Higher order Hochschild cohomology of schemes

TL;DR

This paper develops a theory of higher Hochschild cohomology for schemes over a field of characteristic zero, establishing its properties, decompositions, and relations to classical Hochschild cohomology using advanced homological tools.

Contribution

It introduces two new definitions of higher Hochschild cohomology for schemes and proves a Hodge decomposition for smooth varieties, extending classical results.

Findings

Higher Hochschild complex commutes with localization.

Established Hodge decomposition for smooth algebraic varieties.

Generalized Swan's equivalence to separated schemes.

Abstract

We show that Higher Hochschild complex associated to a connected pointed simplicial set commutes with localization of commutative algebras over a field of characteristic zero. Then, we define in two ways higher order Hochschild cohomology of schemes over a field of characteristic zero. Originally, we can take the hyperext functor of the sheaf associated to Higher Hochschild presheaf. We obtain a Hodge decomposition for higher order Hochschild cohomology of smooth algebraic varieties over a field of characteristic zero which generalizes Pirashvili's Hodge decomposition. We can also define the Higher Hochschild cohomology of order d of a separated scheme by taking the ext functor of its structure presheaf over the Higher Hochschild presheaf of order d -- 1. These two definitions are really close to those of Swanfor classical Hochschild cohomology, but our tools are model categories and derived functors. We also generalize the equivalence of Swan's definitions to any separated schemes over a field.