The notion of a weakly proregular sequence in a commutative ring was first
formally introduced by Alonso-Jeremias-Lipman (though the property that it
formalizes was already known to Grothendieck), and further studied by
Schenzel and Porta-Shaul-Yekutieli: a precise definition of this notion
will be given during the talk. An ideal in a commutative ring is called
weakly proregular if it has a weakly proregular generating set. Every
ideal in a commutative noetherian ring is weakly proregular.
It turns out that weak proregularity is the appropriate context for the
Matlis-Greenlees-May (MGM) equivalence: given a weakly proregular ideal I
in a commutative ring A, there is an equivalence of triangulated
categories (given in one direction by derived local cohomology and in the
other by derived completion at I) between cohomologically I-torsion (i.e.
complexes with I-torsion cohomology) and cohomologically I-complete
complexes in the derived category of A.
In this talk, we will give a categorical characterization of weak
proregularity: this characterization then serves as the foundation for a
noncommutative generalisation of this notion. As a consequence, we will
arrive at a noncommutative variant of the MGM equivalence. This work is
joint with Amnon Yekutieli.