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.