Complete homology over associative rings

Celikbas, Olgur; Christensen, Lars Winther; Liang, Li; Piepmeyer, Greg

doi:10.1007/s11856-017-1562-3

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…Like stable homology, it is a generalization of Tate homology. We compare these two generalizations in [8]. From the point of view of stable homology, it is interesting to know when it agrees with complete homology, because the latter has a universal property.…”

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confidence: 99%

“…From the point of view of stable homology, it is interesting to know when it agrees with complete homology, because the latter has a universal property. In this direction the main results in [8] are that these two homology theories agree over Iwanaga-Gorenstein rings, and for finitely generated modules over Artin algebras and complete commutative local rings. Moreover, the two theories agree with Tate homology, whenever it is defined, under the exact same condition; that is, if and only if every Gorenstein projective module is Gorenstein flat; see Theorem 6.7.…”

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confidence: 99%

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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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Stable homology over associative rings

Celikbas

Christensen

Liang

et al. 2017

Trans. Amer. Math. Soc.

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“…Thus the above definitions of bounded cohomology and stable cohomology are independent of the choices of V-coresolutions. In view of Proposition 2.4, it can be proved similarly as in [16,Theorem 4.4] (see also [6,Appendix B]) that stable cohomology is actually the cohomology given in Definition 3.2.…”

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confidence: 83%

Stable functors and cohomology theory in Grothendieck categories

Guo¹,

Li²

2020

Preprint

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“…Stable homology, as a broad generalization of Tate homology to the realm of associative rings, was introduced by Vogel and Goichot [9], and further studied by Celikbas, Christensen, Liang and Piepmeyer [2,3], and Emmanouil and Manousaki [6]. In their paper [2], it is shown that the vanishing of stable homology over commutative noetherian local rings can detect modules of finite projective (injective) dimension, even of finite Gorenstein dimension, which lead to some characterizations of classical rings such as Gorenstein rings, the original domain of Tate homology.…”

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confidence: 99%