Bokstedt Hsiang Madsen

Bokstedt Hsiang Madsen
Name	Bokstedt Hsiang Madsen
Fields	Algebraic K-theory, Homotopy theory, Topological cyclic homology, Category theory
Known for	Bokstedt trace, relationships among Topological Hochschild homology, Algebraic K-theory, and Cyclotomic spectra
Notable works	"Bokstedt-Hsiang-Madsen trace" (construction)

Bokstedt Hsiang Madsen is a foundational construction and collection of results in Algebraic K-theory and Homotopy theory connecting Topological Hochschild homology and Topological cyclic homology via the trace methods initiated by Morten Brun, Bjorn Dundas, Ib Madsen, Thomas Goodwillie, and Bjorn Ivar Madsen. It provides tools used in calculations involving Quillen, Waldhausen, Dennis trace, and modern approaches to cyclotomic structures arising from work by Angeltveit, Blumberg, Mandell, and Nikolaus.

History and Origins

The construction emerged from efforts to relate Algebraic K-theory of rings and Waldhausen K-theory of categories to computable invariants like Topological Hochschild homology (THH) and Topological cyclic homology (TC). Early antecedents include the Dennis trace map from Quillen K-theory to Hochschild homology, and subsequent advances by Bokstedt, Hsiang, and Madsen built on techniques developed in the milieu of Stable homotopy theory and the study of S1-actions on loop spaces inspired by work of Segal, May, and Boardman–Vogt. Influential conferences and collaborations involving Institute for Advanced Study, Massachusetts Institute of Technology, and Nordic Topology Network catalyzed dissemination, with formalization influenced by categorical frameworks from Mac Lane and Grothendieck-inspired sheaf theory.

Construction and Model Structure

The Bokstedt–Hsiang–Madsen approach constructs a trace map using models of Topological Hochschild homology built from cyclic bar constructions and symmetric spectra frameworks introduced by Hovey, Shipley, and Smith. The construction relies on equivariant model categories such as those developed by Mandell–May–Schwede–Shipley and the cyclotomic spectrum formalism later axiomatized by Nikolaus–Scholze. Central objects include cyclic objects with coherent S1-action coming from the work of Connes on cyclic homology and enriched category techniques related to Kelly and Leinster. The model structure uses cofibrantly generated model categories, fibrant replacement functors from Quillen, and norm maps akin to those in equivariant stable homotopy theory as studied by Lewis, May, and Steinberger.

Homotopy Theoretic Properties

Bokstedt–Hsiang–Madsen constructions exhibit homotopy invariance properties paralleling those of Topological Hochschild homology and are compatible with localization sequences in the spirit of Thomason–Trobaugh and excision phenomena examined by Suslin and Wodzicki. They admit spectral sequences relating Hochschild-type homology to homotopy groups much as the Bockstein and Connes spectral sequences studied by Connes and Loday operate. Fixed-point and geometric fixed-point functors for cyclic and cyclotomic spectra, developed in the tradition of Lewis–May–Steinberger and refined by Hill–Hopkins–Ravenel, interact with transfer and norm maps to produce Frobenius and Verschiebung operators analogous to structures observed in Dieudonné theory and Crystalline cohomology contexts explored by Berthelot and Illusie.

Relationship to Topological Cyclic Homology

The trace map furnishes a comparison map from Algebraic K-theory to Topological cyclic homology (TC) via THH and cyclotomic structures; this comparison is central to the Bokstedt–Hsiang–Madsen program. The relationship was sharpened by subsequent work of Dundas, Goodwillie, and McCarthy, culminating in the Dundas–Goodwillie–McCarthy theorem comparing relative K-theory and relative TC under nilpotent extensions, and by the Nikolaus–Scholze reformulation of cyclotomic spectra that streamlines the passage from THH to TC. Applications leverage input from crystalline comparison theorems via connections to Beilinson and Bhatt perspectives, and to calculations influenced by Hesselholt and Madsen on K-theory of local fields and Witt vector phenomena in the spirit of Hesselholt–Madsen work.

Applications and Examples

Concrete applications include computations of K-theory for rings of integers in local and global fields building on results by Hesselholt, Madsen, and Voevodsky-adjacent motivic techniques; analyses of cyclic fixed points for structured ring spectra used by Lurie in higher algebra; and explicit calculations for matrix algebras and group algebras extending classical results of Swan and Cartan–Eilenberg. Examples appearing in the literature include K-theory computations for truncated polynomial algebras computed by Bokstedt and collaborators, TC-calculations for p-adic rings related to work of Fargues and Fontaine, and applications in chromatic homotopy theory influenced by Ravenel and Hopkins.

Computational Techniques and Calculations

Techniques exploit spectral sequences such as the Bokstedt spectral sequence, cyclotomic spectral sequences, and the homotopy fixed point spectral sequence introduced by Adams and elaborated by Greenlees–May. Practical computations use models of symmetric spectra, orthogonal spectra, and equivariant orthogonal spectra from Mandell–May–Schwede–Shipley, along with software-assisted methods inspired by computational work in Stable homotopy groups of spheres and tools developed in the Homotopy Type Theory community. Important computational innovations include trace methods, cyclotomic Frobenius calculations by Hesselholt, and descent techniques leveraging étale and pro-étale comparisons explored by Scholze and Bhatt–Morrow–Scholze.

Category:Algebraic K-theory Category:Topological Hochschild homology Category:Homotopy theory