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 × k T , and let M be an S-module. The Poincaré series P M R of M has been expressed in terms of P M S , P k S and P k T by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as...

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Bibliographic Details
Published in:Journal of algebra 2009-02, Vol.321 (3), p.758-773
Main Author: Moore, W. Frank
Format: Article
Language:English
Subjects:
Fiber product
Minimal free resolution
Poincaré series
Yoneda algebra
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Summary:Let S and T be local rings with common residue field k, let R be the fiber product S × k T , and let M be an S-module. The Poincaré series P M R of M has been expressed in terms of P M S , P k S and P k T by Kostrikin and Shafarevich, and by Dress and Krämer. 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.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2008.10.015