Enhanced six operations and base change theorem for higher Artin stacks

Abstract

In this article, we develop a theory of Grothendieck's six operations for derived categories in étale cohomology of Artin stacks, for both torsion and adic coefficients. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends many previous theories on this subject, including the one developed by Laszlo and Olsson, in which the operations are subject to more assumptions and the base change isomorphism is only constructed on the level of sheaves. Moreover, our theory works for higher Artin stacks as well. In addition, we define perverse t-structures on higher Artin stacks for general perversity, extending Gabber's work on schemes. Our method differs from previous approaches, as we exploit the theory of stable $\infty$-categories developed by Lurie. We enhance derived categories, functors, and natural isomorphisms to the level of $\infty$-categories and introduce $\infty$-categorical (co)homological descent. To handle the issue of ``homotopy coherence'', we develop a general technique for gluing subcategories of $\infty$-categories and several other $\infty$-categorical techniques. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to construct functors in different contexts. One of our category-theoretical results generalizes Deligne's gluing theory developed in the construction of the extraordinary pushforward operation in étale cohomology of schemes.

188 pages. v4: incorporated arXiv:1211.5294 and arXiv:1404.1128; v3: reformulation; v2: minor improvements

References in corpus (1)

• Higher Topos Theory

Cited by in corpus (25)

• Triangulated categories of mixed motives
• The six operations in equivariant motivic homotopy theory
• A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
• Fundamental classes in motivic homotopy theory
• An introduction to affine Grassmannians and the geometric Satake equivalence
• Good Reduction of K3 Surfaces
• The intersection motive of the moduli stack of shtukas
• Perversely categorified Lagrangian correspondences
• Virtual fundamental classes of derived stacks I
• Enhanced adic formalism and perverse t-structures for higher Artin stacks
• Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited
• Gluing restricted nerves of $\infty$-categories
• Motivic Realizations of Singularity Categories and Vanishing Cycles
• On lax transformations, adjunctions, and monads in $(\infty ,2)$-categories
• Rigidity in automorphic representations and local systems
• Gluing pseudo functors via $n$-fold categories
• Cohomological Hall algebras and vertex algebras
• Geometric Satake, Springer correspondence, and small representations II
• Derivator Six-Functor-Formalisms -- Construction II
• Deformation theory of perfect complexes and traces
• The Atiyah-Bott formula and connectivity in chiral Koszul duality
• Derivator Six Functor Formalisms -- Definition and Construction I
• Spaces with ${\mathbb{G}}_m$-action, hyperbolic localization and nearby cycles
• The cohomology of the moduli space of curves at infinite level
• Absolute Poincaré duality in étale cohomology