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Cohomology of products of local rings

William F Moore, University of Nebraska - Lincoln


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 PRM of M over R has been expressed in terms of PSM,PSk and PTk 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 PS#Tk in terms of PSk and PTk . 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


Recommended Citation

Moore, William F, "Cohomology of products of local rings" (2008). ETD collection for University of Nebraska - Lincoln. AAI3313102.