A Künneth theorem for the cyclic homology of A-infinity algebras

Abstract

The cyclic homology of a Z/2Z-graded, smooth and proper A-infinity category satisfying the Hodge-to-de-Rham degeneration property carries the structure of a polarized semi-infinite Hodge structure or the so-called EP-structure. Given two A-infinity algebras A and B with the above conditions, we construct a Künneth map from the tensor product of their cyclic homologies to the cyclic homology of the A-infinity tensor product A [circled times] B and show that it respects the EP-structures. As an application, we show that if A and B are equipped with weak Calabi-Yau structures, then A [circled times] B also inherits a weak Calabi-Yau structure. Also, we show that the Künneth quasi-isomorphism respects good splittings of the Hodge filtration on A and B compatible with the weak Calabi-Yau structure. Our explicit calculations rely on the combinatorial (tree) description of the tensor product of A-infinity algebras.