Stable homology over associative rings

TL;DR

This paper explores stable homology over associative rings, especially commutative noetherian rings, revealing its role in detecting Gorenstein rings and modules of finite G-dimension, and establishing its unbalanced nature.

Contribution

It introduces stable homology as a tool to characterize Gorenstein rings and modules of finite G-dimension, extending the understanding of homological invariants.

Findings

Stable homology detects Gorenstein rings.

Vanishing of stable homology characterizes modules of finite G-dimension.

Stable homology is unbalanced unless the ring is Gorenstein.

Abstract

We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings---among them Gorenstein rings, the original domain of Tate homology. Closely related to gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone. Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced.