Vanishing of Ext and tor over fiber products

Saeed Nasseh; Sean Sather-Wagstaff

doi:10.1090/proc/13633

Vanishing of Ext and tor over fiber products

Saeed Nasseh
, Sean Sather-Wagstaff

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

Abstract

Consider a non-trivial fiber product R = S ×_k T of local rings S, T with common residue field k. Given two finitely generated R-modules M and N, we show that if Tor^R_i (M, N) = 0 = Tor^R_i+1(M, N) for some i ≥ 5, then pd_R (M) ≤ 1 or pd_R (N) ≤ 1. From this, we deduce several consequences, for instance, that R satisfies the Auslander-Reiten Conjecture.

Original language	English
Pages (from-to)	4661-4674
Number of pages	14
Journal	Proceedings of the American Mathematical Society
Volume	145
Issue number	11
DOIs	https://doi.org/10.1090/proc/13633
State	Published - 2017

Scopus Subject Areas

General Mathematics
Applied Mathematics

Keywords

Auslander-Reiten
Ext-index
Ext-vanishing
Fiber product
Injective dimension
Projective dimension
Semidualizing complexes
Tor-vanishing

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10.1090/proc/13633

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title = "Vanishing of Ext and tor over fiber products",

abstract = "Consider a non-trivial fiber product R = S ×k T of local rings S, T with common residue field k. Given two finitely generated R-modules M and N, we show that if TorRi (M, N) = 0 = TorRi+1(M, N) for some i ≥ 5, then pdR (M) ≤ 1 or pdR (N) ≤ 1. From this, we deduce several consequences, for instance, that R satisfies the Auslander-Reiten Conjecture.",

keywords = "Auslander-Reiten, Ext-index, Ext-vanishing, Fiber product, Injective dimension, Projective dimension, Semidualizing complexes, Tor-vanishing",

author = "Saeed Nasseh and Sean Sather-Wagstaff",

note = "Publisher Copyright: {\textcopyright} 2017.",

year = "2017",

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volume = "145",

pages = "4661--4674",

journal = "Proceedings of the American Mathematical Society",

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TY - JOUR

T1 - Vanishing of Ext and tor over fiber products

AU - Nasseh, Saeed

AU - Sather-Wagstaff, Sean

PY - 2017

Y1 - 2017

N2 - Consider a non-trivial fiber product R = S ×k T of local rings S, T with common residue field k. Given two finitely generated R-modules M and N, we show that if TorRi (M, N) = 0 = TorRi+1(M, N) for some i ≥ 5, then pdR (M) ≤ 1 or pdR (N) ≤ 1. From this, we deduce several consequences, for instance, that R satisfies the Auslander-Reiten Conjecture.

AB - Consider a non-trivial fiber product R = S ×k T of local rings S, T with common residue field k. Given two finitely generated R-modules M and N, we show that if TorRi (M, N) = 0 = TorRi+1(M, N) for some i ≥ 5, then pdR (M) ≤ 1 or pdR (N) ≤ 1. From this, we deduce several consequences, for instance, that R satisfies the Auslander-Reiten Conjecture.

KW - Auslander-Reiten

KW - Ext-index

KW - Ext-vanishing

KW - Fiber product

KW - Injective dimension

KW - Projective dimension

KW - Semidualizing complexes

KW - Tor-vanishing

UR - https://www.scopus.com/pages/publications/85029593939

U2 - 10.1090/proc/13633

DO - 10.1090/proc/13633

M3 - Article

SN - 0002-9939

VL - 145

SP - 4661

EP - 4674

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 11

ER -

Vanishing of Ext and tor over fiber products

Abstract

Scopus Subject Areas

Keywords

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