DS. Then the sequence 0 H is exact, and ( rank G0(R) = #Spec R = #Spec (5/rad 5) GROTHENDIECK GROUPS OF INVARIANT RINGS 541 Proof. Put S' = S/radS and V = f/radS • f = S'*G. Since the canonical map of 5 onto 5' has a nilpotent kernel, one has Co(5) = C()(5'), and similarly G0(f) = GO(T'). In particular, since 5" is central-simple, we have (17) rank G^c = 1. Furthermore, since G acts trivially on k, the Noether-Skolem theorem implies that G acts by inner automorphisms on 5'. Therefore, T' = S' ®kE for some twisted group algebra E = kxG of G over k (cf. [28, Proposition 12.4]). Since S' is a central simple A:-algebra, we deduce that T (18) rank G0(f) = rank G()(k G). By (a), we can filter T = S*G by 7; = S,*G. Then T{) = kG and grT = gr(S) * G are both right Noetherian and have finite global dimension (for gr T see [24, Theorem 7.5.6 (iii)]). Thus [30, Theorem 7] (see also [13, 5.2]) implies that induction yields isomorphisms G,(T) = Gt(kG) for all / 3= 0. The same result also yields G,(5) = G,(A:). In particular, G0{S)= ([S]) = G0(S)a has rank 1 and, by (13), (17) and (18), the estimate of the proposition becomes X (19) rank G0(kG) - rank G0(k G) =£ rank G0(R) - #Spec R ^ 0. But kG and kxG are both semisimple algebras and kG is split. Therefore, the rank of G0(kG) equals the k-dimension of the centre of kG, and the rank of x T G0(k G) is at least bounded above by the A:-dimension of the centre of k G. But the dimension of the centre of kG is equal to the number of all conjugacy classes of G, whereas the dimension of the centre of kxG is the number of conjugacy classes of r-special elements of G (cf. [14, Problem 11.8]). Thus x rank G{)(kG) ^ rank G0(k G), and so equality holds throughout in (19). Exactness of the sequence 0—» G()(T)—»G0(/?)—»0 follows from (8) in 2.3, except for injectivity of the inflation map T(): G()(T)-*G0(T) which follows from the proposition. 3. Examples 3.1. The Weyl algebra in positive characteristic In this section, we let 5 = k{x, y}l{xy — yx — 1) denote the Weyl algebra over a field k of characteristic p>2. Thus 5 is a Noetherian domain of Krull dimension d(S) = 2 [24,6.6.14] and 5 is an Azumaya algebra of rank p2 over its centre C = k[V], where V = kxp + kyp [31]. K 8 The group T = SL2(k) acts on S by x = ax + by, y = cx + dy for g = ( eT. Note that this action respects the canonical filtration of S that is \c dl defined by the generators x and y. The action of T maps the generating subspace V of C to itself and the corresponding endomorphisms of V are given by the (ap bp\ (a b\ p p p p p matrices F(g) = ( Jn for g = , so {x f = a x + b y and similarly for \c' dp I \c d) yp (cf. [1,1.2.4]). The map F defines an injective 'Frobenius' endomorphism of Y. Thus, if 1 =£g e F is an element of finite order not divisible by p then F{g) has no eigenvalue 1, and hence g acts fixed-point-freely on V. Thus D = VC holds for each finite p'-subgroup G=sr. Furthermore, S = S/VS = Mp(k) and the 542 KENNETH A. BROWN AND MARTIN LORENZ operation of F on S is inner (as in the proof of 2.7, Corollary). Hence it is given by a homomorphism y: T^?GLp(k). Now let G be a finite p'-subgroup of F and assume that A: is a splitting field for G. Then assumptions (a) and (b) of 2.7, Corollary are satisfied, and so we obtain that Y(G) rank G0(R) = #Spec Mp(k) . Y< The cyclic group tm. Assume that p \ m and that k contains a root of unity e of order m. Put c = ( _, ] e F and G = (c) = Zm. Then y(c) can be represented (/ +1)/2 by the diagonal matrix dc = diag(e'~ ' : 1=1,..., p). If m>p then the centralizer of dc in Mp{k) consists of exactly the diagonal matrices and so we have Y{G) p Mp(k) = kx kx ... x /c = k . Thus our equality above becomes rank G0(R)= p. The binary dihedral group Ow. This is the subgroup of F that is generated by the above cyclic group Z2m together with the matrix d = ( j. The order of Dm is Am which we assume to be relatively prime to p and greater than 2p. One P checks that y(d) can be represented by the matrix ud = Y. =o (—1)'/! Ep_u+X where ELj has 1 in the (/, y)-position and O's elsewhere, as usual. The diagonal matrices which centralize ud are exactly those whose (1, l)-entry is identical with the (p, p)-entry, the (2, 2)-entry is the same as the (p — 1, p — l)-entry etc., and Y(G) (p+l)/2 so we have Mp(k) = k . Therefore, rank G0(R) = \{p + 1). We remark that the situation for characteristic 0 is easier, because S is a simple ring in this case. It follows that R is Morita equivalent to T, for all G =s r, and so one has G0(R) = G(){kG) (cf. [1]). 3.2. The quantum plane for a root of unity The quantum plane is the algebra S = k{x, y}/(xy - qyz), where qek*. We assume that q is a root of unity of order m. If m > 2 then Aut^S) = k* x k*, with (a,b)ek*xk* acting by JC->«JC, y*-^by. For m = 2, one also has the auto- morphisms x »-> ay, y>-+ bx. Note that all these automorphisms respect the grading of S that is given by total degree in x and y. Thus, with the filtration of S coming from this grading, hypothesis (a) of 2.7, Corollary is satisfied for any finite subgroup G c k* xk*, because 5 is Noetherian and has global dimension 2. The centre of 5 is the polynomial ring C = k[%, rj] with %=xm and r)=ym. Thus, assuming that m does not divide \G\, we obtain D = (§, rj), the ideal of C that is generated by § and rj, and so 5 = S/DS has dimension m2 over k. The radical of S is generated by the images of x and y, and hence we have 5/rad S = k. The corollary therefore implies that rank G{)(R) = 1. GROTHENDIECK GROUPS OF INVARIANT RINGS 543 Since G0(S)= ([S]) (cf. the proof of 2.7, Corollary), this could also have been deduced from 2.6, Theorem 2 because, in view of 2.7, Lemma (iii), the above description of 5/rad S implies that Montgomery's equivalence relation on Spec R is trivial. 3.3. The localized quantum plane for a root of unity Here we consider the case where S is obtained from the quantum plane in 2.9 by inverting x and y. Thus S = k{x±l, y±[}l(xy — qyx), where q e k*. In addition to the automorphisms coming from the quantum plane, one now has a a b c d 'multiplicative' action of SL2(Z) on S, with I ) acting by x*->x y , y>-^x y . \c a) We continue to assume that q is a root of unity of order m. In this case, 5 is an Azumaya algebra of rank m2 over its centre C = k[x±m, y±m] (cf. [18, Lemma 1.1 and its proof]). Our above methods apply to actions of finite subgroups G of SL^Z) (so G is cyclic of order 2, 4, or 6), even though 2.7, Corollary does not cover this type of action. We illustrate this by discussing the case G = (g = -Id) = C2 for k algebraically closed of characteristic not equal to 2. Thus xs =x~l and ys = y~\ and the ideal D of C is generated by the elements x2m - 1 and y2m - 1. Therefore, C = k(Z/2Z)2, the group algebra of (Z/2Z)2 with trivial G-action, and we obtain that 4 (20) G0(S)c = G0(C) = Z . More explicitly, e,6 = ±\ m m 2 with SBt8 = S/(x - e, y — d) a central-simple fc-algebra of dimension m . (In fact, Se,s = (e, 6)m, in the notation of [33, p. 194].) Therefore, f = 0e,6=±, fe>a T T with Te 6 = Se 8* G = Se6®k k G for some twisted group algebra k G, because the Noether-Skolem theorem implies that G acts by inner automorphisms on 2 Se 8- Since H (G, k*) = 0, the twist is actually trivial, and so we conclude that f 2 (21) rankG0(f) = 8. Furthermore, one finds that U if 0 = 1, (22) rankG (£)= 2 rank G (Sj ) = ] 7 if q =-1, () 0 6 U The Grothendieck-Quillen theorem (cf. [29, Theorem 14.7]) implies that Go(S)=([S]) = Go(S)G, and so (23) rank G0(5)c = Z. The group G0(T) can be determined by means of Moody's theorem [27]. Namely, T is a twisted group algebra, T = kT with r = (Z0Z)X|G = {a, P, y: ay = ya~\ fiy = y/5"1, (x$ = $oc, y2 = 1). 544 KENNETH A. BROWN AND MARTIN LORENZ A full set of non-conjugate maximal finite subgroups of T is: //, = (y), H2=(OCY), H3=(PY), and H4=( l T,; = k{xt) with x, = g, x2 = xg, x3 = yg, x4 = r~ xyg, where r ek* is chosen so that q = r2. Therefore, Moody's theorem implies that G0(T) is generated by the elements [etT] (0 ^ i ^ 4) with e0 = 1, e, = ^(1 + *,-). These generators are in fact Z-independent. This can be checked by identifying T in an obvious way with a direct factor of a polycyclic group algebra k^Q, and G0(T) with the corresponding factor of G(,(/c^). Here, «=<£ n, £, y: y%=rlY, Yn = rXY, [I V) = ?> K, §] = [£ V] = h Z2m = l = Y2). Since the rank of G^k^) can be computed from [19] (it is 10m), the generators for GoikW) that are obtained by using Moody's theorem in a similar fashion to above turn out to be Z-independent which in turn yields the independence of the above generators for GQ(T). Therefore, 5 (24) G()(7>Z . In view of (20)-(24), 2.7, Proposition yields the following final result: (l if q = l, rank G0(R) = \ 4 if q =-1, 15 if?#±l. The result for q = 1 also follows from 2.6, Theorem 2(iii), since 5 is commutative in this case. In fact, as a special case of [5, 3.7], we know more precisely that The result for q =£ ±1 shows that R is Morita equivalent with T in this case. (This follows from 2.7, Proposition: the inflation map G0(T)—> G()(T) is injective, with image 0, whence 7 = 0.) Therefore, (24) implies that We conclude by remarking that Morita equivalence of the fixed ring R with T for all but finitely many choices of q is not particular to the inversion action. To see this, note that S is an image of the group algebra kffl of the discrete Heisenberg group via §•-»*, rji-^y. The centre of kW is &[£*'], where £ = [£, 77], and the localization of kW at the non-zero elements of &[£*'] is a simple ring, in fact a localized quantum plane for a non-root of unity, which we will denote by 50. Now let G be a finite subgroup of SL2(Z), with \G\ not zero in k, and let G act on So as above. Then it is easy to see that this action is outer. (The commutator subgroup of the group of units of 50 consists of the powers of £.) Therefore, the skew group GROTHENDIECK GROUPS OF INVARIANT RINGS 545 ring To = So* G is a simple ring [26, Theorem 2.3], and so we have SQtSQ = To. We obtain that for some non-zero ar(£) efc[£±l]. (For the inversion action, ar(£) = £2-l.) 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