Generation in derived categories and singularities

The Department of Mathematics in the College of Arts and Sciences is honored to welcome Dr. Josh Pollitz to deliver the weekly colloquium. Dr. Pollitz is an NSF Postdoctoral Fellow in the Mathematics Department, University of Utah, where he works in commutative algebra, particularly structures on resolutions, triangulated categories, and notions of cohomological support.

Abstract: The use of homological methods to study singularities in commutative algebra has a long and storied history. One effective strategy is to study the derived category which consists of all complexes of modules over a ring and using its triangulated structure (consisting of cones, suspensions, and retracts). For example, a celebrated theorem of Auslander, Buchsbaum and Serre that characterizes smooth points on a variety (over the complex numbers) can be viewed as a statement of generation in the derived category using finitely many cones, suspensions, and retracts. A noteworthy application of this theorem is that it solves the so-called localization problem for regular rings. In this talk, I will explain these ideas and some of my work studying generation in the derived category to obtain ring theoretic applications.