  We study homological multiplicities of spherical varieties of reductive group
$G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the
category of smooth representations of a $p$-adic group, we introduce a sheaf
that measures these multiplicities.
  We show that these multiplicities are finite whenever the usual
mutliplicities are finite, in particular this holds for symmetric varieties,
conjectured for all spherical varieties and known for a large class of
spherical varieties. Furthermore, we show that the Euler-Poincar\'e
characteristic is constant in families induced from admissible representations
of a Levi $M.$ In the case when $M=G$ we compute these multiplicities more
explicitly.
