The Auslander-Reiten translate on monomial rings

For t in Nn, 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, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator xa has a⩽t. We compute the multigraded cohomology and Betti spaces of Ntk(S/I) for every iterate k and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley-Reisner rings.