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Asymptotic Growth of Stable Ext Modules
Let R = Q/(f1,...,fc) where Q is a regular local ring and the fi form a regular sequence, and let M and N be finitely generated R-modules such that the lengths of the R-modules ExtR i (M,N) are finite for i sufficiently large. A result of Gulliksen implies that there exist polynomials such that governing the growth of the lengths of the even and odd Ext modules. Under these assumptions the stable cohomology modules will also have finite length for and be governed by polynomials in even and odd degrees. The focus of this thesis is to examine the relationship between the polynomials governing the lengths of the Ext-modules in large positive degrees and the polynomials governing the lengths of the stable cohomology modules in large negative degrees. We show that the polynomials have the same leading coefficient, depending on the parity of the codimension c. As an application of the result we show that the pairing hc(M,N) defined by Celikbas and Dao as a generalization of the Herbrand difference introduced by Buchweitz has the following properties: • if the embedding dimension of R is odd then hc(M,N) is alternating and hc(M,M) = 0. • if the embedding dimension of R is even then hc(M,N) is symmetric. (no equations included)
Bachmann, Matthew, "Asymptotic Growth of Stable Ext Modules" (2023). ETD collection for University of Nebraska - Lincoln. AAI30574502.