Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence

TL;DR

This paper introduces weak stability as a new condition for torsion classes, extending the MGM equivalence from commutative to noncommutative rings and generalizing previous results on derived torsion.

Contribution

It establishes weak stability as an analogue of weak proregularity for noncommutative rings and proves the noncommutative MGM equivalence under this condition.

Findings

Weak stability characterizes torsion classes in commutative rings.

The noncommutative MGM equivalence holds for weakly stable, quasi-compact, finite-dimensional torsion classes.

Generalization of symmetric derived torsion results, correcting previous errors.

Abstract

Let A be a commutative ring, and let \a = \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and \a-adic completion functors to be nicely behaved is the weak proregularity of \a. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem is that in the commutative case, the ideal \a is weakly proregular if and only if the corresponding torsion class T_{\a} is weakly stable. We then study weak stability of torsion classes in module categories over noncommutative rings. There are three main theorems in this context: - For a torsion class T that is weakly stable, quasi-compact and finite dimensional, the right derived torsion functor is isomorphic to a left derived tensor functor. - The Noncommutative MGM Equivalence, under the same assumptions on T. - A theorem about symmetric derived torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and corrects an error in a paper of Yekutieli-Zhang from 2003.