Grothendieck Duality for Projective Deligne-Mumford Stacks

Abstract

We develop Grothendieck duality for projective Deligne-Mumford stacks, in particular we prove the existence of a dualizing complex for a morphism from a projective stack to a scheme and for a proper representable morphism of algebraic stacks. In the first case we explicitly compute the dualizing complex and prove that Serre duality holds for smooth projective stacks in its usual form. We prove also that a projective stack has dualizing sheaf if and only if it is Cohen-Macaulay, it has a dualizing sheaf that is an invertible sheaf if and only if it is Gorenstein and for local complete intersections we explicitly compute the invertible sheaf. As an application of this general machinery we compute the dualizing sheaf of a nodal projective curve.