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 /?.