Event № 289
The notion of a weakly proregular idea 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. The precise definition is quite technical, but will be given in the talk. 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.
At the beginning of this talk, these ideas will be motivated by studying what happens in a very particular case: power series in one variable over a field. In particular, a portion of this talk will be elementary and accessible to any one with a background in basic commutative and homological algebra.
Time permitting, after a brief survey of the general theory we will proceed to 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.