Differential Graded Rings and Derived Categories of Bimodules

Homological algebra plays a major role in noncommutative ring theory. This interaction is often called "noncommutative algebraic geometry", because these 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 over A. 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 is based on resolutions of A by differential graded rings (better known as DG 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 derived category of A'-bimodules. A recent theorem shows that this category 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 dualizing complexes. When the base ring K is a field, Van den Bergh (1997) discovered a powerful existence result for dualizing complexes. We are now trying to extend Ven 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 one or two examples, to give the flavor of this material.

For those who want to follow the talk smoothly, I recommend reading, in advance, these notes: Introduction to Derived Categories http://arxiv.org/abs/1501.06731 .