Categorical Künneth formulas for cohomological motives
Timo Richarz, Jakob Scholbach
TL;DR
The paper develops a unified, categorical approach to Künneth formulas for motives via abstract six-functor formalisms. By establishing criteria for full faithfulness and essential surjectivity of exterior product functors, it proves categorical Künneth results for ind-dualizable objects in adic sheaves, cohomological motives, and Weil motives, under natural hypotheses such as $Y$ being smooth with $\text{char}\,k=0$ or $Y$ proper. It also formulates conjectures for étale motives and frames these results in a broader ∞-categorical context using enriched mapping spaces and descent techniques. The findings provide structural decompositions of motive categories on products, with potential applications to intersection motives and shtuka-related constructions, and offer a robust framework for Künneth formulas across multiple motivic settings.
Abstract
The manuscript at hand systematically studies Künneth formulas at a categorical level. We give criteria for an abstract six functor formalism to satisfy the categorical Künneth formula, and use this to formulate conjectures for categories of étale motives. As supporting evidence for these conjectures, we prove categorical Künneth formulas for adic sheaves and for cohomological motives, i.e., étale motives modulo the kernel of the adic realization.