Intersection theory on derived stacks

Abstract

We define integral cycle complexes and Chow homology of derived Artin stacks. To any quasi-smooth family of derived Artin stacks we associate a bivariant fundamental class, encoding the virtual fundamental classes of its fibres and their deformation invariance up to coherent homotopy. We extend the Riemann-Roch theorem of Baum-Fulton-MacPherson from quasi-projective schemes to derived Artin stacks, and thereby obtain a virtual Riemann-Roch formula for quasi-smooth derived Artin stacks. As another application of the formalism, we construct intersection products on regular Artin stacks locally of finite type over a field. Finally, we prove an integral comparison of our Chow homology with the cycle-theoretic Chow groups of Kresch.

arXiv comment56 pages, submitted version; v2: new title/abstract, add integral Chow groups, add new appendix on comparison with Kresch

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Cited by in corpus (12)

• Fundamental classes in motivic homotopy theory
• K-theory and G-theory of derived algebraic stacks
• Dimensional reduction in cohomological Donaldson-Thomas theory
• On the rational motivic homotopy category
• Virtual excess intersection theory
• Enumerative invariants and wall-crossing formulae in abelian categories
• Cohomological Hall algebras and vertex algebras
• Absolute Poincaré duality in étale cohomology
• A categorification of the quantum Lefschetz principle
• Cohomological Hall algebras of quivers and Yangians
• Virtual classes of ${\mathbb{G}}_{\text{m}}$-gerbes
• Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty }$-algebras