Two Mayer-Vietoris spectral sequences for $\mathcal{D}$-modules

TL;DR

This paper introduces two spectral sequences inspired by Mayer-Vietoris techniques for computing cohomology in the context of $\

Contribution

It develops two new spectral sequences for $\

Findings

Provides methods to compute cohomology of hyperplane arrangements.

Introduces spectral sequences related to $\

Demonstrates applications in algebraic geometry and $\

Abstract

We provide two Mayer-Vietoris-like spectral sequences related to the localization over the complement of a closed subvariety of an algebraic variety by using techniques from $\mathcal{D}$-modules and homological algebra. We also give, as an application of the previous, a method to calculate the cohomology of the complement of any arrangement of hyperplanes over an algebraically closed field of characteristic zero.