#### Alladi Ramakrishnan Hall

#### A noncommutative Matlis-Greenlees-May equivalence

#### Rishi Vyas

##### Ben Gurion University of the Negev

*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.

Roughly speaking, an element s in a commutative ring A is said to be

weakly proregular if every module over A can be reconstructed from its

localisation at s considered along with its local cohomology at the

ideal generated by s. This notion extends naturally to finite

sequences of elements: a precise definition 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.

Done