Multigraded Tor and local cohomology

Abstract

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.