[Paper Review] Trace theories, Bokstedt periodicity and Bott periodicity

D. Kaledin|arXiv (Cornell University)|Apr 8, 2020

Homotopy and Cohomology in Algebraic Topology17 references4 citations

TL;DR

This paper establishes a conceptual framework linking Topological Hochschild Homology (THH), Bökstedt periodicity, and Bott periodicity via trace theories and stabilization. It proves that THH(A) is isomorphic to the zeroth term of the conjugate filtration on co-periodic cyclic homology, identifying the Bökstedt generator σ with the Bott periodicity element u⁻¹, and provides a new algebraic description of THH in terms of Hochschild-Witt homology.

ABSTRACT

We flesh out the theory of "trace theories" and "trace functors" sketched in arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction theorem claiming that a trace theory is completely determined by the associated trace functor. As an application, we consider Topological Hoshschild Homology $THH(A,M)$ of a algebra $A$ over a perfect field of positive characteristic, with coefficients in a bimodule $M$, and prove two comparison results. Firstly, we give a very simple algebraic model for THH in terms of Hochschild-Witt Homology WHH of arXiv:1604.01588 (and we also identify $TP(A)$ with the periodic version $WHP(A)$ of WHH). Secondly, we prove that $THH(A)$ is identified with the zero term of the conjugate filtration on the co-periodic cyclic homology $\overline{HP}(A)$ of arXiv:1509.08784, and the isomorphism sends the Bokstedt periodicity generator to the Bott periodicity generator. We also give an independent proof of Bokstedt periodicity that is somewhat shorter than the usual ones.

Motivation & Objective

• To provide a conceptually clear and algebraic proof of Bökstedt periodicity in Topological Hochschild Homology.
• To establish a comparison between THH and co-periodic cyclic homology, identifying the Bott periodicity generator.
• To express THH(A,M) in terms of Hochschild-Witt homology WHH(A,M) for any k-algebra A and bimodule M.
• To clarify the role of the conjugate filtration in relating THH to periodic cyclic homology.
• To resolve the mystery of the 'wrong' totalization in p-adic Hodge theory by showing it is equivalent to THH.

Proposed method

• Uses the theory of trace theories to reduce the computation of THH(A,M) to THH(k,M) for a perfect field k.
• Applies stabilization and additivization techniques to functorial constructions on bimodules and their homotopy categories.
• Employs relative Tate cohomology and spectral sequences to analyze the homotopy groups of THH and WHH.
• Constructs a ring map φ: THH(k) → R such that φ(σ) is not nilpotent, proving k[σ] embeds into THH(k).
• Uses the cyclotomic structure and truncation of the cyclotomic trace to define φ without relying on advanced spectral sequence techniques.
• Applies the conjugate filtration and compares it with the spectral sequence from THH to HH[σ], showing compatibility with Bott periodicity.

Experimental results

Research questions

• RQ1 Can Bökstedt periodicity in THH be proven via a purely algebraic and conceptual method rather than spectral sequence arguments?
• RQ2 Is there a direct algebraic isomorphism between THH(A) and the conjugate filtration of co-periodic cyclic homology?
• RQ3 How does Hochschild-Witt homology WHH relate to THH in the context of bimodules over a k-algebra?
• RQ4 What is the precise relationship between the Bökstedt generator σ and the Bott periodicity generator u⁻¹ in periodic homology theories?
• RQ5 Why does the 'wrong' totalization in p-adic Hodge theory correspond to THH, and how is this explained by the conjugate filtration?

Key findings

• THH(A) is canonically isomorphic to the zeroth term of the conjugate filtration on co-periodic cyclic homology, i.e., THH(A) ≅ V₀CP̄(A).
• The Bökstedt periodicity generator σ ∈ THH(k) maps to the Bott periodicity generator u⁻¹ ∈ CP̄(A) under the isomorphism, so THH(A) ⊗k[σ] k(σ,σ⁻¹) ≅ CP̄(A).
• There exists a functorial isomorphism between THH(A,M) and WHH(A,M) for any k-algebra A and A-bimodule M.
• The spectral sequence from HH(A)[σ] to THH(A) is isomorphic to the conjugate spectral sequence, explaining the degeneration pattern in positive characteristic.
• The map φ: THH(k) → R with φ(σ) non-nilpotent proves that k[σ] embeds into THH(k), and dimension reasons imply THH(k) = k[σ].
• The use of relative Tate cohomology instead of Tate homology allows a uniform proof for all primes p, including p=2, resolving a technical gap in earlier work.