Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Let $A$ be a dg algebra over $\mathbb F_2$ and let $M$ be a dg $A$-bimodule. We show that under certain technical hypotheses on $A$, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_A^L M$ and converges to the Hochschild homology of $M$. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.