Analytic Deviation One Ideals and Test Modules

TL;DR

This paper investigates properties of certain Cohen-Macaulay modules over local rings, establishing conditions under which specific Tor modules have polynomial length functions or modules are free, advancing understanding of ideal deviations.

Contribution

It proves that for ideals with analytic deviation one and low reduction number, the length polynomial of Tor modules is either degree d-1 or the associated module is free over the fiber cone.

Findings

The length of Tor^{A}_{1}(M,A/I^{n+1}) is either polynomial of degree d-1 or the module is free.

Identifies conditions linking polynomial degree to module freeness.

Provides new criteria for analyzing Cohen-Macaulay modules over local rings.

Abstract

Let A be a Cohen-Macaulay local ring of dimension d and I an ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. Let I be a locally complete intersection ideal of analytic deviation one and reduction number at most one. We prove that the polynomial given by $length(Tor^{A}_{1}(M,A/I^{n+1}))$ either has degree d-1 or $F_I(M) $ is a free$F(I)-$$module.