[Paper Review] On the Blumberg-Mandell Künneth theorem for TP

Benjamin Antieau, Akhil Mathew+1 more|arXiv (Cornell University)|Oct 16, 2017

Homotopy and Cohomology in Algebraic Topology14 citations

TL;DR

This paper provides a new, stronger proof of the Blumberg-Mandell Künneth theorem for periodic topological cyclic homology (TP) of smooth and proper dg categories over perfect fields of positive characteristic. Using a spectral algebraic approach centered on the regularity of π⁎THH(k), the authors establish that dualizable objects in the category of THH(k)-modules with S¹-action are perfect, which implies a refined Künneth formula and a finiteness theorem for TC⁻(C).

ABSTRACT

We give a new proof of the recent Künneth theorem for periodic topological cyclic homology (TP) of smooth and proper dg categories over perfect fields of characteristic p>0 due to Blumberg and Mandell. Our result is slightly stronger and implies a finiteness theorem for topological cyclic homology (TC) of such categories.

Motivation & Objective

• To strengthen the Blumberg-Mandell Künneth theorem for TP in positive characteristic by proving a more general perfection result.
• To establish a finiteness theorem for topological cyclic homology (TC⁻) of smooth and proper dg categories over perfect fields of characteristic p > 0.
• To show that the Künneth formula holds under weaker assumptions—specifically, that only one of the two categories needs to be smooth and proper.
• To apply the strengthened theorem to derive finiteness results for crystalline cohomology and TF homology in the context of supersingular K3 surfaces.
• To demonstrate that the non-finite generation of certain crystalline cohomology terms is resolved by periodicity in TP, preserving finiteness of TP(X) over TP(k).

Proposed method

• The authors work in the ∞-category of THH(k)-modules with S¹-action, using the Tate construction to define TP and TC⁻ as THH^{tS¹} and THH^{hS¹}.
• They prove that any dualizable object in this category is perfect, which implies the Künneth theorem for TP and TC⁻.
• The key technical input is the regularity of the graded ring π⁎THH(k) ≃ W(k)[x±1] with |x| = 2, which ensures that perfect objects are dualizable.
• The proof uses a thick subcategory argument: perfect objects are those in the thick subcategory generated by the unit, and duality implies compactness in this context.
• The authors generalize the result to relative THH over localizations of rings of integers at primes above p, replacing THH with relative THH over S⁰[q] with q mapped to a uniformizer.
• They apply the Nikolaus–Scholze formula for TC to deduce that non-finite generation in TF homology is resolved by periodicity in TP.

Experimental results

Research questions

• RQ1 Does the Künneth theorem for TP hold when only one of the two smooth and proper dg categories is compact, rather than both?
• RQ2 Can the Blumberg-Mandell theorem be strengthened to imply finiteness results for TC⁻(C) and TF(C) in positive characteristic?
• RQ3 Is the non-finite generation of certain crystalline cohomology terms in the conjugate spectral sequence compatible with the finiteness of TP(X) over TP(k)?
• RQ4 Can the perfection of THH(C) in the category of THH(k)-modules with S¹-action be established via the regularity of π⁎THH(k)?
• RQ5 Does the periodicity of TP(k) stabilize non-finite terms in the local-global spectral sequence for TF(X), ensuring finiteness of TF(X)?

Key findings

• The authors prove that any dualizable object in the ∞-category of THH(k)-modules with S¹-action is perfect, which strengthens the Blumberg-Mandell Künneth theorem.
• The Künneth formula for TP(C ⊗k D) holds when only one of C or D is smooth and proper over k, not necessarily both.
• The result implies a finiteness theorem for TC⁻(C) = THH(C)^{hS¹}, showing that TC⁻(C) is a finitely generated W(k)-module when C is smooth and proper over k.
• For a supersingular K3 surface X over a perfect field k of characteristic p > 0, TF_{-1}(X) is not finitely generated as a W(k)-module, despite non-finite generation in intermediate cohomology terms.
• The non-finite generation in the conjugate spectral sequence is resolved by periodicity in TP, as the periodicization of the spectral sequence ensures that the relevant differentials are balanced across periodic copies.
• The proof shows that the failure of finite generation in H^2(X, W_nΩ^1) is compensated by periodicity in TP, preserving the finiteness of TP(X) over TP(k).