Cohomology of Fiber Products of Local Rings

TL;DR

This paper investigates the cohomology of fiber products of local rings, providing explicit resolutions, structural theorems, and depth computations to deepen understanding of their algebraic properties.

Contribution

It introduces explicit minimal resolutions and structural theorems for Ext modules over fiber product rings, advancing the understanding of their cohomological behavior.

Findings

Explicit minimal resolutions for fiber product rings

Structural theorems for Ext modules

Depth calculations of cohomology modules

Abstract

Let $S$ and $T$ be local rings with common residue field $k$, let $R$ be the fiber product $S \times_k T$, and let $M$ be an $S$-module. The Poincar\'e series $P^R_M$ of $M$ has been expressed in terms of $P^S_M$, $P^S_k$ and $P^T_k$ by Kostrikin and Shafarevich, and by Dress and Kr\"amer. Here, an explicit minimal resolution, as well as theorems on the structure of $\Ext_R(k,k)$ and $\Ext_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product.