Asymptotic behavior of dimensions of syzygies

Abstract

Let R R be a commutative noetherian local ring and M M be a finitely generated R R -module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M M eventually stabilize to the depth of R 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 R is equidimensional and the Betti numbers of M M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M M coincides with the dimension of R R .