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.

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Last updated: 21 Jun 2024