Categorical K\"unneth formulas for cohomological motives

Abstract

The manuscript at hand systematically studies K\"unneth formulas at a categorical level. We give criteria for an abstract six functor formalism to satisfy the categorical K\"unneth formula, and use this to formulate conjectures for categories of \'etale motives. As supporting evidence for these conjectures, we prove categorical K\"unneth formulas for adic sheaves and for cohomological motives, i.e., \'etale motives modulo the kernel of the adic realization.