  Let A be a dg algebra over 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 of M with itself 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.
