Virtual fundamental classes of derived stacks I

Abstract

We construct the étale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base change, excess intersection, and Grothendieck-Riemann-Roch formulas. These classes also satisfy a general cohomological Bézout theorem which holds without any transversity hypotheses. The construction is new even for classical stacks and as one application we extend Gabber's proof of the absolute purity conjecture to Artin stacks.

arXiv comment36 pages, preliminary version

References in corpus (15)

• Higher Topos Theory
• Triangulated categories of mixed motives
• Noetherian approximation of algebraic spaces and stacks
• Fundamental classes in motivic homotopy theory
• Enhanced six operations and base change theorem for higher Artin stacks
• Two-dimensional categorified Hall algebras
• Completions and derived de Rham cohomology
• Modules over algebraic cobordism
• Orientation theory in arithmetic geometry
• Derived algebraic cobordism
• The intersection motive of the moduli stack of shtukas
• The Morel-Voevodsky localization theorem in spectral algebraic geometry
• Bivariant theories in motivic stable homotopy
• The intrinsic stable normal cone
• Algebraic G-theory in motivic homotopy categories

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