Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

Asadollahi, Javad; Salarian, Shokrollah

doi:10.1080/00927870600639815

Cited by 26 publications

(17 citation statements)

References 12 publications

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“…To construct a proper left injective resolution of N , it is sufficient to prove that K has the same properties as N , including (1). Proof.…”

Section: Claimmentioning

confidence: 99%

“…In the treatment by Christensen, Frankild, and Holm [6], these invariants were considered for complexes with bounded homology. It is, however, possible to define Gorenstein projective dimension for unbounded complexes: This was done by Veliche [21], and the dual case of Gorenstein injective dimension was treated by Asadollahi and Salarian [1].…”

Section: Introductionmentioning

confidence: 99%

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

2016

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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

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“…To construct a proper left injective resolution of N , it is sufficient to prove that K has the same properties as N , including (1). Proof.…”

Section: Claimmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2016

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“…We have rid(T A ) = Rid(T A ) = 0 by Lemma 3.1. Now we take T A ∈ mod-A with id(T A ) = m. By the remark after [2], Theorem 2.3, we have 0 = rid(T A ) = Rid(T A ) < Gid(T A ) = id(T A ) < ∞.…”

Section: Finitistic Dimension Of Endomorphism Ringsmentioning

confidence: 99%

Finitistic dimension and restricted injective dimension

2015

Czech Math J

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“…Veliche [10] extended the concept of Gorenstein projective dimension of homologically bounded complexes to the setting of unbounded complexes over associative rings. Dually, Asadollahi and Salarian [1] introduced the concept of Gorenstein injective dimension of unbounded complexes over associative rings.…”

Section: Introductionmentioning

confidence: 99%

Gorenstein Dimensions of Unbounded Complexes Under Base Change

Wu¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. Transfer of homological properties under base change is a classical field of study. Let R → S be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between U ⊗ L R X (or RHom R (X, U )) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities G-dim S (U ⊗ L R X) = G-dim R X + pd S U and Gid S (RHom R (X, U )) = G-dim R X + id S U hold.

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Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

Cited by 26 publications

References 12 publications

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

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

Finitistic dimension and restricted injective dimension

Gorenstein Dimensions of Unbounded Complexes Under Base Change

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