Let K be a regular noetherian commutative ring. I will begin by explaining the theory of rigid residue complexes over essentially finite type K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes over finite type K-schemes. An important feature is that the rigid residue complex over a scheme X is a quasi-coherent sheaf in the etale topology of X. For any map f : X → Y between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map f is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.
Then I will move to finite type Deligne-Mumford K-stacks. Any such stack \X has a rigid residue complex on it, and for any map f : \X → \Y between stacks there is a rigid trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish with the Residue Theorem, which holds when the map f : \X → \Y is proper and coarsely schematic; and the Duality Theorem, which also requires the map f to be tame. If there is time I will outline the proofs.
- Lecture notes are at http://www.math.bgu.ac.il/~amyekut/lectures/resid-stacks/notes.pdf