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

TL;DR

This paper explores the homological properties of absolute integral and perfect closures of Noetherian domains, focusing on projective dimensions, free resolutions, and global dimensions in prime characteristic.

Contribution

It establishes bounded free resolutions for prime and maximal ideals in the closures and computes their global dimensions, advancing understanding of their homological structure.

Findings

Prime ideals of $R_{ ext{infty}}$ have bounded free resolutions.

Maximal ideals of $R^+$ have similar bounded resolutions in prime characteristic.

Global dimensions of $R^+$ and $R_{ ext{infty}}$ are computed in specific cases.

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.