Tor algebra of local rings with decomposable maximal ideal

TL;DR

This paper investigates the algebraic structure of the Tor algebra of a local ring with a decomposable maximal ideal, expressing it in terms of the Tor algebras of related quotient rings, thus advancing understanding of their homological properties.

Contribution

It provides a detailed description of the Tor algebra structure for local rings with decomposable maximal ideals, linking it to the Tor algebras of quotient rings.

Findings

Explicit structure of the Tor algebra in terms of quotients

Connections between maximal ideal decomposition and homological invariants

Enhanced understanding of the algebraic properties of such rings

Abstract

Let $(R,{\frak{m}}_R)$ be a commutative noetherian local ring. Assuming that ${\frak{m}}_R=$$I\oplus J$ is a direct sum decomposition, where $I$ and $J$ are non-zero ideals of $R$, we describe the structure of the Tor algebra of $R$ in terms of the Tor algebras of the rings $R/I$ and $R/J$.