The Auslander Condition and Perverse Sheaves
Abstract: Perverse sheaves arose in the study of singular spaces (intersection cohomology). These are objects in the derived category that behave like sheaves (for instance they can be glued).
On an algebraic scheme the dualizing complex is a perverse sheaf. >From this, and using extra data (local duality or rigidity), one can glue dualizing complexes inside the derived category.
Now let A be a noncommutative algebra, and let R be a dualizing complex over it. Suppose R satisfies the Auslander condition. Then considering Mod A (the category of A-modules) as a space, R is a perverse sheaf.
We propose to use this observation to prove existence of dualizing complexes over A when localization is possible, and also to make some headway towards duality for noncommutative schemes. (Joint work with J.J. Zhang.)