Amnon
Yekutieli, BGU
Residues and Duality for Schemes and
Stacks
Abstract:
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 the ind-rigid trace homomorphism
(that usually does not commute with the differentials). When the map
f is proper, the ind-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 the ind-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/handout.pdf
(November
2013)