A sheaf of Hochschild complexes on quasi-compact opens

TL;DR

This paper constructs a sheaf of complexes on a scheme that captures Hochschild complexes locally, enabling a spectral sequence for Hochschild cohomology in quasi-compact schemes.

Contribution

It introduces a sheaf of Hochschild complexes on schemes, providing a new tool for local-to-global analysis of Hochschild cohomology.

Findings

Constructs a sheaf of complexes matching Hochschild complexes on quasi-compact opens.

Establishes a local-to-global spectral sequence for Hochschild cohomology.

Shows the sheaf is acyclic for sections on quasi-compact opens.

Abstract

For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open subset U of X, C(U) is quasi-isomorphic to the Hochschild complex of the scheme U. Since C is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if X is quasi-compact.