Cycle maps on cohomology theories for dg-categories and their applications

Abstract

In this article, we propose noncommutative versions of Tate conjecture and Hodge conjecture. If we consider these conjectures for a dg-category of perfect complexes over a certain schemes $X$, then they are equivalent to the classical Tate and Hodge conjectures for $X$ respectively. We also propose a strategy of how to prove these conjectures by utilizing a version of motivic Bass conjecture.

References in corpus (3)

• The Tate conjecture over finite fields (AIM talk)
• XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio)
• Parshin's conjecture and motivic cohomology with compact support