Let A be a commutative ring, and let \a = \frak{a} be a finitely generatedideal in it. It is known that a necessary and sufficient condition for thederived \a-torsion and \a-adic completion functors to be nicely behaved is theweak 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 fornoncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a modulecategory: weak stability. Our first main theorem is that in the commutativecase, the ideal \a is weakly proregular if and only if the correspondingtorsion class T_{\a} is weakly stable. We then study weak stability of torsion classes in module categories overnoncommutative rings. There are three main theorems in this context: - For atorsion class T that is weakly stable, quasi-compact and finite dimensional,the right derived torsion functor is isomorphic to a left derived tensorfunctor. - The Noncommutative MGM Equivalence, under the same assumptions on T.- A theorem about symmetric derived torsion for complexes of bimodules. Thislast theorem is a generalization of a result of Van den Bergh from 1997, andcorrects an error in a paper of Yekutieli-Zhang from 2003.
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机译：令A为交换环，令\ a = \ frak {a}为其中的有限生成的理想值。众所周知，派生的\ a扭转和\ a-adic完成函子表现良好的必要和充分条件是\ a的弱正则性。特别是，米高梅等效项成立。由于弱正规则性是根据环的元素定义的（特别是涉及到Koszul络合物的极限），因此不适用于非交换环理论。在本文中，我们在模块类别中引入了有关扭转类别T的新条件：弱稳定性。我们的第一个主要定理是，在可交换的情况下，当且仅当对应的扭转类别T _ {\ a}是弱稳定的时，理想\ a才是弱正规则的。然后，我们研究了非交换环模块类别中扭力类的弱稳定性。在这种情况下，有三个主要定理：-对于微弱稳定，准紧凑和有限维的变调子类T，右派生的扭转函数与左派生的张量函数同构。 -在T.的相同假设下，非交换MGM等价。-关于双模复合体的对称派生扭转定理。该最后定理是1997年Van den Bergh结果的推广，并纠正了2003年Yekutieli-Zhang的论文中的错误。
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