2022/11/25 by Marc Chardin, Chardin, Marc, Rafael Holanda +1 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.14357
openalex publication_date 2022/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Zk-graded module is its support: degrees in which it is not zero. A stabilized version of it is adding -\mathbb Nk, in order for the complement (vanishing region) to be stable by addition of \mathbb Nk. 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 Bi 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 Bi'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 Bi's is also established. We then derive a consequence on linear resolutions for truncations of a graded module.