On the homology of regular quotients

TL;DR

This paper constructs a free resolution for powers of regular ideals over a ring, generalizing the Koszul complex, and analyzes the algebra structure of related Tor groups, showing triviality in certain cases.

Contribution

It introduces a new free resolution for $R/I^s$ that extends the Koszul complex and studies the algebraic properties of associated Tor groups for these quotients.

Findings

The algebra structure of $ ext{Tor}^R_*(R/I,R/I^s)$ is trivial for $s>1$.

The reduction map induces a trivial algebra map between these quotients.

The construction generalizes classical Koszul complexes to higher powers of regular ideals.

Abstract

We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. For $s>1$, we easily deduce that the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction map $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras.