On vanishing of certain Ext modules

2007/01/06 by Shirô Gotô, Shiro Goto, Futoshi Hayasaka +4
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.AC #msc:13D05 #msc:13D07

paper · pdf · doi:10.48550/arxiv.math/0701195

15 pages, minor changes, to appear in Journal of the Mathematical Society of Japan

arxiv created 2008/07/08 · arxiv updated 2009/12/01

Abstract

Let R be a Noetherian local ring with the maximal ideal m and dim R=1. In this paper, we shall prove that the module Ext¹_R(R/Q,R) does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m² kills the 0th local cohomology module H_m⁰(R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

Related

• On the Vanishing and the Finiteness of Supports of Generalized Local Cohomology Modules
• An elementary proof of Grothendieck's Non-vanishing Theorem
• A characterization of modules locally of finite injective dimension
• On the vanishing of local homology modules
• A note on the vanishing of certain local cohomology modules