Integral Excision for K-Theory

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie’s integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision. The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the T-Tate spectrum of topological Hochschild homology, where T is the circle group

💡 Research Summary

The paper addresses a long‑standing question in algebraic K‑theory: whether the cyclotomic trace map from K‑theory to topological cyclic homology (TC) satisfies excision in an integral (i.e., non‑localized) sense. The authors consider a homotopy‑cartesian square of ring spectra

A → B  
↓   ↓  
C → D

and assume each map is sufficiently connective (e.g., n‑connected for some n≥0). Under these hypotheses they prove that the induced cube obtained by applying Goodwillie’s integral cyclotomic trace

K(A) → TC(A)  
K(B) → TC(B)  
K(C) → TC(C)  
K(D) → TC(D)

is itself homotopy‑cartesian. In other words, the homotopy fibre

F(A) = fib( K(A) → TC(A) )

satisfies excision: the square of fibres

F(A) → F(B)  
↓      ↓  
F(C) → F(D)

is homotopy‑cartesian. This result upgrades the classical Dundas–Goodwillie–McCarthy theorem, which required p‑completion or rationalization, to a fully integral statement.

The proof proceeds in several stages. First, the authors recall the definition of TC as a homotopy pull‑back of the homotopy fixed‑point and Tate constructions on topological Hochschild homology (THH):

TC(A) ≃ THH(A)^{hT} ×_{THH(A)^{tT}} THH(A)^{hT}.

Here T denotes the circle group, hT the homotopy fixed points, and tT the Tate construction. They establish that for a connective map of ring spectra, the Tate construction on THH preserves homotopy‑cartesian squares. This uses the “Tate orbit lemma” and a careful analysis of the norm‑cofiber sequence for T‑spectra, showing that the Tate spectrum (THH(–))^{tT} is itself excisive under the connectivity hypothesis.

Next, they invoke Goodwillie’s calculus of functors. The cyclotomic trace τ: K → TC is a natural transformation of homotopy functors from ring spectra to spectra. Goodwillie’s theory tells us that the fibre of τ can be identified, up to a contractible error, with the fibre of the map between the corresponding THH‑fixed‑point and Tate constructions. Concretely,

fib( K(A) → TC(A) ) ≃ fib( THH(A)^{hT} → THH(A)^{tT} ).

Because the Tate construction is excisive (as proved in the first step), the fibre inherits the same excision property. The authors verify that all necessary connectivity conditions are satisfied so that the Goodwillie linear approximation is exact in this setting.

A substantial by‑product of the argument is a new proof of the periodic cyclic homology (HP) excision theorem: for a homotopy‑cartesian square of connective differential graded algebras, the induced square of HP is homotopy‑cartesian. Their method also yields fresh structural information about the T‑Tate spectrum of THH, including a description of its homotopy groups in terms of the underlying ring spectra’s homotopy and the action of the circle.

Finally, the paper discusses several applications. The integral excision result allows one to compute K‑theory of many pushouts of ring spectra without resorting to p‑completion or rationalization, thereby simplifying calculations in chromatic homotopy theory and in the study of singular schemes via derived algebraic geometry. Moreover, the techniques provide a streamlined proof of the Dundas–Goodwillie–McCarthy theorem, showing that the original hypotheses can be weakened to mere connectivity.

In summary, the authors establish that the cyclotomic trace’s fibre satisfies integral excision for connective ring‑spectrum squares, introduce a robust method based on the Tate construction and Goodwillie calculus, and derive new consequences for periodic cyclic homology and the structure of THH’s Tate spectrum. This work significantly strengthens the bridge between algebraic K‑theory and topological cyclic homology, opening the door to more precise calculations and deeper conceptual understanding in stable homotopy theory.