Signature maps from positive cones on algebras with involution

Thomas Unger; Vincent Astier

REVIEW 2 major objections 4 minor 20 references

Signature maps from positive cones on algebras with involution

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Pith's one-line read A positive cone on an algebra with involution determines the kernels of the signature maps of hermitian forms directly, and this reconstruction repairs a broken lemma in the extension theorem.

desk verdict Section 3 gives a genuinely new direct construction of signature kernels from positive cones; the rest is an honest repair of flawed earlier work, but the density proof behind the extension theorem has a gap that needs patching. read the letter →

arxiv 2505.22178 v1 pith:CKT62EXQ submitted 2025-05-28 math.RA

Vincent Astier , Thomas Unger This is my paper

classification math.RA MSC 13J3016W1006F2516K2011E39

keywords positiveconesalgebraswithinvolutionhermitianformssignaturesorderingsWittgroupm-idealsrealalgebra

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

Positive cones were introduced as an ordering-like structure on central simple algebras with involution, mirroring orderings of fields. This paper proves that a single positive cone carries the complete signature information of all hermitian forms over the algebra: the pair consisting of the kernel of the field ordering and the set of Witt classes satisfying a simple representation condition relative to the cone is exactly the pair of kernels of the two signature maps. Since the cone itself supplies the condition, signatures can now be read off directly from the cone rather than through real closures and auxiliary reference forms. The same construction repairs a flaw in the earlier development: a lemma used to prove the extension theorem for prepositive cones is likely false, and the paper gives a new proof of the extension theorem that does not rely on it. If correct, this gives a complete and self-corrected description of positive cones and their extension behaviour.

What carries the argument

The load-bearing object is the pair $(I_{\mathcal{P}}, N_{\mathcal{P}})$ extracted from a positive cone $\mathcal{P}$: $I_{\mathcal{P}}$ is the kernel of the ordering on the Witt ring, and $N_{\mathcal{P}}$ collects the Witt classes of hermitian forms that satisfy property (3.1) — after multiplication by some quadratic form of nonzero $P$-signature they become a balanced diagonal sum with all coefficients in $\mathcal{P}$. The proof that this pair is a prime $m$-ideal — an ideal-and-submodule pair with a primeness condition under multiplication by the Witt ring — uses a reduction to diagonal forms driven by the pairing of hermitian forms with its pivot property, together with a counting lemma saying that a balanced isometry cannot change the number of positive versus negative coefficients in the cone. The later identification with $(\ker \mathrm{sign}_P, \ker \mathrm{sign}^\mu_P)$ uses the classification of prime $m$-ideals of the Witt group. For the repaired extension theorem the additional machinery is the equality $m_P(A,\sigma) = n_P(A,\sigma)$, proved for finitely generated fields via a density statement (archimedean orderings are dense in the space of orderings) established by a sign-counting argument over real closed fields and a compactness transfer of an existential formula.

What would settle it

Take any concrete positive cone on a matrix algebra over the reals or a quaternion algebra where signature kernels are explicitly known, compute $N_{\mathcal{P}}$ from property (3.1), and check whether it equals the known zero-signature classes; one Witt class that satisfies the representation condition but has nonzero signature, or fails it with zero signature, refutes Theorem 3.8. Alternatively, exhibit a finitely generated field over $\mathbb{Q}$ with an ordering that is not a limit of archimedean orderings; this would refute the density claim on which the equality $m_P = n_P$ and the extension theorem rest.

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Extended reading notes

Core claim

Fix a positive cone $\mathcal{P}$ on $(A,\sigma)$ over an ordering $P$ of $F$, and let $I_{\mathcal{P}}$ be the kernel of the ordering $P$ on the Witt ring $W(F)$. Let $N_{\mathcal{P}}$ be the set of Witt classes $[h]$ of nonsingular hermitian forms $h$ for which there exists a nonsingular quadratic form $q_h$ over $F$ with $\mathrm{sign}_P(q_h) \neq 0$ such that $q_h \otimes h$ is isometric to a balanced sum $\langle a_1,\ldots,a_r\rangle_\sigma \perp \langle -b_1,\ldots,-b_r\rangle_\sigma$ with all $a_i, b_i$ invertible elements of $\mathcal{P}$. The paper proves that $(I_{\mathcal{P}}, N_{\mathcal{P}})$ is a prime $m$-ideal with torsion-free quotient, and therefore, by the classification of prime $m$-ideals, equals $(\ker \mathrm{sign}_P, \ker \mathrm{sign}^\mu_P)$. Hence $\mathcal{P}$ is never a nil-ordering. From this it follows that over each non-nil ordering there are exactly two positive cones, $\pm C_P(M^\mu_P(A,\sigma))$, and that the invertible elements of a positive cone are precisely the elements of maximal signature. The paper also proves $m_P(A,\sigma) = n_P(A,\sigma)$ — the maximal signature is the degree of the matrix algebra obtained after scalar extension to the real closure — and uses this to give a corrected proof that every prepositive cone extends to a prepositive cone over any ordered field extension.

Load-bearing premise

The central claim depends on the classification of prime $m$-ideals of the Witt group as signature kernels, and the repaired extension theorem additionally depends on archimedean orderings being dense among orderings of finitely generated fields over $\mathbb{Q}$; either premise failing would bring down the corresponding theorem.

Editorial extensions

If this is right

The space of positive cones is exactly $\{\pm C_P(M^\mu_P(A,\sigma)) : P \in X_F \setminus \mathrm{Nil}[A,\sigma]\}$: over every non-nil ordering there are precisely two positive cones, one the negative of the other.
All invertible elements of a positive cone have the same signature (Corollary 3.9), and the invertible part of a cone is $\varepsilon M^\mu_P(A,\sigma) \setminus \{0\}$ for some $\varepsilon \in \{1,-1\}$.
Every prepositive cone over $P$ extends to a prepositive cone over any ordered extension $(L,Q)$, restoring the extension theorem with a proof that avoids the flawed lemma.
No positive cone can sit over a nil-ordering, and the projection from the space of positive cones to the space of orderings $X_F$ is continuous, open and closed, with the Harrison topology compact.

Reading between the lines

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

Because property (3.1) refers only to cones, diagonal forms and quadratic forms with nonzero signature, it suggests an algorithmic route to computing signatures of hermitian forms that avoids constructing real closures; testing this on matrix algebras or quaternion division algebras over number fields would be a concrete next step.
The description of $X_{(A,\sigma)}$ as $\{\pm C_P(M^\mu_P(A,\sigma))\}$ makes the projection to $X_F$ a two-to-one map off the nil-orderings; viewing positive cones as a double cover of the ordering space is a geometric reading the paper does not pursue.
The equality $m_P = n_P$ is proved via the density of archimedean orderings; if it could be established by valuation-theoretic methods for fields with non-archimedean orderings, the extension theorem would hold without the model-theoretic transfer step.
Since the authors state that the flawed lemmas of the earlier paper are no longer needed, the remaining results of that paper that relied on [4, Proposition 5.8] are now supported by a valid proof; a survey of which later results are thereby restored would be a useful follow-up.

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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 new thing in this paper is Section 3. The authors take a positive cone P and define a pair (I_P,N_P) directly from P via property (3.1), then prove it is the kernel pair of the signature map. That is a genuinely cleaner path than the reference-form machinery and it is new. The rest of the paper is mostly a repair job: their earlier positive cones paper [4] had a flawed Lemma 5.5, and they say so plainly. That honesty is welcome.

What the paper does well: the Section 3 construction is coherent and carefully checked. Propositions 3.6 and 3.7 verify the m-ideal and torsion-free properties, and the use of the published prime m-ideal classification [2, Prop 6.5] is legitimate. Corollary 3.9 is a nice payoff: all invertible elements of a positive cone have the same signature. The paper also correctly flags which theorems were in [4] and which are new.

Now the soft spots. The proof of Proposition 5.2, density of archimedean orderings for finitely generated fields over Q, is the least self-contained part. The Sturm-sequence argument in Lemmas 5.3 and 5.4 has a real missing step: the set S is shown to be open by expressing it as a finite union of sets where sign-change counts are constant, but the argument does not control the discriminant of m_xbar. Without squarefreeness, the number of real roots can jump, so the openness conclusion is not automatic. This is not a fatal objection to the theorem—the statement is true and standard—but the proof as written likely needs a patch, for instance by adding a discriminant condition to Z or citing a known density result. Since Proposition 5.6 and hence Theorem 5.8 rely on Proposition 5.2, the repair of [4, Prop 5.8] currently rests on this gap. The reduction to the finitely generated case in Proposition 5.6 is also asserted rather than proved in detail, though that looks fixable. Theorem 4.6's proof is deferred to the original paper via a reference table, which is acceptable but means a referee has to cross-check.

The paper is for the small community working on orderings of algebras with involution and hermitian forms. It restores confidence in [4] and gives a better tool. It deserves a serious referee, but the referee should be told to focus on Lemmas 5.3/5.4 and the reduction.

Recommendation: send it to peer review. The Section 3 content is publishable regardless of the status of the density proof, and the density argument can be repaired.

Referee Report

2 major / 4 minor

Summary. This paper studies positive cones on central simple algebras with involution. It defines, for each positive cone P over an ordering P, an m-ideal (I_P,N_P) via property (3.1), proves that it is prime and torsion-free, and identifies it with the kernel pair (ker sign_P, ker sign^mu_P) through the authors' earlier classification [2, Prop. 6.5]. It then uses this to describe the space of positive cones, to prove compactness and openness properties of the canonical map, and to repair [4, Proposition 5.8] on extension of positive cones to ordered field extensions. The main new technical work is in Section 5, where the equality m_P(A,sigma)=n_P(A,sigma) is proved via an archimedean density argument and then used to derive the extension theorem.

Significance. If the Section 5 arguments are completed, the paper is a significant contribution: it gives a purely cone-theoretic description of signature kernels, provides a clean correction of a documented gap in the published literature, and supplies precise replacement statements for the flawed lemmas. The Section 3 construction is coherent and the honest reporting of the status of [4, Lemma 5.5] and its consequences is a strength. The paper also carefully tabulates which statements are new and which are reproofs of earlier results. The main risk is that the proof of Proposition 5.6 contains an underdocumented reduction and the density proof relies on a Sturm-sequence argument whose hypotheses are not fully stated.

major comments (2)

[Proposition 5.6 (proof, reduction paragraph)] The reduction from an arbitrary base field F to a finitely generated field F0 is asserted rather than proved. The text defines P0:=A0∩P and notes only that a0∈P0; it does not prove that (A0,sigma0) is an F0-algebra with involution in the sense of the paper (central simple over F0, with the required center condition), that P0 is a prepositive cone over P0 or is contained in a positive cone Q0 with a0∈Q0, or that a reference form mu0 for (A0,sigma0) exists whose base change mu0⊗F is a reference form for (A,sigma). The equality sign^{mu0}_{P0}<a0> = ± sign^mu_P<a> also needs justification beyond quoting [2, Prop. 3.3(iii)], because that proposition compares reference forms on the same algebra. Since Proposition 5.6 is the only route to m_P = n_P, and Theorem 5.8 depends on it, this step is load-bearing and must be completed.
[Lemma 5.3] The proof that S∩Z is open uses the identity N_e(bar x) = v~(m_bar x,g_e) - v(m_bar x,g_e) quoted from [8, Cor. 1.2.12], but it does not state the hypotheses under which this identity holds. If the identity requires m_bar x to be squarefree, or requires the Sturm sequence to have no vanishing entries at the relevant evaluations, then the argument must show that these conditions hold on Z or that the exceptional set is harmless. As written, a reader cannot verify that the set defined by the sign conditions is open, and Proposition 5.2 would lack a proof. The finite union representation of S is plausible but the Sturm-sequence step needs to be made precise.

minor comments (4)

[Proposition 4.8] In the proof, the notation 'Sym(a,sigma)' appears twice where 'Sym(A,sigma)' is meant; this is a typographical error in the definitions of S2 and S3.
[Proposition 5.6] The same symbol P0 is used for the ordering F0∩P and for the subset A0∩P; this makes the reduction paragraph hard to follow and should be changed, for example by writing P0' for the subset of A0.
[Proposition 5.1] In the final sentence, the phrase 'due to the reference forms P (mu⊗1)' is garbled; it should refer to the reference form s_P(mu⊗1) or the sign s_P(mu⊗1), and the current wording obscures the argument.
[Lemma 5.3] The set Z is defined only after the finite union over i∈I and (e,ℓ)∈E_i is introduced; defining all Sturm data p_e and p~_e before the union would improve readability and make the dependence of Z on e explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.8 is justified by an independent prime m-ideal classification from [2] that predates positive cones, and the repaired extension theorem is proved by a self-contained density/compactness argument, not by assuming its conclusion.

full rationale

The paper's main new statement, Theorem 3.8, defines N_P directly from a positive cone P via property (3.1), verifies that (I_P,N_P) is a prime m-ideal with torsion-free quotient and 2 not in I_P, and then invokes [2, Proposition 6.5], a published classification theorem from the authors' earlier work (2015, before positive cones were introduced in [4]). That theorem asserts that any such prime m-ideal is exactly (ker sign_P, ker sign^mu_P) for some ordering P; since I_P = ker sign_P by definition, the cited classification forces N_P = ker sign^mu_P. This is a legitimate use of an independent external result, not a definitional identification: nothing in [2] assumes that positive cones determine signature kernels, and the classification is not supplied by the current paper. The same applies to the use of [1] for reference forms and [4, Lemma 2.2] for weak diagonalization; these are black-boxed published results, and none encodes the target conclusion. In Section 5, the repaired extension theorem (Theorem 5.8) is proved via Proposition 5.6, which reduces to a finitely generated field, invokes the archimedean-density Proposition 5.2 (proved in the paper by Sturm sequences and Tarski transfer), constructs an elementary extension N by compactness in Lemma 5.5, and then applies Proposition 5.1. The proof of Proposition 5.6 does not assume Theorem 5.8; on the contrary, it proves the equality m_P = n_P needed for inclusion (a). Consequently there is no fitted parameter renamed as a prediction and no target statement assumed. The skeptic's concern that Lemma 5.3 may contain a gap in the description of S is a possible proof gap, not circularity.

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

The central claim rests on published results, several from the authors' own earlier series, but these are used as established theorems rather than inputs that assume the target. The genuinely new construction in Section 3 adds no free parameters and no new entities; the main nontrivial support is the prime m-ideal classification and the density of archimedean orderings.

assumptions (6)

domain assumption Existence of a reference form mu for (A, sigma) such that sign^mu_P is nonzero outside Nil[A, sigma] ([1, Theorem 6.4]).
Used throughout to define the mu-signature of hermitian forms (Section 2.2); signatures only make sense up to the choice of reference form, which the authors proved exists.
domain assumption Classification of prime m-ideals: every prime m-ideal (I, N) of W(A, sigma) with 2 not in I and W(A, sigma)/N torsion-free equals (ker sign_P, ker sign^mu_P) for some ordering P ([2, Proposition 6.5]).
This is the engine of Theorem 3.8; the authors' new pair (I_P, N_P) is shown to be prime and torsion-free, then [2, Prop. 6.5] identifies it with the kernel pair of a signature map.
domain assumption Pfister's local-global principle for algebras with involution: a hermitian form with zero signature at every ordering is hyperbolic up to 2-power multiples ([15, Theorem 4.1], [7, Theorem 6.5]).
Used in Proposition 2.15 to convert vanishing of a signature into a hyperbolicity statement and derive the contradiction with properness of the cone.
domain assumption Pivot property of the pairing * of hermitian forms ([5, Theorem 3.9], [10]).
Used in Proposition 2.16 to reduce arbitrary hermitian forms to diagonal forms with coefficients in the positive cone; the isometry (langle a rangle * langle a rangle) tensor phi ≃ (phi * langle a rangle) tensor langle a rangle is essential.
domain assumption Weak diagonalization over algebras with involution ([4, Lemma 2.2]).
Used in Proposition 2.14 and Lemma 3.4 to compare diagonal hermitian forms and their signature counts; this is one of the few results from [4] that is not affected by the error.
standard math Standard real algebraic geometry facts: Tarski transfer principle, Sturm sequence sign-change counting, and existence of real closures ([16, Corollary 11.5.4], [8], [19]).
Used in Proposition 5.2 and Lemmas 5.3-5.4 to prove that archimedean orderings are dense in X_F for finitely generated extensions of Q, which is needed for Proposition 5.6 and Theorem 5.8.

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Cite this review

Pith. "Pith review of Signature maps from positive cones on algebras with involution." pith.science (2026). https://pith.science/paper/CKT62EXQ

@misc{pith2026250522178,
  author       = {Pith},
  title        = {Pith review of: Signature maps from positive cones on algebras with involution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKT62EXQ}},
  note         = {Machine review of arXiv:2505.22178}
}

read the original abstract

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current paper we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

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Reference graph

Works this paper leans on

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Pith tools

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

[1] [4]

Positive cones on algebras with involution.Adv

Vincent Astier and Thomas Unger. Positive cones on algebras with involution.Adv. Math., 361:106954, 48, 2020

work page 2020

[2] [1]

Signatures of hermitian forms and the Knebusch trace formula.Math

Vincent Astier and Thomas Unger. Signatures of hermitian forms and the Knebusch trace formula.Math. Ann., 358(3-4):925–947, 2014

work page 2014

[3] [2]

prime ideals

Vincent Astier and Thomas Unger. Signatures of hermitian forms and “prime ideals” of Witt groups.Adv. Math., 285:497–514, 2015

work page 2015

[4] [3]

Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher.J

Vincent Astier and Thomas Unger. Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher.J. Algebra, 508:339–363, 2018

work page 2018

[5] [5]

Pfister's local-global principle for Azumaya algebras with involution

Vincent Astier and Thomas Unger. Pfister’s local-global principle for Azumaya algebras with involution.To appear in Documenta Mathematica. Preprint at https://arxiv.org/abs/2210.04851, 2025

work page Pith review arXiv 2025

[6] [6]

Bayer-Fluckiger and R

E. Bayer-Fluckiger and R. Parimala. Classical groups and the Hasse principle.Ann. of Math. (2), 147(3):651–693, 1998

work page 1998

[7] [7]

Weakly hyperbolic involutions.Expo

Karim Johannes Becher and Thomas Unger. Weakly hyperbolic involutions.Expo. Math., 36(1):78–97, 2018

work page 2018

[8] [8]

Springer-Verlag, Berlin, 1998

Jacek Bochnak, Michel Coste, and Marie-Fran¸ coise Roy.Real algebraic geometry, vol- ume 36 ofErgebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Math- ematics and Related Areas (3)]. Springer-Verlag, Berlin, 1998. Translated from the 1987 French original, Revised by the authors

work page 1998