  For a multidegree t in N^n, E.Miller has defined a category of positively
t-determined modules over the polynomial ring S in n variables. We consider the
Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A
monomial ideal I is positively t-determined if every generator x^a has a \leq
t. We compute the multigraded cohomology- and betti spaces of Na_t^k(S/I) for
every iterate k, and also the S-module structure of these cohomology modules.
This comprehensively generalizes results of Hochster and Gr\"abe on local
cohomology of Stanley-Reisner rings.
