Multigraded Tor and local cohomology

TL;DR

This paper extends notions of regularity and invariants to multigraded modules over products of projective spaces, analyzing the relationship between free resolutions and local cohomology supports.

Contribution

It establishes a connection between stabilized cohomology supports and shifts in graded free resolutions in the multigraded setting.

Findings

Stabilized support of cohomology corresponds to union of supports for individual factors.

Shifts in free resolutions lie within the intersection of stabilized supports.

A bijection between stabilized supports of Tor modules and local cohomologies is proven.

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.