  Notions of Castelnuovo-Mumford regularity and of $a^*$ invariant were
extended from standard graded algebras to the toric setting.
  We here focus our attention on the standard multigraded case, which
corresponds to a product of $k$ projective spaces. A natural notion for a
$\mathbb Z^k$-graded module is its support: degrees in which it is not zero. A
stabilized version of it is adding $-\mathbb N^k$, in order for the complement
(vanishing region) to be stable by addition of $\mathbb N^k$.
  Cohomology of twists of a sheaf on a product of projective spaces, provided
by a graded module, are given by local cohomologies with respect to the product
$B$ of the ideals $B_i$ generated by the $k$ sets of variables.
  Our results shed some light on a central issue, the relation between shifts
in graded free resolution and cohomology vanishing: it shows that stabilized
support of cohomology with respect to $B$ corresponds to the union of
stabilized supports for cohomologies in the $B_i$'s, while shifts in (some of
the) graded free resolutions are inside the intersection of these stabilized
supports. A one-to-one correspondence between stabilized supports of Tor
modules and of local cohomologies with respect to the sum of the $B_i$'s is
also established.
  We then derive a consequence on linear resolutions for truncations of a
graded module.
