Representability of continuous K-theory in rigid analytic motivic $\mathbb{A}^1$-homotopy theory

Can Yaylali; Christian Dahlhausen; Yicheng Zhou

REVIEW 3 major objections 3 minor 15 references

Representability of continuous K-theory in rigid analytic motivic $\mathbb{A}^1$-homotopy theory

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Continuous K-theory of rigid spaces is represented by the motivic object Z × BGL.

desk verdict The paper proves Nisnevich descent for continuous K-theory and a conditional representability theorem for its A1-localization; the descent result is solid and new, but the headline is tied to an open desingularization assumption. read the letter →

arxiv 2608.06209 v1 pith:2SDRNOI6 submitted 2026-08-06 math.KT math.AG

Christian Dahlhausen , Can Yaylali , Yicheng Zhou This is my paper · ORCID

classification math.KTmath.AG MSC 19E9914F4214G22

keywords continuousK-theoryanalyticrigidspacesNisnevichdescentA1-homotopytheorycondensedspectrapro-spectraBGL

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Continuous K-theory, the pro-spectrum-valued invariant built from reductions of a rigid space modulo powers of a uniformizer, is shown to satisfy Nisnevich descent, and under a regular-model assumption it is also $\mathbb{A}^1$-invariant. Because of this, the infinite loop space of connective analytic K-theory is represented in the rigid analytic motivic $\mathbb{A}^1$-homotopy category by the same motivic object that represents algebraic K-theory for schemes, namely $L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL})$. The paper also proves Weibel vanishing, identifies continuous homotopy K-theory with analytic K-theory, and establishes an unconditional $\mathbb{A}^1$-invariance statement for continuous K-theory on local Tate pairs with no regularity assumption. If these results stand, the motivic package of schemes—Grassmannian models, Bass delooping, enriched Hom-spaces—applies to K-theory of rigid analytic spaces.

What carries the argument

The central mechanism is the comparison between continuous K-theory and the K-theory of nuclear modules: the underlying spectrum of $K^{\mathrm{cont}}(A)$ is identified with $K(\mathrm{Nuc}(A))$, and this is lifted to an enriched equivalence of sheaves $K^{\mathrm{nuc}} \simeq \gamma_\kappa K^{\mathrm{cont}}$ with values in condensed spectra. Nuclear K-theory is already known to satisfy Nisnevich descent, and the limit-preserving comparison functor $\gamma_\omega \colon \mathrm{Pro}^\omega(\mathrm{Sp}_+)\to\mathrm{Cond}^\omega(\mathrm{Sp})$, which is conservative on bounded-below objects, transfers that descent back to pro-spectra. The representing object is then fixed by the classical presentation of connective algebraic K-theory as $L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL})$, combined with the equivalence $K^{\mathrm{an}}(A)\simeq (L_{\mathbb{A}^1}K^{\mathrm{cont}})(A)$.

What would settle it

Choose a local Tate pair $(A,A^+)$ with pseudo-uniformiser $\varpi$ that is not regular, and compare the pro-spectra $K^{\mathrm{cont}}(A)$ and $\mathrm{lim}_{t\mapsto\varpi^t} K^{\mathrm{cont}}(A\langle t\rangle)$; Theorem 4.27 asserts these are equivalent, so a single prime where $K_0$ or $K_{-1}$ differs would refute that claim. For the representability theorem itself, a sharper test is to find a regular analytic adic space $X$ for which $K^{\mathrm{cont}}(X)\to K^{\mathrm{cont}}(X\times\mathbb{A}^1)$ is not an equivalence, which would disprove the paper's Conjecture 4.24.

Watch

Extended reading notes

Core claim

Under the assumption $(\spadesuit)^{\mathrm{an}}_R$ that every smooth analytic adic space over the base is analytically locally $\mathrm{Spa}(A,A^+)$ with $A$ admitting a noetherian ring of definition and a proper regular model over it, there is a canonical equivalence $\Omega^\infty K^{\mathrm{an}}_{\ge 0} \simeq L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL})$ in $\mathrm{RigH}(B,\mathrm{Pro}^\omega(\mathrm{Spc}))$. For every smooth $X$ over $B$, after passage to light condensed spectra this gives the functorial formula $\Omega^\infty\gamma_\omega KH^{\mathrm{cont}}_{\ge 0}(X) \simeq \mathrm{Hom}_{\mathrm{Cond}^\omega(\mathrm{Spc})}(L_{\mathrm{mot}}X, L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL}))$. The route to this identification goes through the equivalence of continuous K-theory with K-theory of nuclear modules, Nisnevich descent for that nuclear K-theory, and the conservative comparison functor from pro-spectra to condensed spectra. Separately, the paper proves Nisnevich descent for $K^{\mathrm{cont}}$ and $KH^{\mathrm{cont}}$, generalized Weibel vanishing, and an $\mathbb{A}^1$-invariance theorem for continuous K-theory on local Tate pairs that needs no regularity hypothesis.

Load-bearing premise

The proof needs the resolution-of-singularities input that every smooth rigid space over the base is locally covered by rings that admit a proper regular model over a noetherian ring of definition extending the original ring; this is currently known only in certain characteristic-zero cases and remains an open problem in general.

Editorial extensions

If this is right

Nisnevich descent makes continuous K-theory a sheaf on rigid spaces, so covers and local-to-global arguments can be used to compute it.
Under the regular-model assumption, connective analytic K-theory is represented by $L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL})$, transferring motivic descriptions such as Grassmannian models to the rigid analytic setting.
The enriched condensed statement gives functorial Hom-space formulas for continuous homotopy K-theory on smooth rigid spaces with coefficients in light condensed spectra.
The analytic Bass construction produces a commutative algebra object $\mathrm{KGL}^{\mathrm{an}}$ in the rigid analytic motivic stable category representing non-connective analytic K-theory.
Weibel vanishing holds for continuous K-theory of finite-dimensional qcqs rigid spaces, with the explicit bound $N=d$ in the discretely valued or excellent cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the local Tate pair result can be glued along stalks of the Nisnevich sheaf, the regular-model assumption could be removed from the representability theorem; the paper leaves the necessary stalkwise comparison as an explicit open step.
The condensed-spectra enrichment invites an analytic analogue of motivic cohomology, defined as enriched Hom-groups into $L_{\mathrm{mot}}(\mathbb{Z}\times\mathrm{BGL})$, which would give a new computational tool for K-theory of rigid spaces.
One could test the dependence on regularity by computing continuous K-theory of a semistable or singular rigid space and comparing with the local Tate pair statement; Theorem 4.27 predicts $\mathbb{A}^1$-invariance even without regularity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's real content is Nisnevich descent for continuous K-theory (Thm 4.14) and a representability theorem for analytic K-theory in the rigid motivic category (Thm 5.1). The descent result is genuinely new and the route through Andreychev's nuclear K-theory and the conservative comparison functor is convincing. The A1-invariance for local Tate pairs without regularity (Thm 4.27) is a nice, unexpected result. The paper is also careful: it gives extended proofs for some terse statements in Andreychev's thesis and fixes a false lemma of his. That is real service to the community.

The soft spots are the following. First, the headline representability is conditional on (spade)^an_R, which is an open resolution-of-singularities assumption in general; the paper says this honestly in Remark 4.1, but the abstract's phrase 'continuous K-theory is representable' overstates the theorem, which actually represents the A1-localization KH^cont (identified with analytic K-theory) under that assumption. The stress-test note suggests the BGL identification might not actually need (spade)^an_R, since K^an is already an A1-invariant Nisnevich sheaf under milder hypotheses; as written the proof uses it, so the theorem's true hypotheses need checking. Second, the proof of Theorem 4.14 only explicitly treats Nisnevich squares of qcqs objects, while the statement is for all lft adic spaces; this is likely fixable by the same quasi-separated patching used in Lemma 4.23, but it should be spelled out. Third, Lemma 4.12 implicitly assumes the functor commutes with finite products; that is true for the K-theory functors in play, so it's a minor presentation issue.

None of this undermines the main line. The paper is serious, the mathematics looks sound, and the limitations are disclosed. It deserves a real referee round. I'd bring it to reading group if the group works in motivic homotopy or K-theory; I'd cite it for the descent result.

Referee Report

3 major / 3 minor

Summary. The paper studies continuous K-theory K^cont and analytic K-theory K^an on adic spaces locally of finite type over a base B which is either a nonarchimedean field or a Tate ring with a noetherian finite-dimensional ring of definition. The main unconditional result is Theorem 4.14: K^cont satisfies Nisnevich descent. This is proved by comparing K^cont, via a conservative comparison functor to condensed spectra, with the nuclear K-theory of Andreychev, which is known to satisfy Nisnevich descent. The paper also proves Weibel vanishing (Proposition 4.17), an A1-invariance statement for local Tate pairs without regularity assumptions (Theorem 4.27), and a comparison of the A1-localisation of K^cont with K^an (Proposition 4.21). Under an assumption (♠)^an_R on the existence of regular models locally in the analytic topology, which is an open desingularisation problem, Theorem 5.1 identifies the Ω^∞ of the connective cover of K^an (equivalently of KH^cont) with L_mot(Z×BGL) in the rigid analytic motivic homotopy category, with an enriched consequence for smooth X. The paper also includes an appendix correcting a lemma from Andreychev's thesis.

Significance. If the identified gaps are addressed, the unconditional Nisnevich descent theorem would be a significant contribution, as Nisnevich descent was a missing ingredient for motivic representability. The paper provides detailed foundational work on pro-spectra and condensed spectra, including a proof (credited to Scholze) of conservativity of the comparison functor, and a clarification of localisation sequences in dualisable categories. The conditional representability theorem is a natural analogue of Morel–Voevodsky's K≃Z×BGL and would be a strong result, though its current dependence on an open resolution-of-singularities hypothesis limits its scope. The paper is careful in many places, acknowledges the conditional nature in Remark 4.1, and corrects a false statement in the literature (Appendix A).

major comments (3)

[Abstract and Theorem 5.1] The abstract and title claim representability of continuous K-theory, but Theorem 5.1 represents the A1-localisation KH^cont, identified with analytic K-theory K^an via Proposition 4.21. On smooth X, the comparison K^cont(X) → KH^cont(X) is an equivalence only under Conjecture 4.24 (A1-invariance), which is known only under (♠)^an_R (Proposition 4.15). As stated, the abstract's 'continuous K-theory is representable' overstates the proved theorem; the paper should either adjust the title and abstract or explicitly state that representability is proved for the A1-localisation of continuous K-theory.
[Proposition 4.21 and Lemma 4.23] The interchange of the geometric realisation colimit over n∈Δ^op with the pro-limit over j is justified by 'uniformly bounded below' citing [KST23, Lem. 2.8]. However, the affinoid algebras A⟨Δ^n_{π^j}⟩ have dimension dim(A)+n, so the Weibel-vanishing bound from Proposition 4.17 depends on n; there is no evident uniform bound independent of n. The manuscript should spell out how [KST23, Lem. 2.8] applies in this situation, or restrict the statement to cases where such a uniform bound holds. This is load-bearing for the identification KH^cont≃K^an and for the Nisnevich sheaf property of KH^cont.
[Theorem 5.1, proof of BGL identification] The identification Ω∞τ≥1 K^an(A) ≃ 'lim'_ρ BGL(A⟨Δ⟩_ρ) is attributed to [KST19a, Lem. 7.5] and said to be available under (♠)^an_R. Since (♠)^an_R is an open desingularisation assumption (Remark 4.1), the representability theorem is conditional on a major open problem. The paper does not state the precise hypotheses of [KST19a, Lem. 7.5] nor discuss whether they can be weakened; indeed, K^an is already an A1-invariant Nisnevich sheaf under the milder hypotheses of Theorem 4.20 and Lemma 4.23. The authors should clarify whether (♠)^an_R is necessary for the BGL-identification or an artifact of the proof route, and at minimum state the exact input needed from [KST19a, Lem. 7.5].

minor comments (3)

[Theorem 3.24] The codomain of γω is written Condκ(Sp), but since γω = j∗ ◦ γκ with j∗ : Condκ → Condω, the codomain should be Condω(Sp); the subsequent usage in Theorem 4.14 uses the light version.
[Lemma 4.12] In the display and proof, the expression 'colim_i ∏_i F(R)' is ambiguous; it should be 'colim_i ∏_{T_i} F(R)' where T_i is the finite set in the presentation of T.
[Definition 4.2] The phrase 'as the pushout of the diagram , up to weak equivalence' contains a missing diagram; the intended diagram, similar to (4.8.2), should be displayed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the descent and representability theorems reduce to external results (KST, Andreychev, Morel-Voevodsky) and to stated prior lemmas, not to their own conclusions.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 4.14 reduces Nisnevich descent for K^cont to the equivalence γκK^cont ≃ K^nuc (Proposition 4.11) and to Andreychev's Nisnevich descent for K^nuc [And23, Satz 5.12], with conservativity of the comparison functor (Theorem 3.24, attributed to Clausen-Scholze); none of these inputs is the target descent statement. Proposition 4.21 compares L_A1 K^cont with K^an using [DY24, Lem. 3.6], [KST23, Lem. 2.8] and [KST23, Cor. 2.9], and K^an is not defined as L_A1 K^cont, so the comparison is a proved equivalence, not a tautology. Theorem 5.1 identifies L_mot(Z×BGL) with Ω^∞ K^an_{≥0} using the external BGL identification [KST19a, Lem. 7.5] under (♠)^an_R and the classical Morel-Voevodsky description; the proof constructs the comparison map and then checks 1-connective covers and degree zero, so it does not assume the conclusion. The self-citations [DY24], [Dah24], [Dah25] are used as stated prior results (category construction, analytic descent, semi-valuation ring input), each with proofs outside this paper and none equivalent to the representability or A^1-invariance claims; this is ordinary use of the literature, not load-bearing circularity. The conditional assumption (♠)^an_R, which Remark 4.1 admits is an open desingularisation problem in general, makes Theorem B conditional but does not make it circular: the paper honestly states the hypothesis and does not derive it from the conclusion.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central theorems are deductions from a network of external results: Andreychev's thesis, Efimov's continuity theorem, Kelly-Saito-Tamme pro-cdh descent, KST analytic K-theory, and the authors' own rigid motivic framework. There are no fitted numerical parameters or invented entities. The most consequential qualitative input is (spade)^an_R, a resolution-of-singularities-type assumption that is open in general.

assumptions (7)

domain assumption Nisnevich descent for nuclear K-theory K(Nuc(-)), cited as Andreychev, Satz 5.12.
Theorem 4.14 reduces Nisnevich descent for K^cont to this external theorem; the proof is not reproduced in the paper.
domain assumption Efimov continuity theorem: K_nuc(Lambda^wedge_I) is equivalent to lim_n K(Lambda/I^n) for weakly proregular ideals.
Used in Corollary 4.8 and Proposition 4.13 to identify nuclear and continuous K-theory; the argument given is a proof sketch that relies on Efi25b.
domain assumption Pro-cdh descent for arbitrary qcqs derived schemes, cited as Kelly-Saito-Tamme, KST26 Theorem A.
Used in Proposition 4.16 and in the generalized Weibel vanishing statement, Proposition 4.17, to go beyond noetherian rings of definition.
domain assumption Resolution-of-singularities assumption (spade)^an_R: every smooth adic space over B is analytically locally Spa(A, A+) with A satisfying (dagger)_A, meaning a noetherian ring of definition A0 and a proper regular model over Spec(A0).
This is the main load-bearing hypothesis for Theorem 5.1 and Proposition 4.15. It is open in general and only known for quasi-excellent characteristic-zero rings of definition, as stated in Remark 4.1.
ad hoc to paper F preserves finite products in Lemma 4.12.
The proof identifies F(Cont(T_i, R)) with a product of F(R), which requires F to preserve finite products. This is not stated as an assumption in the lemma, although it is satisfied by the applications K and F(B) = fib(K(B) to K(B[varpi^-1])).
domain assumption Properties of the rigid A1-homotopy category RigH(B, V) from DY24, including the A1-localization formula [DY24, Lemma 3.6].
Used in Proposition 4.21 and Lemma 4.23. This framework was introduced by two of the authors in an earlier paper and is external to the new results.
domain assumption Analytic K-theory is a pro-B1-invariant analytic sheaf satisfying the Bass fundamental theorem, from KST23.
External foundational results from Kerz-Saito-Tamme are used in Propositions 4.15, 4.21, and Corollary 5.12.

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Pith. "Pith review of Representability of continuous K-theory in rigid analytic motivic $\mathbb{A}^1$-homotopy theory." pith.science (2026). https://pith.science/paper/2SDRNOI6

@misc{pith2026260806209,
  author       = {Pith},
  title        = {Pith review of: Representability of continuous K-theory in rigid analytic motivic $\mathbbA^1$-homotopy theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SDRNOI6}},
  note         = {Machine review of arXiv:2608.06209}
}

abstract

We prove that both continuous K-theory and analytic K-theory of rigid analytic spaces (\`a la Kerz--Saito--Tamme) satisfiy descent with respect to the Nisnevich topology. Together with the fact that it is $\mathbb{A}^1$-invariant assuming resolutions of singularities, we deduce that it is representable in the $\mathbb{A}^{1}$-homotopy category of rigid spaces (\`a la Dahlhausen--Yaylali). We identifiy the representing object with both $\mathbb{Z}\times\mathrm{BGL}$ and the analytification of algebraic K-theory. As a consequence, we get a representability statement for coefficients in light condensed spectra. Moreover, we show Weibel vanishing and that continuous K-theory is $\mathbb{A}^1$-invariant on local Tate pairs (without any regularity assumption).

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

[FK18] Kazuhiro Fujiwara and Fumiharo Kato.Foundations of Rigid Geometry I, volume 7 ofMonographs in Mathematics

Topology, 35(1):267–271, 1996.doi:10.1016/0040-9383(94)00056-5. [FK18] Kazuhiro Fujiwara and Fumiharo Kato.Foundations of Rigid Geometry I, volume 7 ofMonographs in Mathematics. EMS,

work page doi:10.1016/0040-9383(94)00056-5 1996

[2] [4]

Princeton University Press, Princeton, NJ, 2009.doi:10.1515/ 9781400830558

[Lur09] Jacob Lurie.Higher topos theory, volume 170 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009.doi:10.1515/ 9781400830558. [Lur17] Jacob Lurie. Higher algebra.https://www.math.ias.edu/~lurie/ papers/HA.pdf,

work page 2009

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[Hin20] Vladimir Hinich

Schemes with examples and exercises.doi:10.1007/978-3-8348-9722-0. [Hin20] Vladimir Hinich. Yoneda lemma for enriched∞-categories.Advances in Mathematics, 367:107129, 2020.doi:10.1016/j.aim.2020.107129. [Hub93] Roland Huber. Continuous valuations.Mathematische Zeitschrift, 212(1):455–477, 1993.doi:10.1007/BF02571668. [Hub94] R. Huber. A generalization of ...

arXiv 2020

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arXiv:2206.02022v1

[Man22] LucasMann.Ap-adic6-functorformalisminrigid-analyticgeometry,2022. arXiv:2206.02022v1. 50 [Mat80] HideyukiMatsumura.Commutativealgebra.2nded,volume56ofMath.Lect. Note Ser.The Benjamin/Cummings Publishing Company, Reading, MA,

arXiv 2022

[5] [337]

On thek-theory of pullbacks.Annals of Mathematics, 190(3), November 2019.doi:10.4007/annals.2019.190.3

[LT19] Markus Land and Georg Tamme. On thek-theory of pullbacks.Annals of Mathematics, 190(3), November 2019.doi:10.4007/annals.2019.190.3

work page doi:10.4007/annals.2019.190.3 2019

[6] [1980]

A historical overview of pro cdh descent in algebraic k-theory and its relation to rigid analytic varieties, 2016.arXiv:1612

[Mor16] Matthew Morrow. A historical overview of pro cdh descent in algebraic k-theory and its relation to rigid analytic varieties, 2016.arXiv:1612. 00418v1. [MR25] Shubhodip Mondal and Emanuel Reinecke. On postnikov completeness forrepletetopoi.Homology,HomotopyandApplications,27(1):179–196,2025. doi:10.4310/hha.2025.v27.n1.a10. [MV99] Fabien Morel and ...

work page doi:10.4310/hha.2025.v27.n1.a10 2001

[7] [1996]

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