Maximal Cohen-Macaulay complexes and their uses: A partial survey

TL;DR

This paper introduces maximal Cohen-Macaulay complexes over local rings, explores their constructions, and demonstrates their applications in proving homological conjectures and results in birational geometry.

Contribution

It defines maximal Cohen-Macaulay complexes, links their existence to Hochster's conjectures, and applies them to prove key homological and geometric results.

Findings

Existence of maximal Cohen-Macaulay complexes is established.

New proofs of homological conjectures are provided.

Applications to birational geometry are demonstrated.

Abstract

This work introduces a notion of complexes of maximal depth, and maximal Cohen-Macaulay complexes, over a commutative noetherian local ring. The existence of such complexes is closely tied to the Hochster's ``homological conjectures", most of which were recently settled by Andr\'e. Various constructions of maximal Cohen-Macaulay complexes are described, and their existence is applied to give new proofs of some of the homological conjectures, and also of certain results in birational geometry.