Cohomology of Fiber Products of Local Rings

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.