Hochschild homology and the derived de Rham complex revisited
Arpon Raksit
TL;DR
The paper develops a universal-property framework for two central invariants in derived algebraic geometry: the derived de Rham complex and HKR-filtered Hochschild homology. It introduces homotopy-coherent cochain complexes and a filtered circle action to encode the extra structures these invariants carry, and proves that the HKR filtration’s associated graded recovers derived de Rham data. A key innovation is the filtered circle construction, which yields natural filtrations on cyclic, negative cyclic, and periodic cyclic homology and explains Adams-operations behavior in this integral setting. By connecting Koszul duality, Beilinson t-structures, and derived commutative algebra in a common framework, the work provides a characteristic-free, conceptual bridge between Hochschild-type invariants and derived de Rham cohomology with broad implications for nonconnective contexts and later applications to Adams operations.
Abstract
We characterize two objects by universal property: the derived de Rham complex and Hochschild homology together with its Hochschild-Kostant-Rosenberg (HKR) filtration. This involves endowing these objects with extra structure, built on notions of "homotopy-coherent cochain complex" and "filtered circle action" that we study here. We use these universal properties to give a conceptual proof that the associated graded of the HKR filtration identifies with the derived de Rham complex, as well as to give a new construction of the filtrations on cyclic, negative cyclic, and periodic cyclic homology that relate these invariants to derived de Rham cohomology.