Grothendieck Duality for Deligne-Mumford Stacks

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.

arXiv commentTitle has changed a little bit. The first chapter has been almost completely rewritten. Numerous bug fix

References in corpus (5)

• Smooth toric DM stacks
• Good moduli spaces for Artin stacks
• Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks
• Gromov-Witten theory of Deligne-Mumford stacks
• Periodic twisted cohomology and T-duality

Cited by in corpus (18)

• Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks
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• Witten's top Chern class via cosection localization
• General GLSM Invariants and Their Cohomological Field Theories
• Stable rationality and conic bundles
• Serre functors and dimensions of residual categories
• Relative Orbifold Donaldson-Thomas Theory and the Degeneration Formula
• $A_{\infty }$-structures associated with pairs of 1-spherical objects and noncommutative orders over curves
• Weak Akizuki-Nakano vanishing theorem for globally $F$-split 3-folds
• Poisson Structures on Moduli Spaces of Higgs Bundles over Stacky Curves
• Categorical crepant resolutions for quotient singularities
• Moduli spaces of semistable pairs on projective Deligne-Mumford stacks
• Deformations of weak $\mathbb{Q}$-Fano 3-folds
• Geometry and cohomology of compactified Deligne--Lusztig varieties
• Functional equations of Nekrasov functions proposed by Ito-Maruyoshi-Okuda
• A compactification of the moduli space of self-maps of ${\mathbb{CP}}^1$ using stable maps