Weak proregularity, weak stability, and the noncommutative MGM equivalence
Let \(R\) be a commutative ring, and let \(\mathfrak{a}\) be a finitely generated ideal in \(R\). There are two functors on the category of \(R\)-modules that the ideal \(\mathfrak{a}\) determines: the \(\mathfrak{a}\)-torsion functor \(\Gamma_{\mathfrak a}\), and the \(\mathfrak{a}\)-adic completion functor \(\Lambda_{\mathfrak a}\). The most known general condition under which the functors \(\Gamma_{\mathfrak a}\) and \(\Lambda_{\mathfrak a}\) are adjoint to each other is when the ideal \(\mathfrak{a}\) is \textit{weakly proregular}. In particular, the \(MGM\) \textit{Equivalence} holds. The goal of this paper is to find a noncommutative analogue of weak proregularity, and to prove a suitable version of the \(MGM\) Equivalence. To aim that, the authors define a new condition called \textit{weak stability} on a torsion class \(T\) in a module category. Then, the first main theorem says that in the commutative case, the ideal \(\mathfrak{a}\) is weakly proregular if and only if the corresponding torsion class \(T\) is weakly stable. Weak stability of torsion classes in module categories over noncommutative rings is studied and there are three main theorems in this context: Theorem 7.13, Theorem 8.4 and Theorem 9.7 (it is a correction of Theorem 1.23 by \textit{A. Yekutieli} and \textit{J. J. Zhang} [J. Algebra 259, No. 2, 451--493 (2003; Zbl 1035.16004)]).
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