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

Lie Fu, Mauro Porta, Sarah Scherotzke, Nicolò Sibilla

TL;DR

This work extends HKR theory to derived Deligne–Mumford stacks in characteristic 0 by introducing the orbifold inertia stack $^{ m DM}X$ and proving a canonical HKR equivalence: $ ext{HH}_*(X)\, ow ext{equiv}\, ext{Γ}(^{ m DM}X, ext{Sym}( ext{L}_{^{ m DM}X}[1]))$, with an $S^1$-action matching the de Rham differential. It also develops Hochschild cohomology via $ ext{HH}^*(X)\, ow ext{equiv}\, ext{Γ}( extsf{T}[-1]^{ m DM}X,q^!( ext{O}_X))$ and extends HKR to cyclic, negative cyclic, and periodic cyclic homology using the derived de Rham complex of the inertia. The paper establishes foundational results on mapping stacks and inertia for derived DM stacks, proves structural properties of orbifold inertia, and provides extensive examples (global quotients, weighted projective lines, root stacks, group quotients, and mapping stacks) illustrating new computability for Hochschild invariants in singular or non-global-quotient settings. The results have implications for deformation theory of DM stacks and enable explicit computations beyond classical HKR regimes. Overall, the framework unifies loop-space, inertia, and derived de Rham techniques to broaden HKR applicability in derived geometry.

Abstract

We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-Căldăraru-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.

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

TL;DR

This work extends HKR theory to derived Deligne–Mumford stacks in characteristic 0 by introducing the orbifold inertia stack and proving a canonical HKR equivalence: , with an -action matching the de Rham differential. It also develops Hochschild cohomology via and extends HKR to cyclic, negative cyclic, and periodic cyclic homology using the derived de Rham complex of the inertia. The paper establishes foundational results on mapping stacks and inertia for derived DM stacks, proves structural properties of orbifold inertia, and provides extensive examples (global quotients, weighted projective lines, root stacks, group quotients, and mapping stacks) illustrating new computability for Hochschild invariants in singular or non-global-quotient settings. The results have implications for deformation theory of DM stacks and enable explicit computations beyond classical HKR regimes. Overall, the framework unifies loop-space, inertia, and derived de Rham techniques to broaden HKR applicability in derived geometry.

Abstract

We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-Căldăraru-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.

Paper Structure

This paper contains 29 sections, 49 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Historical context.
  3. Main results
  4. Applications
  5. Some results on mapping stacks
  6. Categorical finiteness properties
  7. Étale codescent
  8. Orbifold inertia for derived Deligne-Mumford stacks
  9. (Derived) orbifold inertia
  10. Deformation theory of orbifold inertia
  11. A computation in homotopy theory
  12. Proof of \ref{['thm:basics_inertia']}
  13. HKR isomorphism for derived Deligne--Mumford stacks
  14. Unipotent loops
  15. Shifted tangents
  16. ...and 14 more sections

Key Result

Theorem 1.2

For every derived Deligne--Mumford stack $X$ locally almost of finite presentation over a commutative $\mathbb{Q}$-algebra, there is a canonical equivalence of derived DM stacks over $X$: \xymatrix{ \mathsf T[-1] \mathsf I^{\mathsf{DM}} X \ar[rr]^{\widehat{\mathsf{aff}}^\ast}_{\simeq} \ar[dr

Theorems & Definitions (122)

  • Definition 1.1
  • Theorem 1.2: HKR for derived DM stacks, Theorem \ref{['thm:HKR_DM']}
  • Corollary 1.3: HKR for Hochschild homology, \ref{['thm:HKR-HH']} and \ref{['corS^1=dR']}
  • Corollary 1.4: HKR for Hochschild cohomology, \ref{['Hochcoh']}
  • Corollary 1.5: HKR for cyclic, negative cyclic and periodic cyclic homology, \ref{['cor:HCHNHP']}
  • Remark 1.1
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • ...and 112 more