# Cohen-Macaulay homological dimensions

## TL;DR

This paper introduces new Cohen-Macaulay homological dimensions for complexes, characterizes Cohen-Macaulay rings, and relates these dimensions to existing invariants, advancing the understanding of homological properties in commutative algebra.

## Contribution

It defines Cohen-Macaulay projective, injective, and flat dimensions for complexes and explores their relationships with existing homological invariants.

## Key findings

- Cohen-Macaulay dimensions characterize Cohen-Macaulay rings.
- Cohen-Macaulay flat dimension lies between Gorenstein flat and large restricted flat dimensions.
- Cohen-Macaulay injective dimension lies between Gorenstein injective dimension and Chouinard invariant.

## Abstract

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.