Cohomological properties of non-standard multigraded modules

In this paper we study some cohomological properties of non-standard multigraded modules and Veronese transforms of them. Among others numerical characters, we study the generalized depth of a module and we see that it is invariant by taking a Veronese transform. We prove some vanishing theorems for the local cohomology modules of a multigraded module, as a corollary of these results we get that the depth of a Veronese module is asymptotically constant.