The six operations in equivariant motivic homotopy theory

Abstract

We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel-Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoub's and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.

Vanishing theorems for the negative K-theory of stacks

“…Proof. The proof is the same as [21,Lemma 2.13], the key point being that every quasicoherent sheaf on a quasi-compact quasi-separated stack is the colimit of its finitely generated quasi-coherent subsheaves [37].…”

“…Descent for abstract blow-ups is more difficult and uses several non-trivial properties of the category Stk ′ . The proof ultimately relies on the proper base change theorem in stable equivariant motivic homotopy theory, proved in [21].…”

Motivic Euler Characteristics and Witt-Valued Characteristic Classes

“…Following [4,10,18], the functor SH(−) admits the Grothendieck six operations; we briefly review the aspects of this structure that we will be needing here. We note that the theory in [4,18] is for quasi-projective S-schemes; the extension to separated S-schemes of finite type is accomplished in [10,Theorem 2.4.50].. For each X, SH(X) is the homotopy category of a closed symmetric stable model category [19], which makes SH(X) into a closed symmetric monoidal category. We denote the symmetric monoidal product in SH(X) by ∧ X and the adjoint internal Hom by Hom X (−, −).…”

Unstable motivic homotopy theory

“…The original proof is [MV99, Section 3 Theorem 2.23 p.115], and a particularly accessible exposition, done over an algebraically closed field, may be found in [AE17, Section 7], using unpublished notes of A. Asok and [Hoy17]. 1 We use the algebre-geometric term "closed immersion" for a map isomorphic to the inclusion of a closed subscheme.…”