A noncommutative Matlis-Greenlees-May equivalence

Seminar

Speaker

Dr. Rishi Vyas (Ben-Gurion University)

Date

09/03/2016 - 11:30 - 10:30Add to Calendar 2016-03-09 10:30:00 2016-03-09 11:30:00 A noncommutative Matlis-Greenlees-May equivalence 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. Third floor seminar room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public

Place

Third floor seminar room

Abstract

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.

Last Updated Date : 28/02/2016