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Essential Dimension, Symbol Length and $p$-rank

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Pith's one-line read In characteristic p, defining central simple p-algebras of degree $p^{\ell m}$ and exponent $p^m$ costs at least $\ell+1-r$ parameters, and at least $\ell+1$ when the base field is perfect.

desk verdict Solid lower-bound theorem with a load-bearing citation; the upper-bound section is broken by a plainly false lemma. read the letter →

arxiv 1908.08844 v3 pith:VBCFGUDI submitted 2019-08-23 math.RA math.KT

Adam Chapman , Kelly McKinnie This is my paper · ORCID
classification math.RAmath.KT MSC 16K2013A3519D4520G10
keywords essentialdimensionsymbollengthp-rankcentralsimplealgebrasBrauergroupKato-Milnecohomologypositivecharacteristicindecomposablep-algebras
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 proves a lower bound on how hard it is to define central simple algebras in characteristic p. Over a base field k of characteristic p and p-rank r, every algebra of degree $p^{\ell m}$ and exponent $p^m$ has essential p-dimension at least $\ell+1-r$; when k is perfect, this reads at least $\ell+1$. The argument is short: the p-rank of a field bounds the symbol length in its Brauer $p^m$-torsion, and there exist indecomposable p-algebras whose symbol length stays large after every prime-to-p extension. The paper also bounds the symbol length in the higher Kato-Milne cohomology groups $H^{n+1}_{p^m}(F)$ by the binomial coefficient $\binom{r}{n}$, and gives upper and lower bounds for the essential dimension of Brauer classes of fixed symbol length.

What carries the argument

The load-bearing objects are the p-rank and the symbol length. The p-rank of F is the integer r with $[F:F^p]=p^r$; the symbol length of a Brauer class is the smallest number of $p^m$-symbols, meaning cyclic algebras of degree $p^m$, whose tensor product is Brauer-equivalent to the class. Proposition 3.2 shows that a p-basis of F of size r lets every class in $\mathrm{Br}_{p^m}(F)$ be written as a sum of at most r symbols. Lemma 5.3 converts this into a lower-bound engine: if a class descends to a finitely generated field E of transcendence degree t over k, then E has p-rank $r+t$, so its symbol length is at most $r+t$, contradicting the existence of algebras that keep symbol length at least $\ell+1$ after prime-to-p extension.

What would settle it

Find a perfect field k of characteristic p and positive integers $\ell\ge 2$, m such that some algebra A of degree $p^{\ell m}$ and exponent $p^m$ over a field $F\supseteq k$ has essential p-dimension at most $\ell$; equivalently, show that after some prime-to-p extension of F, A becomes Brauer-equivalent to a tensor product of at most $\ell$ cyclic algebras of degree $p^m$. Such an example would contradict Theorem 5.4, while proving that no such algebra exists for any perfect k would confirm it.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 5.4: for a field k of characteristic p with $\mathrm{rank}_p(k)=r$ and integers $\ell\ge 2$, $m\ge 1$, the essential p-dimension of the functor $\mathrm{Alg}_{p^{\ell m},p^m}$ satisfies $\mathrm{ed}(\mathrm{Alg}_{p^{\ell m},p^m};p)\ge \ell+1-r$. In the perfect-field case this is $\mathrm{ed}\ge \ell+1$. The proof runs by contradiction: if an algebra could be descended over a prime-to-p extension to a field of transcendence degree below $\ell+1-r$, then Lemma 5.3 plus the p-rank bounds on symbol length would force the algebra's symbol length to be at most $\ell$; Proposition 3.1 supplies an algebra whose symbol length is at least $\ell+1$ even after every prime-to-p extension. Along the way the paper proves that the p-rank r bounds the symbol length in $\mathrm{Br}_{p^m}(F)$ by r, and in $H^{n+1}_{p^m}(F)$ by $\binom{r}{n}$.

Load-bearing premise

The proof rests on the imported fact that, for each relevant p, $\ell$, m, there is an algebra of the required degree and exponent whose smallest number of cyclic factors stays at least $\ell+1$ even after enlarging the field by any extension of degree prime to p; this existence result is cited, not proved here.

Editorial extensions

If this is right

  • For every perfect base field k of characteristic p, the essential p-dimension of the functor of central simple algebras of degree $p^{\ell m}$ and exponent $p^m$ is at least $\ell+1$, recovering the known $m=1$ bound and extending it to all exponent heights m.
  • For a base field of finite p-rank r, the lower bound degrades by r: $\mathrm{ed}(\mathrm{Alg}_{p^{\ell m},p^m};p)\ge \ell+1-r$, so the same parameter-counting obstruction persists unless r is large.
  • Because $\mathrm{ed}(\mathrm{Alg}_{p^t,p^m};p)\ge \mathrm{ed}(\mathrm{Alg}_{p^{\ell m},p^m};p)$ for $\ell=\lfloor t/m\rfloor$, the theorem gives lower bounds for degrees that are not pure powers $p^{\ell m}$.
  • The p-rank symbol-length bound implies that any Brauer class in $\mathrm{Br}_{p^m}(F)$ with symbol length s has essential dimension at most $s+m$ over an infinite perfect field, and combined with known symbol-length upper bounds this yields $\mathrm{ed}_{\mathrm{Br}_2}([A])\le 5$ for degree-8 exponent-2 algebras.
  • The higher-cohomology bound says the generic sum of $\ell$ symbols in $H^{n+1}_{p^m}$ has essential p-dimension at least $\ell+n$, extending symbol-counting lower bounds beyond the Brauer group.

Reading between the lines

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

  • The method's bottleneck is the imported existence result for indecomposable algebras with stable high symbol length; improving that construction would raise the lower bounds directly.
  • If the conjectured lower bound $\mathrm{sl}_{p^m}([A])+1\le \mathrm{ed}_{\mathrm{Br}_{p^m}}([A])$ in Remark 6.9 holds, then over fields of transcendence degree d over an algebraically closed field every p-primary algebra of exponent $p^m$ would have index at most $p^{d-1}$, a special case of the period-index conjecture.
  • The same p-rank device may apply to other cohomological functors in characteristic p, not just Brauer groups: whenever a class is expressed by $p^m$-symbols and the base field's p-rank controls the size of a p-basis, the symbol-length upper bound yields an essential-dimension lower bound by descent.
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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

3 major / 4 minor

Summary. The paper studies the essential dimension of central simple algebras of degree p^{ℓm} and exponent p^m over fields of characteristic p. Its main result (Theorem 5.4) asserts that if k has p-rank r, then ed(Alg_{p^{ℓm},p^m}; p) ≥ ℓ+1−r, and in particular ℓ+1 when k is perfect. The proof combines a bound on the symbol length of Brauer classes by the p-rank (Proposition 3.2 and Lemma 5.3) with the existence of p-algebras of degree p^{ℓm} and exponent p^m whose symbol length remains ≥ ℓ+1 after every prime-to-p scalar extension (Proposition 3.1). The paper also gives an upper bound on the symbol length in H^{n+1}_{p^m}(F) in terms of the p-rank (Corollary 3.4), and offers upper and lower bounds for the essential dimension of Brauer classes of a given symbol length (Section 6).

Significance. If Theorem 5.4 is correct, it is a significant strengthening of the earlier lower bound from [McK17], extending it from m=1 and algebraically closed base fields to all m and all perfect base fields, with a concise argument. The symbol-length bounds for Kato-Milne cohomology in Corollary 3.4 are natural and likely useful. However, the paper's central lower bound rests on an unproved existence statement imported from [Kar95], and the upper-bound section relies on a lemma that appears to be false as stated. These issues must be resolved before the paper's results can be accepted.

major comments (3)
  1. [§3, Proposition 3.1] This proposition is the sole source of the contradiction in Theorem 5.4, yet it is not proved. The proof is a vague citation to [Kar95] ("see the introduction... Sections 2 and 3") with no theorem number. The required property is stronger than the existence of an indecomposable p-algebra over F: the algebra must have symbol length at least ℓ+1 after every prime-to-p scalar extension L/F. Indecomposability over F does not by itself imply that A_L is not Brauer-equivalent to a tensor product of ℓ cyclic p^m-algebras for such L. A precise statement and proof, or an accurate reference establishing this exact stability property, must be supplied for the main theorem to be supported.
  2. [§6, Lemma 6.3] Lemma 6.3 is false as stated. For m=1 and G=(Z/p)^ℓ with ℓ≥2, the lemma claims ed(G) ≤ 1. But if the base field k is perfect, any field E of transcendence degree at most 1 over k has p-rank at most 1 by Lemma 5.2, and a G-Galois extension with G=(Z/p)^ℓ corresponds to ℓ linearly independent classes in E/℘(E) under the Artin-Schreier correspondence. Thus ed((Z/p)^ℓ) ≥ ℓ, contradicting the lemma. This error invalidates the proof of Proposition 6.5 and the bounds in Corollaries 6.6–6.8, which are advertised in the abstract. The authors should verify the original statement in [Led04] and either replace Lemma 6.3 with a correct version or provide an alternative proof of the upper bounds.
  3. [§6, Lemma 6.3 and Proposition 6.5] Even beyond the falsity of Lemma 6.3, the descent argument in Proposition 6.5 is not justified: Remark 6.4 asserts that a subgroup G′ of W_m(F)/℘(W_m(F)) generated by ℓ elements descends to an intermediate field of transcendence degree ≤ m. For m=1 this would imply that any ℓ Artin-Schreier classes can be defined over a field of p-rank at most 1, which is impossible for ℓ≥2 by the same p-rank argument. The upper bound ed ≤ m+ℓn therefore needs a fundamentally different proof or a corrected statement.
minor comments (4)
  1. [§5, proof of Theorem 5.4] The text says "The case of p = n = 2 and m = 1" but the variable n is not otherwise used in the theorem; this should be p = ℓ = 2 and m = 1.
  2. [§3, proof of Proposition 3.1] The sentence "If the symbol length of A is at most ℓ then A decomposes (as an algebra) as a tensor product of ℓ cyclic algebras of degree p^m" is imprecise: symbol length is about Brauer equivalence, so one obtains a Brauer decomposition, not necessarily an algebra isomorphism.
  3. [General] The manuscript contains several typographical artifacts (e.g., "n /greaterorequalslantm", "F 4", "ed(Algpn,pm ; p) /greaterorequalslant3") that should be cleaned up in the final version.
  4. [§6, Lemma 6.3] The reference to [Led04] should be checked carefully, since the stated lemma appears to be a misquotation of the original example.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main lower-bound proof is assembled from independent known results and does not reduce to its inputs.

full rationale

No circular step was found. Theorem 5.4 derives the lower bound on essential p-dimension from Proposition 3.1, which imports an existence result from Karpenko [Kar95], and Lemma 5.3, which bounds symbol length by p-rank using Albert's theorem and Bourbaki's p-rank lemma. The contradiction in Theorem 5.4 does not define, fit, or rename any quantity in terms of the target lower bound. The self-citations that appear, [McK17] in Theorem 6.1 and [CM19] in Lemma 3.3 and Section 4, are auxiliary prior results with independent content; they are not used to force the paper's central claim. Proposition 3.1 is stated by citing [Kar95] without a theorem number, and the required stability under arbitrary prime-to-p extensions may deserve closer verification, but this is a correctness or external-support concern, not a circular definition or fitted prediction. The derivation is self-contained relative to established external theorems, with no ansatz smuggled in through self-citation and no known result renamed as a new prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central lower-bound claim rests on standard algebra facts plus Karpenko's indecomposability theorem. No free parameters or invented entities appear. The upper-bound section adds a questionable assumption about p-groups.

assumptions (6)
  • standard math The exact sequence of Kato-Milne cohomology groups with Shift and Exp maps (Theorem 2.1).
    Used in Lemma 3.3 and Theorem 6.1 to relate H^{n+1}_{p^m} to H^{n+1}_p and H^{n+1}_{p^{m-1}}.
  • standard math Bourbaki's lemma: the p-rank of a finitely generated extension of transcendence degree t over k is rank_p(k)+t.
    Used in Lemma 5.3 to convert essential dimension into an upper bound on p-rank.
  • standard math Albert's theorem ([Alb68, Theorem 28]) on decomposition of classes split by a p-basis extension into p^m-symbols.
    Used in Proposition 3.2 to prove that p-rank r bounds symbol length by r.
  • domain assumption Existence of indecomposable p-algebras of degree p^{ℓm} and exponent p^m whose symbol length stays at least ℓ+1 after prime-to-p extensions.
    Imported from [Kar95] and used in Proposition 3.1; this is the main external input for the lower-bound theorem.
  • domain assumption [McK17, Theorem 5.8]: the generic sum of ℓ symbols in H^{n+1}_p over an algebraically closed field has essential p-dimension at least ℓ+n.
    Used in Theorem 6.1 as the base case m=1. It is a prior result by one of the authors.
  • domain assumption Ledet's bound on essential dimension of p-groups, as stated in Lemma 6.3.
    Used in Remark 6.4 and Proposition 6.5 for the upper-bound section; the statement appears suspect as written.

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Cite this review

Pith. "Pith review of Essential Dimension, Symbol Length and $p$-rank." pith.science (2026). https://pith.science/paper/VBCFGUDI

@misc{pith2026190808844,
  author       = {Pith},
  title        = {Pith review of: Essential Dimension, Symbol Length and $p$-rank},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBCFGUDI}},
  note         = {Machine review of arXiv:1908.08844}
}
abstract

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^m$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell+1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\operatorname{Br}_{p^m}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^m$. We also prove that the symbol length of the Milne-Kato cohomology group $\operatorname H^{n+1}_{p^m}(F)$ is bounded from above by $\binom rn$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

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