Vanishing of local homology

“…In case M is bounded above, the next Proposition is [8,Proposition 2.7]. We however wish to apply this result to the Hochschild complex which is bounded below, so we give a proof that works for complexes without any boundedness condition.…”

Section: Then There Is a Functorial Isomorphismmentioning

confidence: 99%

Hochschild cohomology commutes with adic completion

Shaul¹

2016

Algebra Number Theory

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“…I would like to thank the referee for some very useful remarks, and Anders Frankild for showing me [1].…”

Section: Acknowledgementmentioning

confidence: 99%

Ext vanishing and infinite Auslander-Buchsbaum

Jørgensen

2004

Proc. Amer. Math. Soc.

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Abstract. A vanishing theorem is proved for Ext groups over non-commutative graded algebras. Along the way, an "infinite" version is proved of the non-commutative Auslander-Buchsbaum theorem. IntroductionLet R be a noetherian local commutative ring, and let X be a finitely generated R-module of finite projective dimension. The classical Auslander-Buchsbaum theorem states pd X = depth R − depth X.This can also be phrased as an Ext vanishing theorem, namely, if M is any Rmodule, then(1) Ext i R (X, M ) = 0 for i > depth R − depth X. A surprising variation of this is proved in [1]: Suppose that R is complete in the m-adic topology. Then equation (1) remains true if X is any R-module of finite projective dimension, provided M is finitely generated. In other words, the condition of being finitely generated is shifted from X to M .In Theorem 2.3 below, this result will be generalized to the situation of a noncommutative noetherian N-graded connected algebra.The route goes through an "infinite" version of the non-commutative AuslanderBuchsbaum theorem, given in Theorem 1.4. This result is a substantial improvement of the original non-commutative Auslander-Buchsbaum theorem, as given in [3, thm. 3.2], in that the condition of dealing only with finitely generated modules is dropped.The notation of this paper is standard and is already on record in several places such as [2] or [3]. So I will not say much, except that throughout, k is a field, and A is a noetherian N-graded connected k-algebra. However, let me give one important word of caution: Everything in sight is graded. So for instance, D(A) stands for D(Gr A), the derived category of the abelian category Gr(A) of Z-graded A-left-modules and graded homomorphisms of degree zero.

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Vanishing of local homology

Cited by 26 publications

References 18 publications

Detecting completeness from Ext-vanishing

Detecting completeness from Ext-vanishing

Hochschild cohomology commutes with adic completion

Ext vanishing and infinite Auslander-Buchsbaum

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