Proceedings of the Edinburgh Mathematical Society (2005) 48, 75–90 c DOI:10.1017/S0013091503001056 Printed in the United Kingdom DUALIZING MODULES AND n-PERFECT RINGS EDGAR E. ENOCHS1, OVERTOUN M. G. JENDA2 AND J. A. LOPEZ-RAMOS´ 3 1Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA ([email protected]) 2Department of Mathematics, Auburn University, AL 36848-5310, USA ([email protected]) 3Departamento de Algebra y An´alisis Matem´atico, Universidad de Almer´ıa, 04120 Almer´ıa, Spain ([email protected]) (Received 2 December 2003) Abstract In this article we extend the results about Gorenstein modules and Foxby duality to a non- commutative setting. This is done in § 3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote § 2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of § 3, i.e. when we have a dualizing module. Keywords: n-perfect ring; dualizing module; Gorenstein modules 2000 Mathematics subject classification: Primary 16D20 1. Introduction Grothendieck in [11] introduced the notion of a dualizing complex for a commutative Noetherian ring R. A dualizing module (or canonical module) for R is a module whose deleted injective resolution is a dualizing complex for R. Gorenstein local rings are pre- cisely those local rings for which R itself is a dualizing module for R. All these notions have been extensively developed and applied in commutative algebra and algebraic geometry. In recent years, it has become clear that all these notions have useful non-commutative versions which can be applied in various areas of algebra such as the theory of modular group representations. Bass had noted that Gorenstein local rings are precisely those whose self injective dimension is finite (see [2]). But integral group rings of finite groups (and, in fact, of many infinite groups) have the same property. So these rings can also be said to serve as their own dualizing modules. In this article we consider a general notion of a dualizing bimodule for a pair of rings S and R. We use this terminology since we want to generalize the main result of [9]to 75 Downloaded from https://www.cambridge.org/core. 04 Oct 2021 at 02:26:12, subject to the Cambridge Core terms of use. 76 E. E. Enochs, O. M. G. Jenda and J. A. L´opez-Ramos the non-commutative setting, that is, we want to characterize the Auslander and Bass classes which arise in this situation in terms of the Gorenstein injective and projective dimensions. We note that the language of cotilting theory is also appropriate in this setting. An important property of modules over rings admitting a dualizing module is that flat modules have finite projective dimensions (cf. [14, Corollary 3.2.7] or [15, Theorem 4.2.8]). In § 2 we introduce a class of rings with this property. 2. n-perfect rings Left perfect rings R are characterized in Bass’s Theorem P as those rings such that every left flat R-module is projective (cf. [2]). In this sense the following definition is an extension of that of left perfect rings. Definition 2.1. A ring R is said to be left (right) n-perfect if every left (right) flat R-module has projective dimension less or equal than n. Example 2.2. As we have noted before, it is immediate that left perfect rings are left 0-perfect, and so a ring may be left n-perfect and not right n-perfect (see [1, Exercise 2, p. 322]). Example 2.3. If R is a local commutative Noetherian ring of Krull dimension d, then R is a left (and right) d-perfect ring (cf. [15, Theorem 4.2.8]). Example 2.4. Let R be a left Noetherian ring such that id(RR) n. Then, by [6, Proposition 9.1.2], R is left n-perfect. In particular, if R is n-Gorenstein, i.e. R is left and right Noetherian and id(RR), id(RR) n, then R is left and right n-perfect. Example 2.5. If R is a left n-perfect ring, then R[x]is(n + 1)-perfect. To show this, ∈ let F be a flat left R[x]-module. Then F =− lim→Pi, where Pi R[x]-Mod are projective and so are direct summands of a free R[x]-module. Now since R[x] is a free left R-module, it follows that F =− lim→Pi, with Pi projective in R-Mod. But then F [x] =− lim→Pi[x], where Pi[x] are projective in R[x]-Mod, and so F [x] is flat in R[x]-Mod. Now suppose that pd(F ) n and let 0 → Pn →···→P1 → P0 → F → 0, with each Pi projective. Then 0 → Pn[x] →···→P1[x] → P0[x] → F [x] → 0 shows that pd(F [x]) n in R[x]-Mod, and since we have an exact sequence 0 → F [x] → F [x] → F → 0, it follows that pd(F ) n +1inR[x]-Mod. Example 2.6. Let R be a filtered ring and G(R) be its associated graded ring. If G(R) ∈ is left n-perfect, then R is n-perfect. Let F R-Mod be a flat module. Then F =− lim→Pi, Downloaded from https://www.cambridge.org/core. 04 Oct 2021 at 02:26:12, subject to the Cambridge Core terms of use. Dualizing modules and n-perfect rings 77 with Pi finitely generated projective R-modules. Now, by [12, Proposition I.4.2.2], if we take a good filtration on each Pi, we get that G(F )=G(lim−→Pi) =− lim→G(Pi), and so, using Proposition I.6.5 and Lemma I.5.4 of [12], G(F ) is flat in G(R)-gr. Therefore, there exists an exact sequence in G(R)-gr 0 → Pn →···→P1 → P0 → G(F ) → 0, where each Pi is projective in R-Mod. Now we use Eilenberg’s trick, i.e. if P is pro- jective, then there is a free module X such that P ⊕ X is free (if P ⊕ P is free, take X = P ⊕ P ⊕ P ⊕···), and so we can take the direct sum of the preceding complex with complexes of the form 0 →···→0 → X −id→ X → 0 →···→0, where X is a gr-free graded G(R)-module, to get a projective resolution 0 → Fn →···→F1 → F0 → G(F ) → 0, where every Fi is gr-free. Now, by [12, Lemma 6.2.4], there exist filtered free R-modules Hi such that G(Hi)=Fi for all i =1,...,n, and so, by [12, Lemma 6.2.6], we get an exact sequence in R-Mod 0 → Hn →···→H1 → H0 → F → 0, which shows that pd(F ) n. Now, from Examples 2.5 and 2.6, it is possible to get numerous examples of classical rings that are left n-perfect. 3. Gorenstein modules In this section we characterize Gorenstein modules in terms of the so-called Auslander and Bass classes and generalize results obtained in [9]. We point out that similar results have been recently obtained for the commutative case in [4]. We recall from [5] that a left R-module N is said to be Gorenstein injective if there 0 1 exists an exact sequence of injective left R-modules ··· → E1 → E0 → E → E → ··· 0 1 such that N = Ker(E → E ) and that remains exact when HomR(E,·) is applied for every injective left R-module E. Gorenstein projective modules are defined dually. Definition 3.1. Let R and S be right and left Noetherian rings, respectively, and let SVR be an S–R-bimodule such that EndS(V )=R and EndR(V )=S. Then V is said to be a dualizing module if it satisfies the following three conditions. (i) id(SV ) r and id(VR) r. i i (ii) ExtR(V,V ) = ExtS(V,V ) = 0 for all i 1. (iii) SV and VR are finitely generated. Downloaded from https://www.cambridge.org/core. 04 Oct 2021 at 02:26:12, subject to the Cambridge Core terms of use. 78 E. E. Enochs, O. M. G. Jenda and J. A. L´opez-Ramos Example 3.2. If R is a Cohen–Macaulay local ring of Krull dimension d admitting a dualizing module Ω (see [9]), then Ω is a dualizing module in this sense. Example 3.3. If R is an n-Gorenstein (not necessarily commutative), then RRR is a dualizing module. Example 3.4. Let R and S be strongly graded rings over finite groups, right and left Noetherian, respectively, and let Se VRe be a dualizing module (for Re and Se). Then ⊗ ⊗ W = S Se V Re R is a dualizing module (for R and S). Let us show this: EndS(W ) = HomS(W, W ) ∼ ⊗ = HomS-gr(W, W ) Re R ∼ ⊗ = HomSe (V,V ) Re R ∼ ⊗ = Re Re R ∼ = R. 0 r Analogously, EndR(W )=S.If0→ V → E → ··· → E → 0 is an injective resolution → ⊗ → 0 ⊗ →···→ r ⊗ → of V , then 0 V Re R E Re R E Re R 0 is an injective resolution of WR and, analogously, id(SW ) n. i ∼ i ⊗ ∼ i ExtS(W, W ) = ExtS-gr(W, W ) Re R = ExtSe (V,V )=0 i for all i 1 and, analogously, ExtR(W, W ) = 0 for all i 1.