Let K be a regular noetherian commutative ring. I consider finite type commutative
K-algebras and K-schemes. I will begin by explaining the theory of rigid residue
complexes on 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 on
K-schemes and their functorial properties. For any map between K-schemes there is a
rigid trace homomorphism (that usually does not commute with the differentials). When
the map of schemes 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 has a rigid
residue complex on it, and for any map between stacks there is a trace homomorphism.
These facts are rather easy consequences of the corresponding facts for schemes,
together with etale descent. I will finish by presenting two conjectures, that refer
to Grothendieck Duality for proper maps between DM stacks. A key condition here is
that of tame map of stacks.
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