# Event № 1527

Event № 1527

WI
Algebraic Geometry and Representation Theory Seminar
- Amnon Yekutieli, Ben Gurion University
22/07/2015, Wednesday, 11:00

Type: Lecture

Name: Algebraic Geometry and Representation Theory Seminar

Title: Differential Graded Rings and Derived Categories of Bimodules

Speaker: Amnon Yekutieli, Ben Gurion University

Place:
Room 261 ,Ziskind Building, Weizmann Institute of Science

*Abstract:*

Homological algebra plays a major role in noncommutative ring theory. This =Homological algebra plays a major role in noncommutative ring theory. This interaction is often called "noncommutative algebraic geometry", because the homological methods allow us to treat, in an effective way, a noncommutative ring A as "the ring of functions on a noncommutative affine algebraic variety". Some of the most important homological constructs related to a noncommutative ring A are dualizing complexes and tilting complexes. These are special kinds of complexes of A-bimodules. When A is a ring containing a central field K, these concepts are well-understood now. However, little is known about dualizing complexes and tilting complexes when the ring A does not contain a central field (I shall refer to this as the noncommutative arithmetic setting). The main technical issue is finding the correct derived category of A-bimodules. In this talk I will propose a promising definition of the derived category of A-bimodules in the noncommutative arithmetic setting. Here A is a (possibly) noncommutative ring, central over a commutative base ring K (e.g. K = Z). The definition relies on DG rings (better known as differential graded algebras). We choose a DG ring A', central and flat over K, with a DG ring quasi-isomorphism A' -> A. Such resolutions exist. Our candidate for the "derived category of A-bimodules" is the category D(A'^{en}), the derived category of DG bimodules over the enveloping ring A'^{en}, which is the tensor product over K of A' and its opposite. A recent theorem shows that the category D(A'^{en}) is independent of the resolution A', up to a canonical equivalence. This justifies our definition. Now we can define what are tilting complexes and dualizing complexes over A, in the noncommutative arithmetic setting. It seems that most of the standard properties of dualizing complexes (proved by Grothendieck for commutative rings in the 1960's, and by myself for noncommutative rings over a field in the 1990's), hold also in this more complicated setting. We can also talk about rigid dualizing complexes in the noncommutative arithmetic setting. A key problem facing us is that of existence of rigid dualizing complexes. When the base ring K is a field, Van den Bergh (1997) discovered a powerful existence result for rigid dualizing complexes, that relies on a good filtration of A (something that often exists). We are now trying to extend Van den Bergh's method to the noncommutative arithmetic setting. This is work in progress, joint with Rishi Vyas. In this talk I will explain, in broad strokes, what are DG rings, DG modules, and the associated derived categories and derived functors. Also, I will try to go into the details of a few results and examples, to give the flavor of this material. For those who want to follow the talk smoothly, I recommend reading, in advance, these notes (18 pages): Introduction to Derived Categories, http://arxiv.org/abs/1501.06731 .

SubmittedBy:
Gizel Maimon , gizel.maimon@weizmann.ac.il

EventLink: Event № 1527