A ring of cohomological operators on Ext and Tor

TL;DR

This paper extends the construction of cohomological operators on Ext and Tor modules to more general ring homomorphisms, providing a more natural framework for their algebraic structure.

Contribution

It generalizes the existing theory of cohomological operators to non-surjective ring homomorphisms, broadening the applicability of these algebraic tools.

Findings

Extended the construction of cohomological operators to non-surjective homomorphisms.

Provided a natural perspective on the algebraic structure of Ext and Tor modules.

Enhanced understanding of the module structure over symmetric algebras.

Abstract

Let $f \colon R \to B$ be a surjective homomorphism of rings with kernel $I$. Gulliksen (when $I$ is generated by a regular sequence) and later Mehta (in general) showed that for any $B$-modules $M$ and $N$, $\mathrm{Ext}_B^{\ast}(M,N)$ has a structure of graded $\mathrm{S}_B^{\ast}\left(\widehat{I/I^2}\right)$-module, where $\widehat{\quad}$ denotes dual and $\mathrm{S}$ denotes symmetric algebra. This construction is extended to the case where $f$ is not necessarily surjective in a way that allows one to regard these operators from a more natural perspective.