Homological aspects of perfect algebras

TL;DR

This paper explores the homological properties of perfect algebras in prime characteristic, demonstrating their role in resolving singularities and analyzing module dimensions, with specific computations and applications.

Contribution

It provides new results on the finite flat and projective dimensions of modules over perfect rings, and computes their weak and global dimensions in various cases.

Findings

Modules over the ring of absolute integral closure have finite flat dimension.

Under mild conditions, modules have finite projective dimension.

Computed weak and global dimensions for specific perfect rings.

Abstract

We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has finite flat dimension. Under some mild conditions, we show any module over that ring has finite projective dimension. We compute weak dimension and global dimension of perfect rings in a series of nontrivial cases. Some interesting applications are given. In particular, we answer some questions asked by Shimomoto.