  We describe the Tate resolution of a coherent sheaf or complex of coherent
sheaves on a product of projective spaces. Such a resolution makes explicit all
the cohomology of all twists of the sheaf, including, for example, the
multigraded module of twisted global sections, and also the Beilinson monads of
all twists. Although the Tate resolution is highly infinite, any finite number
of components can be computed efficiently, starting either from a Beilinson
monad or from a multigraded module.
