Harvard-MIT Algebraic Geometry Seminar
        	    February 5, 2002 at 3:00 p.m.
			   Harvard Room 507



	       Perverse sheaves and dualizing complexes
		  over noncommutative ringed schemes


	   Amnon Yekutieli (Ben Gurion University, Israel)



Abstract:

In this talk I will discuss an attempt at Grothendieck Duality on
noncommutative spaces. Since in the case of affine noncommutative
spaces (i.e. rings) Grothendieck Duality is pretty well understood,
and on the other hand we don't even know what is a noncommutative
space in general, we consider an intermediate case: a noncommutative
space Y that's an affine fibration over a commutative scheme X. That's
a fancy way to say that Y=(X,A) where A is a sheaf of quasi coherent
noncommutative rings on X. We call (X,A) a quasi coherent ringed
scheme.

As usual in such circumstances, we encounter the problem of gluing. On
each affine open set U in X we have a rigid dualizing complex for A|_U
from the ring construction, and these are compatible on intersections.
But how to glue these complexes globally? One should note that Cousin
complexes, the solution devised by Grothendieck for gluing dualizing
complexes, will not work in the noncommutative world due to well known
obstructions.

Instead we choose to use perverse sheaves. This is a gluing method
invented by Bernstein-Beilinson-Deligne-Gabber in the context of
geometry of singular spaces. We discovered that the Auslander
condition of dualizing complexes over noncommutative rings (a very
algebraic property) is exactly what is needed to define perverse
modules over a noncommutative ring. And furthermore using a few nice
features of the theory, we can also extend the definition from rings
to noncommutative ringed schemes. Finally it turns out that rigid
dualizing complexes are themselves perverse sheaves of bimodules
(namely on the product X^2), so we can glue the local pieces together.

I will explain what are dualizing complexes and what they are good for
(concentrating on the noncommutative side). Then I'll discuss perverse
sheaves, the Auslander condition and how they interact. I'll finish by
sketching our construction.

The work is joint with J.J. Zhang (Seattle).