Absolute, Gorenstein, and Tate Torsion Modules

Iacob, Alina

doi:10.1080/00927870601169275

Cited by 26 publications

(22 citation statements)

References 6 publications

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“…3] Tate homology was defined for modules over Iwanaga-Gorenstein rings and shown to agree with stable homology. Iacob [24] generalized Tate homology further to a setting that includes the case where M is a finitely generated R • -module of finite G-dimension. We prove: Since R is not Gorenstein, E has infinite G-dimension; see [22, thm.…”

Section: Introductionmentioning

confidence: 99%

Stable homology over associative rings

Celikbas

Christensen

Liang

et al. 2017

Trans. Amer. Math. Soc.

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Abstract. We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here.Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings-among them Gorenstein rings, the original domain of Tate homology. Closely related to gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone.Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced.

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Section: Introductionmentioning

confidence: 99%

Stable homology over associative rings

Celikbas

Christensen

Liang

et al. 2017

Trans. Amer. Math. Soc.

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“…Tate (co)homology was originally defined for modules over finite group algebras. Through works of Iacob [14] and Veliche [21] the theories have been generalized to the extent that one can talk about Tate homology Tor R (M, N ) and Tate cohomology Ext R (M, N ) for modules over any ring, provided that the first argument, M , has a complete projective resolution; see 3.1. Stable cohomology and the P-completion of covariant Ext always agree, and they coincide with Tate cohomology whenever the latter is defined; see Appendix B.…”

Section: Comparison To Tate Homologymentioning

confidence: 99%

Complete homology over associative rings

et al. 2017

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“…We start by recalling several definitions and terminology from [3,7,17]. Throughout we use homological notation for complexes of R-modules.…”

Section: A G-relative Derived Depth Formulamentioning

confidence: 99%

Vanishing of relative homology and depth of tensor products

2017

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For finitely generated modules M and N over a Gorenstein local ring R, one has depth M + depth N = depth(M ⊗ R N ) + depth R, i.e., the depth formula holds, if M and N are Tor-independent and Tate homology Tor i (M, N ) vanishes for all i ∈ Z. We establish the same conclusion under weaker hypotheses: if M and N are G-relative Tor-independent, then the vanishing of Tor i (M, N ) for all i 0 is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of G-relative homology.2010 Mathematics Subject Classification. 13D07; 13D02.

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Absolute, Gorenstein, and Tate Torsion Modules

Cited by 26 publications

References 6 publications

Stable homology over associative rings

Stable homology over associative rings

Complete homology over associative rings

Vanishing of relative homology and depth of tensor products

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