Asymptotic behavior of dimensions of syzygies

TL;DR

This paper studies the asymptotic behavior of the dimensions of syzygy modules over certain local rings, showing that under specific conditions, these dimensions stabilize to that of the ring.

Contribution

It establishes conditions under which the dimension of high syzygy modules matches the ring's dimension, extending known depth stabilization results.

Findings

Dimension of high syzygies equals the ring's dimension under given conditions.

Betti numbers eventually non-decreasing implies dimension stabilization.

Results apply to equidimensional local rings with finitely generated modules.

Abstract

Let R be a commutative noetherian local ring, and M 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.