# Deletion-restriction for sheaf homology of graded atomic lattices

## TL;DR

This paper develops a long exact sequence for sheaf homology on graded atomic lattices, enabling computation of hyperplane arrangement lattice homology and generalizing Lusztig's classical result.

## Contribution

It introduces a deletion-restriction long exact sequence for sheaf homology on graded atomic lattices, extending previous work to new classes of arrangements.

## Key findings

- Derived a long exact sequence relating homology of lattices and their deletions/restrictions.
- Computed the sheaf homology of hyperplane arrangement lattices using the new sequence.
- Generalized Lusztig's classical result to a broader context.

## Abstract

We give a long exact sequence for the homology of a graded atomic lattice equipped with a sheaf of modules, in terms of the deleted and restricted lattices. This is then used to compute the homology of the arrangement lattice of a hyperplane arrangement equipped with the natural sheaf. This generalises an old result of Lusztig.