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

## TL;DR

This paper investigates the vanishing patterns of Ext and Tor modules over deformations of Cohen-Macaulay local rings with minimal multiplicity, establishing conditions for their vanishing to imply finiteness of projective or injective dimensions and characterizing regularity or Gorenstein properties.

## Contribution

It provides new criteria linking Ext and Tor vanishing to the finiteness of homological dimensions and characterizes regular and Gorenstein rings via vanishing conditions involving syzygies.

## Key findings

- Vanishing of Ext over certain consecutive degrees implies all higher Ext vanish.
- Either projective dimension of M or injective dimension of N is finite under Ext vanishing.
- Characterizations of regular and Gorenstein rings based on Ext and Tor vanishing involving syzygies.

## Abstract

Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,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 \geqslant 2 $, then $ \mathrm{Ext}_S^i(M,N) = 0 $ for all $ i \geqslant 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.