Vanishing of (co)homology over deformations of Cohen-Macaulay local rings of minimal multiplicity

Abstract

Let $R$ be a $d$-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $S:=R/(\mathbf{f})$, where $\mathbf{f}:=f_1,\dots ,f_c$ is an $R$-regular sequence. Suppose $M$ and $N$ are maximal CM $S$-modules. It is shown that if ${\mathrm{Ext}}_S^i(M,N)=0$ for some $(d+c+1)$ consecutive values of $i⩾2$, then ${\mathrm{Ext}}_S^i(M,N)=0$ for all $i⩾1$. Moreover, if this holds true, then either ${\mathrm{projdim}}_R(M)$ or ${\mathrm{injdim}}_R(N)$ is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

20 pages, Final version after revision, To appear in Glasgow Mathematical Journal

References in corpus (2)

• Local rings with quasi-decomposable maximal ideal
• Some criteria for regular and Gorenstein local rings via syzygy modules

Cited by in corpus (1)

• Complexity and rigidity of Ulrich modules, and some applications