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Rigidity of Ext and Tor with Coefficients in Residue Fields of a Commutative Noetherian Ring

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press:  16 November 2018

Lars Winther Christensen ,
Srikanth B. Iyengar  and
Thomas Marley
Lars Winther Christensen
Affiliation:
Texas Tech University, Lubbock, TX 79409, USA ([email protected])
Srikanth B. Iyengar
Affiliation:
University of Utah, Salt Lake City, UT 84112, USA ([email protected])
Thomas Marley
Affiliation:
University of Nebraska-Lincoln, Lincoln, NE 68588, USA ([email protected])
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Abstract

Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k (𝔭), M) = 0 for some n ⩾ R𝔭, where k(𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $(k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.

Keywords

ExtTorrigidityflat dimensioninjective dimension

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13D05: Homological dimension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 305 - 321
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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