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Cohomology of products of local rings
Abstract
Let S and T be local rings with common residue field k. We study the structure of the cohomology of rings defined in terms of S and T, and how the cohomology of S and T play a role in this structure. We first study the fiber product of S and T, R = S × kT. For an S-module M, the Poincaré series [special characters omitted] of M over R has been expressed in terms of [special characters omitted] and [special characters omitted] by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the algebra 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. We also study the connected sum S#T of S and T, which is defined only when S and T are artinian Gorenstein rings. We give a formula that expresses [special characters omitted] in terms of [special characters omitted] and [special characters omitted]. Our proof requires us to develop some results on Golod rings, modules, and homomorphisms. We also give theorems on the structure of the Koszul homology algebra of both S × kT and S#T.
Subject Area
Mathematics
Recommended Citation
Moore, William F, "Cohomology of products of local rings" (2008). ETD collection for University of Nebraska-Lincoln. AAI3313102.
https://digitalcommons.unl.edu/dissertations/AAI3313102