Cohomological and motivic inclusion-exclusion

TL;DR

This paper extends the inclusion-exclusion principle to a categorical setting for topological spaces and schemes, leading to new spectral sequences and algebraic proofs of homological stability conjectures.

Contribution

It introduces a categorification of inclusion-exclusion for sheaves, resulting in functorial spectral sequences and an algebraic proof of a homological stability conjecture.

Findings

Derived category filtrations generalize classical spectral sequences

Spectral sequences connect stratified spaces and Vassiliev-type invariants

Provides an algebraic proof of Vakil and Wood's homological stability conjecture

Abstract

We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.