Chandrasekhar Hall

Local Cohomology and Completion

Rishi Vyas

Ben Gurion University, Israel

Fix an ideal in a commutative noetherian ring. There is an interesting
relationship between the Local Cohomology and Completion functors at this
ideal: their (right and left, respectively) derived functors are adjoint to
each other, and this adjunction induces an interesting equivalence of
categories. This relationship has been independently studied by a number of
mathematicians over the last few decades: protagonists for this story
include Grothendieck, Matlis, Greenlees-May, Alonso-Jeremias-Lipman,
Schenzel, Porta-Shaul-Yekutieli, and Positselski.

In this expository talk, we will try and describe this relationship by
focusing on the simplest example: an ideal generated by a single element.
The only prerequisites required are a basic understanding of commutative
algebra, and some, again basic, familiarity with chain complexes of modules
over a commutative ring. In particular, most of this talk will not assume
any familiarity with the derived category of a ring.