Hochschild cohomology with support

TL;DR

This paper studies the functoriality of map-graded Hochschild complexes, showing they form a sheaf over a certain topology, and introduces Hochschild complexes with support and a general Grothendieck construction.

Contribution

It establishes the sheaf property of map-graded Hochschild complexes and introduces a framework for Hochschild complexes with support using a Grothendieck construction.

Findings

Map-graded Hochschild complexes form a sheaf over the infinity-cover topology.

Hochschild complexes with support are derived from functoriality related to injections.

A new Grothendieck construction unifies presheaves of algebras and arrow categories.

Abstract

In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to "injections" between map-graded categories, we obtain Hochschild complexes "with support". We revisit Keller's arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.