Grothendieck Duality for Deligne-Mumford Stacks

TL;DR

This paper develops a duality theory for Deligne-Mumford stacks, establishing the existence of dualizing functors, computing dualizing complexes, and proving Serre duality for smooth compact stacks, with applications to tame curves.

Contribution

It introduces a duality framework for algebraic stacks, explicitly computes dualizing complexes, and extends Serre duality to a broad class of stacks, including Gorenstein cases.

Findings

Dualizing functor exists for separated morphisms with affine diagonal

Serre duality holds for smooth compact Deligne-Mumford stacks

Dualizing sheaf computed for tame nodal curves

Abstract

We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact Deligne-Mumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We explicitly compute the dualizing complex for a smooth stack over an algebraically closed field and prove that Serre duality holds for smooth compact Deligne-Mumford stacks in its usual form. We prove also that a proper Cohen-Macaulay stack has a dualizing sheaf and it is an invertible sheaf when it is Gorenstein. As an application of this general machinery we compute the dualizing sheaf of a tame nodal curve.