Homological properties of the perfect and absolute integral closures of Noetherian domains

Abstract

For a Noetherian local domain $R$ let $R^{+}$ be the absolute integral closure of $R$ and let $R_{\infty }$ be the perfect closure of $R$, when $R$ has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of $R^{+}$ and $R_{\infty }$. In particular, we show that any prime ideal of $R_{\infty }$ has a bounded free resolution of countably generated free $R_{\infty }$-modules. Also, we show that the analogue of this result is true for the maximal ideals of $R^{+}$, when $R$ has residue prime characteristic. We compute global dimensions of $R^{+}$ and $R_{\infty }$ in some cases. Some applications of these results are given.

We correct and add an appendix to the published version

Cited by in corpus (1)

• F-coherent rings with applications to tight closure theory