Some Homological Properties of Modules over a Complete Intersection, with Applications

Dao, Hailong

doi:10.1007/978-1-4614-5292-8_10

Cited by 16 publications

(10 citation statements)

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“…Theorem B.1 and its variations have been analyzed, used, and studied in the literature; see [10] and [16] for some history and many consequences of the theorem. The following result [25,Theorem 2.7] played an important role in its proof.…”

Section: Appendix a An Application Of Pushforwardsmentioning

confidence: 99%

Criteria for vanishing of Tor over complete intersections

et al. 2015

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In this paper we exploit properties of Dao's η-pairing [12] as well as techniques of Huneke, Jorgensen, and Wiegand [24] to study the vanishing of Tor i (M, N ) for finitely generated modules M , N over complete intersections. We prove vanishing of Tor i (M, N ) for all i ≥ 1 under depth conditions on M , N , and M ⊗ N . Our arguments improve a result of Dao [13] and establish a new connection between the vanishing of Tor and the depth of tensor products.

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Section: Appendix a An Application Of Pushforwardsmentioning

confidence: 99%

Criteria for vanishing of Tor over complete intersections

et al. 2015

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“…Tor-rigidity is a subtle property, but examples of such modules are abundant in the literature. Here we record a few examples and refer the reader to [12] The key ingredient of our argument is the following result; it allows us to tackle the problem on hand by using the Tor-rigidity property; see 2.7.…”

Section: 7mentioning

confidence: 99%

An extension of a depth inequality of Auslander

Celikbas¹,

Le²,

Matsui³

2021

Preprint

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In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for n-Tor-rigid modules, and obtain an affirmative answer for 2-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

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“…For instance, we do not know if they hold even for one-dimensional Gorenstein or complete intersection domains. In fact, it may be true that over a one-dimensional complete intersection domain, a rigid MCM module is free, but it is known only when R is a hypersurface (see [Dao13], especially Section 9).…”

Section: Questions and Conjecturesmentioning

confidence: 99%

Representation schemes and rigid maximal Cohen–Macaulay modules

Dao

Shipman

2016

Sel. Math. New Ser.

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Let k be an algebraically closed field and A be a finitely generated, centrally finite, nonnegatively graded (not necessarily commutative) k-algebra. In this note we construct a representation scheme for graded maximal Cohen-Macaulay A modules. Our main application asserts that when A is commutative with an isolated singularity, for a fixed multiplicity, there are only finitely many indecomposable rigid (i.e, with no nontrivial self-extensions) MCM modules up to shifting and isomorphism. We appeal to a result by Keller, Murfet, and Van den Bergh to prove a similar result for rings that are completion of graded rings. Finally, we discuss how finiteness results for rigid MCM modules are related to recent work by Iyama and Wemyss on maximal modifying modules over compound Du Val singularities.

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Some Homological Properties of Modules over a Complete Intersection, with Applications

Cited by 16 publications

References 50 publications

Criteria for vanishing of Tor over complete intersections

Criteria for vanishing of Tor over complete intersections

An extension of a depth inequality of Auslander

Representation schemes and rigid maximal Cohen–Macaulay modules

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