  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.
