Cohomological Operators on Quotients by an Exact Zero Divisor

TL;DR

This paper explicitly constructs and analyzes cohomological operators on quotients by an exact zero divisor, revealing their properties and providing examples of non-trivial action.

Contribution

It introduces explicit methods to compute cohomological operators on quotients by exact zero divisors and studies their properties with concrete examples.

Findings

Cohomological operators can act non-trivially on such quotients.

Explicit construction of endomorphisms of free modules over the quotient.

Properties of these operators are characterized and exemplified.

Abstract

Let S be a commutative ring, x, y $\in$ S a pair of exact zero divisors, and R = S/(x). Let F be a complex of free R-modules. In this paper we explicitly compute cohomological operators of R over S by constructing endomorphisms of F. We consider some properties of these cohomological operators, as well as provide an example in which these cohomological operators act non-trivially.