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The mapping cone of an Eisenbud operator and applications to exact zero divisors

Liana M. Şega, Deepak Sireeshan

Abstract

Let $(Q,\mathfrak{m},{\sf k})$ be a local ring that admits an exact pair of zero divisors $(f,g)$, $M$ a $Q$-module with $fM = 0$ and $U$ a free resolution of $M$ over $Q$. We construct a degree $-2$ chain map, which we call an Einsenbud operator, on the complex $U \otimes_Q Q/(f,g)$ and use the mapping cone of the operator to study two exact sequences that relate homology over $Q$ to homology over $Q/(f)$. Several applications are given.

The mapping cone of an Eisenbud operator and applications to exact zero divisors

Abstract

Let be a local ring that admits an exact pair of zero divisors , a -module with and a free resolution of over . We construct a degree chain map, which we call an Einsenbud operator, on the complex and use the mapping cone of the operator to study two exact sequences that relate homology over to homology over . Several applications are given.
Paper Structure (4 sections, 16 theorems, 124 equations)

This paper contains 4 sections, 16 theorems, 124 equations.

Table of Contents

  1. dg algebras and Eisenbud operators
  2. The Mapping cone of the Eisenbud Operator
  3. Complexity and Curvature
  4. Vanishing of homology and other applications

Key Result

Lemma 1.5

Let $(Q,\mathfrak{m},\sf{k})$ be a local ring, $f\in Q$ and set $R = Q/(f)$. Let $(F,\partial)$ be a complex of free $R$-modules and let $(\widetilde{F}, \widetilde{\partial})$ be a lifting of $F$ to $Q$. If $\tau$ is the Eisenbud operator associated to the data $(f,F,\widetilde{F})$, then the follo

Theorems & Definitions (35)

  • Lemma 1.5
  • proof
  • Lemma 1.7
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 25 more