Cohomology over Fiber products of local rings

Let S and T be local rings with common residue field k, let R be the fiber product S×kT, and let M be an S-module. The Poincaré series of M has been expressed in terms of , and by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the structure of ExtR(k,k) and ExtR(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.