3,675 research outputs found
Cohomology of Fiber Products of Local Rings
Let and be local rings with common residue field , let be the
fiber product , and let be an -module. The Poincar\'e
series of has been expressed in terms of , and
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.Comment: 13 page
Noncommutative Kn\"{o}rrer periodicity and noncommutative Kleinian singularities
We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of
noncommutative invariant theory. Namely, let be a left noetherian
AS-regular algebra, let be a normal and regular element of of positive
degree, and take . Then there exists a bijection between the set of
isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules
over and those over (a noncommutative analog of) its second double branched
cover . Our results use and extend the study of twisted matrix
factorizations, which was introduced by the first three authors with Cassidy.
These results are applied to the noncommutative Kleinian singularities studied
by the second and fourth authors with Chan and Zhang.Comment: Numerous typos fixed, removed unnecessary finite order hypothesi
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