Asymptotic behavior of dimensions of syzygies

Beck, Kristen; Leamer, Micah

doi:10.1090/S0002-9939-2013-11510-8

Asymptotic behavior of dimensions of syzygies
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by Kristen A. Beck and Micah J. Leamer PDF

Proc. Amer. Math. Soc. 141 (2013), 2245-2252 Request permission

Abstract:

Let $R$ be a commutative noetherian local ring and $M$ be a finitely generated $R$-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of $M$ eventually stabilize to the depth of $R$. In this paper, we investigate the conditions under which a similar statement can be made regarding dimension. In particular, we show that if $R$ is equidimensional and the Betti numbers of $M$ are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of $M$ coincides with the dimension of $R$.

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