On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements

TL;DR

This paper links Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements with the cohomology of sheaves on affine and projective spaces, providing new insights into module freeness and Terao's conjecture.

Contribution

It establishes a Künneth formula connecting different cohomology theories, offering a new proof of Yuzvinsky's freeness criterion and a reformulation of Terao's conjecture.

Findings

Derived a Künneth formula for cohomology theories

Provided a new proof of Yuzvinsky's freeness criterion

Reformulated Terao's freeness conjecture

Abstract

We establish the relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space, respectively projective space associated to the module of logarithmic vector fields along the arrangement. Our main result gives a K\"unneth formula connecting the cohomology theories, answering a question by Yoshinaga. This, in turn, provides a characterization of the projective dimension of the module of logarithmic vector fields and yields a new proof of Yuzvinsky's freeness criterion. Furthermore, our approach affords a new formulation of Terao's freeness conjecture and a more general problem.