Topological Hochschild homology (THH)

Topological Hochschild homology (THH)
Name	Topological Hochschild homology
Abbreviation	THH
Field	Algebraic topology
Introduced	1980s
Key figures	Bökstedt, Goodwillie, Hesselholt, Madsen, McCarthy

Topological Hochschild homology (THH) is an invariant in algebraic topology connecting stable homotopy theory, algebraic K-theory, and derived algebraic geometry. Originating from work by Bökstedt, Goodwillie, Hesselholt, and Madsen, THH refines classical Hochschild homology to the setting of ring spectrums and monoidal model categorys, providing a homotopy-theoretic trace that feeds into calculations for Quillen's algebraic K-theory of rings and modern approaches to motivic homotopy theory.

Definition and basic properties

THH is defined for a ring spectrum or an associative algebra object in a symmetric monoidal model category such as EKMM S-modules, symmetric spectrums, or orthogonal spectrums, producing a cyclotomic spectrum that encodes circle actions by S^1 and Frobenius lifts related to Witt vectors. It satisfies invariance properties under Morita equivalence and derived equivalences that mirror classical Hochschild–Kostant–Rosenberg theorem phenomena, and admits a Bökstedt spectral sequence converging from Hochschild-like homology of homotopy groups to the homotopy groups of THH. Key formal properties include functoriality for maps of E_1 algebras, multiplicative structures for E_2 algebras, and compatibility with localization and completion as studied by Lurie, Toen, and Vezzosi.

Constructions and models

Constructions of THH use cyclic bar constructions adapted to spectra: the Bökstedt construction, the cyclic nerve in S-modules, and the geometric realization of cyclic objects in simplicial model categorys. Models include Bökstedt's original point-set level approach in orthogonal spectra and modern ∞-categorical formulations via ∞-categories and algebraic topology of higher categories developed by Lurie and Barwick. Alternate presentations use the bar construction in the category of dg-algebras for differential graded enhancements, or via topological operads and A∞-algebra techniques as employed by Stasheff-inspired frameworks. For structured ring spectra, equivariant refinements produce cyclotomic structures studied by Nikolaus and Scholze.

Relation to algebraic Hochschild homology and cyclic homology

THH generalizes classical Hochschild homology of associative algebras over a ring by replacing chain complexes with spectra, and it admits comparison maps to Hochschild homology via passage to Eilenberg–MacLane spectra associated to rings studied by Eilenberg and Mac Lane. It interacts with cyclic homology through cyclotomic trace maps linking THH to negative cyclic homology and periodic cyclic homology in the work of Connes, Goodwillie, and McCarthy. These relationships are central in trace methods comparing THH-based invariants to Dennis trace and Chern character style constructions used by Quillen and Waldhausen in algebraic K-theory contexts.

Computational techniques and examples

Computations of THH exploit spectral sequences, equivariant homotopy fixed point methods, and descent techniques such as Tate spectral sequences and Dundas–Goodwillie–McCarthy excision theorems. Classic computations include THH of finite fields and p-adic integers analyzed by Hesselholt and Madsen, THH of complex K-theory and Morava K-theory studied in work by Angeltveit and Rognes, and calculations for group algebras and matrix algebras illustrating Morita invariance as in papers by Bokstedt and Hess. Tools such as the Bökstedt spectral sequence, the homotopy fixed point spectral sequence for S^1-actions, and the Tate construction are standard in explicit computations appearing in literature by Ausoni, Dundas, Rognes, and Nikolaus.

Trace methods and algebraic K-theory

THH forms the input to the cyclotomic trace map from algebraic K-theory to TC, bridging to THH via the cyclotomic structure exploited by Bökstedt, Hsiang, Madsen, Dundas, Goodwillie, and McCarthy. The resulting trace methods yield powerful computational approximations for K-theory of rings, schemes, and ring spectra; milestone results include the Dundas–Goodwillie–McCarthy theorem on relative K-theory, the work of Hesselholt and Madsen on K-theory of local fields, and advances by Clausen, Mathew, and Nikolaus on cyclotomic spectra and TC in an ∞-categorical setting.

Variants and generalizations

Variants include topological cyclic homology (TC), negative and periodic versions (TC^-, TP), relative THH for maps of ring spectra, and equivariant refinements such as genuine cyclotomic spectra studied by Blumberg and Mandell. Generalizations extend to THH-type invariants for spectral algebraic geometry objects studied by Lurie and Toen, and to motivic or logarithmic refinements appearing in the work of Kato, Hesselholt, and Rognes on log THH and log TC for log schemes and logarithmic geometry contexts.

Applications in topology and algebraic geometry

Applications span computations in stable homotopy theory such as understanding structure of Picard groups of ring spectra, analyzing chromatic homotopy theory phenomena via Morava theories, and informing calculations in moduli of formal groups and elliptic cohomology studied by Hopkins and Miller. In algebraic geometry, THH and TC inform K-theory of schemes, p-adic Hodge theory connections explored by Faltings and Fontaine, and derived de Rham and crystalline comparisons in works by Bhatt, Morrow, and Scholze in the context of perfectoid spaces and p-adic geometry.

Category:Algebraic topology