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.