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On the integral algebraic K-theory of Morava K-theory

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The integral algebraic K-theory groups of connective Morava K-theory have their cardinalities determined in every degree not congruent to 0 or 1 modulo 2p−2.

desk verdict Real result, honest paper: first integral K-theory computations for a non-Eilenberg-MacLane ring spectrum in infinite families, but the proof of Lemma 5.20 is too terse and the finite-field theorem leans on [AKHW24], a preprint by the first author. read the letter →

arxiv 2607.21567 v1 pith:AHY3Z37P submitted 2026-07-23 math.AT math.KT

Gabriel Angelini-Knoll , Haldun Özgür Bay{i}nd{i}r This is my paper · ORCID
classification math.ATmath.KT MSC 19D5555P4355N22
keywords algebraicK-theoryMoravatopologicalcyclichomologyHochschildorbitfiltrationMayWittvectorscardinality
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

The paper determines the sizes of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in every degree not congruent to 0 or 1 modulo $2p-2$. Over a finite field $\mathbb{F}_q$, the odd-degree orders are $|W_{\lfloor r/(p^n-1)\rfloor}(\mathbb{F}_q)|(q^{r+1}-1)$, and the even groups in the resolved range are either 0 or cyclic p-groups $\mathbb{Z}/p^{m(r)}$. After base change to the algebraic closure $\bar{\mathbb{F}}_p$, all even groups vanish and the positive odd groups are countably infinite with bounded p-torsion. The proof runs through a new orbit filtration on topological cyclic homology, induced by the May filtration on topological Hochschild homology, which makes the Frobenius correction map finite enough to count. This matters as the first infinite-family integral K-theory computation for a ring spectrum that is not an Eilenberg–MacLane spectrum.

What carries the argument

The orbit filtration on topological cyclic homology: starting from the multiplicative Whitehead filtration $\tau_{\ge \bullet} A$, the May filtration on THH is promoted to a filtration of cyclotomic spectra by postcomposing the twisted Frobenius with the weight-decreasing map $R_p \Rightarrow \mathrm{id}$; applying TC levelwise yields the orbit filtration $\mathrm{fil}^*_{\mathrm{orb}} TC(A)$. Its associated graded is $\Sigma TC^+(\pi_* A)$, and a quotient $Q_n$ of the filtration recovers TC in degrees $\le n$ while having finite homotopy groups when $\pi_* A \cong \mathbb{F}_q[x_{2m}]$. This finiteness, together with the odd-concentration of $TC^-/TP$ for formal polynomial DGAs (Theorem 2.10, which identifies $\pi_{2r+1} TC(F[x_{2m}])$ with $W_{\lfloor r/m \rfloor}(F)$), is what allows cardinality counting,

What would settle it

Test Lemma 5.20 for $p=3$, $k=2$, $f_1=x_2$, $f_2=2x_1+1$: if $x_1^3-x_2=0$ and $x_2^3-2x_1-1=0$ has no solution in the algebraic closure of $\mathbb{F}_3$, the lemma is false. More generally, search for random linear $f_i$ over $\mathbb{F}_{3^N}$ for increasing $N$; any system with no solution in any finite extension would falsify the lemma. A direct check of the predicted vanishing, e.g. computing $K_2(k(1)_{\bar{\mathbb{F}}_3})$ via the orbit spectral sequence and finding a nonzero group, would also falsify the central even-vanishing claim.

Watch

Extended reading notes

Core claim

The central claim is that the integral algebraic K-theory of connective Morava K-theory, previously known only in low degrees or with finite coefficients, satisfies a sharp numerical law in nearly all degrees. For finite fields $\mathbb{F}_q$, Theorem F gives isomorphisms $K_{2r}(k(n)_{\mathbb{F}_q}) \cong 0$ or $\mathbb{Z}/p^{m(r)}$ in the resolved even cases and cardinalities $|K_{2r+1}(k(n)_{\mathbb{F}_q})| = |W_{\lfloor r/(p^n-1)\rfloor}(\mathbb{F}_q)|(q^{r+1}-1)$ in the resolved odd cases, leaving only two p-power ambiguities in periodic families. For algebraically closed fields, Corollary C states $K_{2k}(k(n)_{\bar{\mathbb{F}}_p}) = 0$ and $K_0$ and $K_{2k-1}$ are countably infinite with bounded p-torsion. The paper derives these from a theorem about $E_1$-rings with hom

Load-bearing premise

The load-bearing premise is Lemma 5.20: over an algebraically closed field of characteristic $p$, any system $x_i^p - f_i(x_1,\ldots,x_k)=0$ with $f_i$ linear has a solution; the proof given is a terse induction with a 'without loss of generality' step, and if that lemma fails, the surjectivity of the Frobenius correction—and with it the even-vanishing theorem—collapses.

Editorial extensions

If this is right

  • For every finite field F_q, the full cardinality of K_t(k(n)_{F_q}) is now determined for all t not congruent to 0 or 1 mod 2p−2; the two pending p-powers m(r) and |A_r| are the only gaps in the resolved period.
  • Base-changing k(n) to F̄_p forces all even K-groups to vanish, so the chromatic analogue of Quillen's computation for F_p has the same even parity.
  • The quotient |K_{2r+1}(k(n)_{F_q})|/|K_{2r}(k(n)_{F_q})| equals |W_{⌊r/(p^n−1)⌋}(F_q)|, a clean ratio law that relates to Lichtenbaum-style quotient formulas for number fields.
  • For W(F)/p^n with F algebraically closed, relative K-theory vanishes in even degrees and is infinite in odd degrees; over F̄_p these odd groups are countably infinite with bounded p-torsion.
  • Relative K-theory of the arithmetic coordinate axis W(F̄_p) ×_{F̄_p} W(F̄_p) (the Burnside-ring base change) is trivial in odd degrees ≥1 and infinite in even degrees ≥1.

Reading between the lines

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

  • The orbit filtration should apply to other connective complex-oriented theories whose associated graded homology is polynomial, such as truncated Brown–Peterson spectra; if the Frobenius surjectivity lemma extends, even vanishing and Witt-vector cardinalities would follow without new homotopy-group computations.
  • The undetermined p-powers m(r) and A_r likely arise from v_n-Bockstein differentials in Adams weight 2; resolving the differentials d_1(v_{n+1}^j ∂λ_{n+1} ε_i) (i ≤ n) would make Theorem F completely explicit.
  • The algebraically closed case suggests a general principle: for p-complete ring spectra with polynomial F̄_p-homology, even integral K-theory is controlled by the surjectivity of F⊗φ − id⊗c on finite quotient modules; this may unify known even-vanishing theorems for rings such as Z/p^n.
  • The countably infinite odd K-groups with bounded p-torsion over F̄_p are a new chromatic phenomenon: unlike finite-field cases, the integral K-groups are infinite yet have only bounded torsion in each degree.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a new 'orbit filtration' on topological cyclic homology, derived from the May filtration on THH, and uses it to compute cardinalities of the integral algebraic K-theory groups of connective Morava K-theory. For a finite field F_q, Theorem F determines |K_t(k(n)_{F_q})| in all degrees not congruent to 0 or 1 modulo 2p-2, leaving undetermined p-groups only in the excluded congruence classes. For algebraically closed fields, Theorem A and Corollary C give even vanishing and countably infinite odd K-theory groups for k(n) over \bar F_p. The paper also proves analogous results for truncated Witt vectors W(F)/p^n and for certain motivic pullback squares.

Significance. If correct, this is the first infinite-family integral algebraic K-theory computation for a non-Eilenberg-MacLane ring spectrum, and the orbit filtration is a promising new structural tool. The paper is explicit about which quantities are determined and which are left as undetermined p-groups, and it gives concrete falsifiable cardinality formulas. The main theorems are substantial, but the proof as written contains a load-bearing gap in Lemma 5.20 and a central dependency on an unpublished preprint [AKHW24].

major comments (2)
  1. [§5.2, Lemma 5.20] The proof of Lemma 5.20 is incomplete in the induction step. After solving (5.22) for x_{k−1} when f_k depends nontrivially on x_{k−1}, substituting into the equations (5.21) for i<k−1 introduces terms of the form c_i g_k(x_k^p) in addition to the original g_i(x_k^p). The induction hypothesis requires deg(g_i)<deg(g_k) for all i<k−1, but the new coefficients may have degree exactly deg(g_k), so the induction template (5.21)–(5.22) is not satisfied. The sentence 'After this, one uses (5.21) for i<k−1 to deduce...' does not explain how these terms are absorbed. Since Lemma 5.20 is exactly what makes W(F)⊗M→W(F)⊗N surjective in Lemma 5.23, and that surjectivity is the key step in the even-vanishing claim of Theorem A, this is a load-bearing gap. The lemma is plausible (homogenizing the equations shows no common zero at Z=0, so the associated polynomial map is finite and surjective), but the
  2. [§8.2–8.4, Theorem 8.13 / Theorem 8.33] Theorem F's even-degree K-theory bounds for k(n)_{F_q} are derived from the mod (p,...,v_{n+1}) syntomic cohomology computation of k(n)_F, stated as Theorem 8.13 and attributed to [AKHW24], an unpublished preprint co-authored by the first author. This is a substantial external dependency for a core advertised result. The manuscript should either reproduce the proof of the imported theorem or state explicitly that Theorem F is conditional on the correctness of [AKHW24]. As written, the reader cannot verify a load-bearing input from within the paper.
minor comments (4)
  1. [Notation 1.4 and §5] The notation F_p is used in Theorem A and Section 5 where A has π_*A≅F_p[x_{2m}]. If F_p denotes the prime field, Proposition 4.39 applies to A; if it denotes the algebraic closure, Proposition 4.39 does not. Please clarify the convention explicitly.
  2. [Theorem F statement] The theorem phrase 'W_n denotes the truncated big Witt vectors of length n' is confusing because Notation 1.4 uses W_n(F) for p-typical Witt vectors and a separate symbol for big Witt vectors. Please disambiguate.
  3. [Remark 9.2 / Proposition 9.1] Minor typos: 'Neverless' in Remark 9.2; missing closing parenthesis in π_{2k}(Σ(THH(K(1))^{hS1}) in Proposition 9.1.
  4. [§4.2, Lemma 4.23] The construction Q_n is defined for N-filtered spectra, but later used for Z-filtered objects in Lemma 4.23 and elsewhere. The comparison via Remark 2.2 is mentioned only parenthetically; please make the convention explicit where Q_n is applied to Z-graded objects.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the self-cited syntomic computation is an independent external input, not a restatement of the target results.

full rationale

The paper's main derivation chain is largely self-contained. Theorem A and the even-vanishing results are proved via the newly constructed orbit filtration and the May spectral sequence, with the key algebraic input Lemma 5.20 being a separate polynomial-equation statement that is structurally independent of the K-theory conclusions. Theorem E is derived from Theorem 2.10, which rests on external computations of [BM22] and [ABM22], not on the paper's own target. Theorem F imports the syntomic cohomology computation of k(n)_F from [AKHW24], a preprint co-authored by the first author; this is a load-bearing self-citation for the finite-field cardinality theorem. However, [AKHW24]'s syntomic cohomology computation is a parameter-free computation of a different invariant--the motivic associated graded of TC--and it does not assume or contain the K-theory cardinalities being derived. Under the stated rules, this is real independent evidence and does not constitute circularity. No fitted parameter is disguised as a prediction: the undetermined groups m(r) and A_r are explicitly left open. The paper does contain a proof gap in Lemma 5.20, where the induction's key 'without loss of generality' step is not fully demonstrated, and Proposition 3.16's claim that a cofiber commutes with homotopy fixed points is terse; these are correctness risks, not circular reductions. The central results do not reduce to their inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No empirical fitting or hand-chosen numerical inputs occur. The theorems contain undetermined placeholders m(r) and A_r in excluded congruence classes, but these are unresolved outputs, not fitted parameters. The main external load is the [AKHW24] syntomic computation and the [BM22] TC computation. The orbit filtration is a new construction but not a postulated entity.

assumptions (4)
  • standard math ∞-category formalism of Lurie and the [NS18] model of cyclotomic spectra/TC are consistent.
    The paper works throughout in ∞-categories and relies on the definition of TC as an equalizer of φ and can.
  • domain assumption The computation of TC^-, TP and TC of the free E1-algebra F[x_2] from [BM22] is correct, and its extension to F[x_{2m}] for p∤m (Theorem 2.10) is valid.
    Theorem 2.10 underlies Theorems D and E. The authors prove it using [ABM22] and [BM22], and state it was independently known to Land, Tamme and Speirs (Remark 2.17).
  • domain assumption The syntomic cohomology of k(n)_F is as described in [AKHW24, Theorem 3.5.7 and 4.5.3] (Theorem 8.13 here).
    Theorem F's Bockstein analysis starts from this computation; [AKHW24] is a preprint co-authored by the first author. If that computation is incorrect, the finite-field cardinality theorem is not supported.
  • standard math p-completeness criteria identify (THH^{tCp})^{hS1} with TP (Remark 3.22), following [BMS19, NS18].
    Used throughout to replace Tate-fixed-point constructions with TP.

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Pith. "Pith review of On the integral algebraic K-theory of Morava K-theory." pith.science (2026). https://pith.science/paper/AHY3Z37P

@misc{pith2026260721567,
  author       = {Pith},
  title        = {Pith review of: On the integral algebraic K-theory of Morava K-theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHY3Z37P}},
  note         = {Machine review of arXiv:2607.21567}
}
abstract

We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn--Raksit--Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.