📝 SummarXiv

Abstract

We prove a Hochschild-Konstant-Rosenberg theorem for general 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 coincide with the classical truncation of the free loop space. We show that, in characteristic 0, the shifted tangent bundle of the orbifold inertia stack is equivalent to the free loop space. This yields an explicit HKR equivalence between Hochschild homology and differential forms on the orbifold inertia stack, as algebras. We also construct a stacky filtered circle, leading to a filtration on the Hochschild homology of a derived DM stack whose associated graded complex recovers the de Rham theory of its orbifold inertia stack. This provides a generalization of recent work of Moulinos-Robalo-To\"en to the setting of derived DM stacks.