  We show that the maximal shifts in the minimal free resolution of the
quotients of a polynomial ring by a monomial ideal are subadditive as a
function of the homological degree. This answers a question that has received
some attention in recent years. To do so, we define and study a new model for
the homology of posets, given by the so called synor complex. We also introduce
an Eilenberg-Zilber type shuffle product on the simplicial chain complex of
lattices.
  Combining these concepts we prove that the existence of a non-zero homology
class for a lattice forces certain non-zero homology classes in lower
intervals. This result then translates into properties of the minimal free
resolution. In particular, it implies a generalization of the original
question.
