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Algebraic properties of Ext-modules over complete intersections

Jason A Hardin, University of Nebraska - Lincoln


We investigate two algebraic properties of Ext-modules over a complete intersection R of codimension c. Given an R-module M, Ext(M,k) can be viewed as a graded module over a polynomial ring in c variables with an action given by the Eisenbud operators. We provide an upper bound on the degrees of the generators of this graded module in terms of the regularities of two associated coherent sheaves. In the codimension two case, our bound recovers a bound of Avramov and Buchweitz in terms of the Betti numbers of M. We also provide a description of the differential graded (DG) R-module RHom(M,N) in terms of well-known DG Q-modules. When M=N, this has the structure of a differential graded algebra (DGA) over Q. In the case where M =Q/I with I generated by a Q-regular sequence, we provide explicit generators and relations for the DGA REnd(M) using the theory of Clifford algebras. This description generalizes a result of Dyckerhoff, who obtains a similar result in a special case. In the case where M= k, our result implies a classical result of Sjodin on the algebraic structure of Ext(k,k) over complete intersections.

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Recommended Citation

Hardin, Jason A, "Algebraic properties of Ext-modules over complete intersections" (2014). ETD collection for University of Nebraska - Lincoln. AAI3630871.