A general descendability criterion for topological six-functor formalisms yields h-descent for étale motivic spectra and rational motivic cohomology, plus arc-descent in weights ≤1.
Lisse extensions of weaves
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abstract
Any sheaf theory on schemes extends canonically to Artin stacks via a procedure called lisse extension. In this paper we show that lisse extension preserves the formalism of Grothendieck's six operations: more precisely, the lisse extension of a weave on schemes determines a weave on (higher) Artin stacks. The setup is general enough to apply to the stable motivic homotopy category with the six functor formalism of Voevodsky-Ayoub-Cisinski-Deglise, for instance, and is not specific to algebraic geometry: for example, it also applies to sheaves of spectra on topological stacks.
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Descendability and descent in topological weaves
A general descendability criterion for topological six-functor formalisms yields h-descent for étale motivic spectra and rational motivic cohomology, plus arc-descent in weights ≤1.