Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks

TL;DR

This paper extends the Hochschild-Kostant-Rosenberg theorem to derived Deligne-Mumford stacks, establishing a canonical isomorphism between Hochschild homology and the cohomology of differential forms on an orbifold inertia stack, with applications to various derived stacks.

Contribution

It introduces the orbifold inertia stack for derived DM stacks and proves a HKR isomorphism linking Hochschild homology to differential forms, generalizing previous results.

Findings

HKR theorem extended to derived DM stacks

Orbifold inertia stack provides a refined geometric object

Explicit computations for weighted projective lines and quotient stacks

Abstract

We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-C\u{a}ld\u{a}raru-Hablicsek in the smooth, global quotient case, although with different methods. To formulate our result, we introduce the notion of orbifold inertia stack of a derived DM stack; this supplies a finely tuned derived enhancement of the classical inertia stack, which does not always coincide with the classical truncation of the free loop space. We show that, in characteristic 0, given a derived DM stack, the shifted tangent bundle of its orbifold inertia stack is equivalent to its free loop space. This yields a canonical HKR isomorphism of algebras between the Hochschild homology of a derived DM stack and the cohomology of differential forms on its orbifold inertia stack. Moreover, this isomorphism intertwines the natural circle action and the de Rham differential. Similarly, HKR theorems for derived DM stacks are established for Hochschild cohomology, cyclic homology, negative cyclic homology, and periodic cyclic homology. As applications, we provide a rich supply of computations of Hochschild homology and Hochschild cohomology for interesting derived DM stacks, such as weighted projective lines, root stacks, quotients by algebraic groups, and mapping stacks, among others.