Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

Abstract

For a commutative ring R and a faithfully flat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S ⊗ R M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom R (S, N ) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change. Date: 12 May 2016. 2010 Mathematics Subject Classification. 13D05; 13D02. Key words and phrases. Gorenstein injective dimension, faithfully flat co-base change, Gorenstein flat dimension, faithfully flat base change.This result compares to the statement about modules in Theorem 1.2.By significantly relaxing the conditions on the rings, the results of this paper improve results obtained by Christensen and Holm [7], by Christensen and Sather-Wagstaff [11], and by Liu and Ren [22]. Details pertaining to Theorems 1.1 and 1.2 are given in Remarks 4.13 and 6.8; the trend is that the rings in [11,22] are assumed to be commutative noetherian and, more often than not, of finite Krull dimension.The paper is organized as follows: In section 2 we set the notation and recall some background material. Sections 3-4 focus on the Gorenstein injective dimension, and Sections 5-6 deal with the Gorenstein flat dimension. Section 7 has some closing remarks and, finally, an appendix by Bennis answers a question raised in an earlier version of this paper. ComplexesLet R be a ring with identity. We consider only unitary R-modules, and we employ the convention that R acts on the left. That is, an R-module is a left R-module, and right R-modules are treated as modules over the opposite ring, denoted R • .2.1 Complexes. Complexes of R-modules, R-complexes for short, is our object of study. Let M be an R-complex. With homological grading, M has the formone switches to cohomological grading by settingFor n ∈ Z the symbol M n denotes the quotient complex of M with (M n ) i = M i for i n and (M n ) i = 0 for i < n.The subcomplexes B(M ) and Z(M ) of boundaries and cycles, the quotient complex C(M ) of cokernels, and the subquotient complex H(M ) of homology all have

A refinement of Gorenstein flat dimension via the flat--cotorsion theory

“…For an R-complex N and i ∈ Z set Ext i FC (M, N ) = H −i (Hom R (T, N )) . For a complex of finite Gorenstein flat dimension, every complete flat-cotorsion resolution is per the next lemma a Tate flat resolution in the sense of [19,7].…”

Five theorems on Gorenstein global dimensions

“…A (F, G) = 0 holds this sequence splits, which means that F is a summand of I and hence injective. This argument can be souped up to yield the next theorem, which is dual to [7, Theorem 1.1] and subsumes [18,Theorem 2.1] Recall, for example from [8,Proposition 3.6], that the Gorenstein injective dimension of an A-complex M can be defined as the least integer n such that (1) H v (M ) = 0 holds for all v < −n and (2) there exists a semi-injective A-complex I, isomorphic to M in the derived category, such that the cycle submodule Z −n (I) is Gorenstein injective.…”

Gorenstein weak global dimension is symmetric

“…Let M be an A ‐complex with for some integer d . Let be a semi‐projective resolution; the module is Gorenstein flat, see Christensen, Köksal, and Liang [9, Prop. 5.12] 1 , and it suffices to show that holds, as this implies that is Gorenstein projective.…”