Local-global principle for triangularizability and diagonalizability of matrices

Kai Huang; Yufan Liu

REVIEW 4 major objections 5 minor 1 cited by

Local-global principle for triangularizability and diagonalizability of matrices

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that when the ring of integers of a number field is a principal ideal domain, a matrix that is triangularizable or diagonalizable over every local completion is already triangularizable or diagonalizable over the ring of i

desk verdict Sections 3–4 are solid and genuinely new, but Theorem 5.4 is carried by a good-fibration lemma whose target is mis-specified as written; worth refereeing, not acceptable as is. read the letter →

arxiv 2511.15827 v3 pith:BJBRC7D4 submitted 2025-11-19 math.NT math.AG

Kai Huang , Yufan Liu This is my paper · ORCID

classification math.NTmath.AG MSC 14G1215A20

keywords local-globalprincipleHassetriangularizabilitydiagonalizabilityBrauer–ManinobstructionstratifiedJordanblockprincipalidealdomain

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 establishes a local-global principle for two basic matrix properties over number fields: if the ring of integers is a principal ideal domain, then a matrix that is triangularizable (or diagonalizable) over every completion—or even over almost all residue fields—is triangularizable (resp. diagonalizable) over the number field and over its ring of integers. The proof is elementary and uses unimodular completion plus the Chebotarev density theorem to pass from local splitting of the characteristic polynomial to global splitting. When the ring of integers is not a PID, the principle can fail, and the paper translates the problem into algebraic geometry by associating to each matrix a triangularization variety X_M and a diagonalization variety Y_M. For diagonalization, Y_M is shown to be a disjoint union of products of general linear groups, so the classical Brauer–Manin obstruction is the only obstruction to rational and integral points. For triangularization, the paper introduces a stratified Brauer–Manin obstruction and proves that it is the only obstruction for matrices similar to a diagonal block plus a single Jordan block, with every irreducible component of X_M smooth and k-rational.

What carries the argument

The central mechanism is the translation of a linear-algebra question into the arithmetic of an auxiliary variety. For a fixed M, the triangularization variety X_M ⊂ GL_n is defined by w·det(T)=1 and (T^*MT)_{ij}=0 for i>j, so its rational or integral points are exactly the transition matrices that achieve triangularization; the diagonalization variety Y_M is defined analogously by off-diagonal entries vanishing. For diagonalization, the proof uses the fact that Y_{M,red} splits as a disjoint union of products of general linear groups, reducing the Brauer–Manin statement to known theorems for linear algebraic groups. For triangularization, the load-bearing tool is the 'good fibration' techni

What would settle it

Compute the morphism in Lemma 5.12 for the smallest nontrivial case m=1, n=2: check whether the map f: W_s → Spec k[X,Y]/(ΣXY−1) is smooth and surjective on O_v-points for almost all v, and whether each fiber is isomorphic to SL_{2,s_2−1}×A^{3−s_1}; if the fiber structure or the codomain is wrong, the induction in Theorem 5.4 loses its foundation.

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Extended reading notes

Core claim

The central claim is that the local-global principle holds for triangularizability and diagonalizability of matrices over rings of integers that are PIDs: if M is triangularizable (resp. diagonalizable) over O_v for every place v—or even over the residue fields O_v/𝔭O_v for almost all v—then M is triangularizable (resp. diagonalizable) over k and over O_k. The key reduction is Lemma 3.1: over a PID, a matrix is integrally triangularizable if and only if it is triangularizable over the fraction field. To explain failures over non-PIDs, the authors define the triangularization variety X_M and diagonalization variety Y_M as closed subvarieties of GL_n cut out by requiring T^*MT to be triangular

Load-bearing premise

The proof of Theorem 5.4 rests on the claim that certain morphisms are 'good fibrations'—smooth, surjective on integral points for almost all places, with fibers isomorphic to affine spaces or special linear groups—and these claims are verified only by 'one checks' rather than by detailed proof; if any of those fibrations is not good (or if the apparent codomain error in Lemma 5.12 changes the fiber structure), the induction showing that the stratified Brauer–Manin set is cov

Editorial extensions

If this is right

Over principal ideal domains such as the integers, a matrix triangularizable modulo almost every prime is triangularizable over the ring itself, giving an effective integral Hasse principle for triangularizability.
The local-global principle for integral diagonalizability and triangularizability holds exactly when the ring of integers is a PID; for every non-PID ring of integers, the paper constructs explicit counterexamples.
For diagonalizability, the Brauer–Manin obstruction is always the only obstruction, so any failure of the local-global principle for diagonalizability is explained by the Brauer group of the associated variety.
For triangularizability, the stratified Brauer–Manin obstruction is the only obstruction for matrices with a single nontrivial Jordan block, and according to the paper's final remark also for all matrices of size at most 5.
If the paper's Conjectures 5.2 and 5.3 hold in full generality, the stratified Brauer–Manin obstruction would be the only obstruction to rational and integral points for every triangularization variety X_M.

Reading between the lines

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

The paper's approach suggests a general recipe for proving local-global principles for matrix properties: encode the property as a variety, decompose it via eigenvalue permutations, and use the fibration method; the Jordan-block case appears to be the hardest core, so proving the conjectures for a matrix with several Jordan blocks of the same eigenvalue would be a natural next test.
The 'good fibration' assertions in Lemma 5.12 are the point most worth scrutinizing: the codomain is written as a localization at ΣXY, but the image lies in the hypersurface ΣXY=1, so a reader extending this work should first verify that the morphism to Spec k[X,Y]/(ΣXY−1) is indeed smooth and has the claimed fibers SL_{s,r}×A^d.
Because the diagonalization variety is a union of algebraic groups, the Brauer group of Y_M is computable in explicit examples; one could use the non-PID counterexamples from Section 3 to compare the classical Brauer–Manin set with the actual rational points and see exactly which Brauer classes account for the failures.
The reduction to a single eigenvalue in Theorem 5.4 means that matrices with multiple Jordan blocks for the same eigenvalue form the immediate test case for Conjectures 5.2 and 5.3, and a concrete computation there would either confirm the conjectural picture or provide a counterexample.

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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 has two real accomplishments. The PID local-global statements (Propositions 3.3 and 3.5) are clean, correctly proved, and new, and the non-PID counterexamples in Remarks 3.4 and 3.7 look right. Theorem 4.2, showing that diagonalization varieties break into connected components each isomorphic to a product of general linear groups, is a genuinely nice structural result and the proof holds up. The introduction of the stratified Brauer–Manin obstruction is also a natural idea and worth developing.

The trouble is concentrated in the proof of the advertised special-case theorem, Theorem 5.4. That proof is carried by Lemma 5.12, and the stress-test concern is accurate. The morphism f: W_s -> Spec k[X,Y,1/(ΣXY)] is not well-posed as a good fibration because on W_s the determinant is 1, so ΣXY = 1 identically. The stated target contains points with ΣXY ≠ 1, and those points have empty fiber. The second condition of Definition 5.5 therefore fails for the morphism as written. The fix is plausibly just to change the target to Spec k[X,Y]/(ΣXY−1), and the rest of the fiber computation might then go through, but the lemma as written is not proved, and Theorem 5.4 currently rests on it.

This is a load-bearing gap, not a cosmetic typo. The same proof also leans on Corollary 5.7 and Lemma 5.11, where the crucial fiber descriptions and surjectivity on integral points are asserted with 'one checks' and 'arises naturally'; those need real arguments too. Lemma 5.14, by contrast, is intricate but is at least actually argued in the text, so I would not put it in the same category. Remark 5.16's claim of order-≤5 results without proof should be removed or supported.

I want to be clear: the earlier sections are good, and the special-case theorem is plausible and probably repairable. But the paper should not be accepted with Theorem 5.4 in its current form. The right move is to send it to a serious referee with a request to focus on Lemma 5.12 and the fibration lemmas; the author can then fix the target and fill in the missing verifications.

Who is this for? Arithmetic geometers working on Hasse principles for matrix varieties, and linear algebraists interested in local-global questions. A careful reader gets real value from Sections 3–4 even if Theorem 5.4 needs repair. I would bring it to a reading group and would cite the PID results if I worked in the area.

Referee Report

4 major / 5 minor

Summary. The paper studies a local-global principle for triangularizability and diagonalizability of an integral matrix over a number field. It proves that when the ring of integers is a PID, local triangularizability/diagonalizability over completions or residue rings implies global. It also constructs counterexamples over non-PID rings of integers. The paper then translates the problem into rational and integral points on triangularization and diagonalization varieties. For diagonalization, Theorem 4.2 shows that every connected component of the reduced diagonalization variety is a product of general linear groups, so the Brauer–Manin obstruction is the only obstruction to rational and integral points. For triangularization, the paper introduces a stratified Brauer–Manin obstruction and claims in Theorem 5.4 that for M similar to diag(λ I_m, J_n(λ)), every irreducible component of the triangularization variety is smooth and k-rational, and the stratified obstruction is the only obstruction.

Significance. The local-global statements for PIDs (Propositions 3.3 and 3.5) and the geometric description of the diagonalization variety (Theorem 4.2) are clean, elementary, and appear correct. These parts have independent value. The stratified Brauer–Manin obstruction is a novel notion, and Theorem 5.4, if fully proved, would be a meaningful advance toward Conjectures 5.2 and 5.3. However, the proof of Theorem 5.4 as written is not rigorous: it depends on several asserted good-fibration properties, an apparently incorrect codomain in Lemma 5.12, and a sketched ideal equality in Lemma 5.14. The advertised special-case result is therefore not yet established. The manuscript is likely repairable, but a substantial revision is needed.

major comments (4)

[Section 5, Lemma 5.12] The morphism f is defined with target Spec k[X_{s_{n-2}+1},...,X_{s_{n-1}}; Y_{...}; 1/(Σ X_α Y_α)]. On W_s the determinant is 1 and the displayed expansion gives det(T) = Σ X_α Y_α, so the image of f lies in the hypersurface Σ X_α Y_α = 1. The stated target contains points with Σ X_α Y_α = c ≠ 1, over which the fiber is empty. Thus the second condition of Definition 5.5 (surjectivity on O_v-points for almost all v) fails for the stated codomain, and the induction proving W_s(k)=W_s(A_k)_• collapses. The target should presumably be Spec k[X,Y]/(ΣXY−1); with that correction, the surjectivity and fiber-description arguments still need to be supplied.
[Section 5, Corollary 5.7, Lemma 5.11, Lemma 5.12] Several load-bearing good-fibration claims are merely asserted. Corollary 5.7 says 'One can verify that f is a good fibration' and 'surjectivity on integral points arises naturally from the construction', without proof; the target V also contains a notational typo (T instead of Z). Lemma 5.11 says 'One checks that f is a good fibration whose fibers are always isomorphic to A^{s-r}' with no coordinate verification. Lemma 5.12 asserts that the fibers are SL_{m+1,s_2−1}×A^{m+2−s_1} (n=2) or W'_{s'}×A^{m+n−2−s_{n−1}} (n≥3) without proof. These fiber identities are exactly what makes f good and drives the induction; they must be proved explicitly.
[Section 5, Lemma 5.14] The proof of the claimed ideal equality is not complete. The cofactor-expansion argument relies on informal statements such as 'we observe', 'if there is any surviving expansion term', and 'the expansion process must stop', without a precise induction invariant or termination argument. Since Lemma 5.14 is used to replace X_M by X'_M and Theorem 5.4 concerns the irreducible components of X_M, this equality is load-bearing. Moreover, the assertion that 'same k-points and thus same reduced subscheme structure' needs justification: for non-reduced schemes of finite type over k, the set of k-points does not determine the reduced structure in general.
[Section 5, proof of Theorem 5.4 and Remark 5.16] The final step of Theorem 5.4 states 'one can easily check that V_r ≃ GL_{m+1}×G_m×A^{m+1}_k' and 'V_r ≃ V_{r'}×G_m×A^{m+n−1}_k' without giving the isomorphisms; these claims carry the induction. Remark 5.16 also asserts that the authors have established Conjectures 5.2 and 5.3 for all M of order ≤5, but no proof or reference is provided. Such an unsupported result should either be proved or explicitly labelled as work in progress.

minor comments (5)

[Section 2, Lemma 2.1] The citation [SZ14, Thmeorem C] contains a typo: 'Thmeorem' should be 'Theorem'.
[Section 3, Proposition 3.5] 'enries' should be 'entries'.
[Section 5, Corollary 5.7] In part (1), the closed subscheme □ is defined using T_{i,j} but the ambient ring is in variables Z_{i,j}. This should be Z_{i,j}.
[Section 5, Lemma 5.12] W_s is defined modulo det(T_{ij})_{n×n}−1; the notation should be det(T)−1 with T the (m+n)×(m+n) matrix. Also, 'W'_s is isomorphic to W_s×G^m' should read W_s×G_m.
[Section 5, Remark 5.16] The claim about order ≤5 needs either a proof or a clear statement that it is work in progress.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper's central local-global results (Propositions 3.3 and 3.5) are proved by classical linear algebra and standard number-theoretic ingredients: unimodular completion over a PID, reduction modulo primes, compactness of GL_n(O_p), strong approximation, and the Chebotarev density theorem. These arguments do not presuppose the geometric conclusions. The geometric reformulation defines the triangularization and diagonalization varieties by the transition-matrix equations, and Theorem 4.2 derives the diagonalization case from the independently established local-global principle plus the structure of Y_M as a disjoint union of products of general linear groups; the Brauer-Manin statement is then imported from external results on connected linear algebraic groups. The stratified Brauer-Manin obstruction is introduced as a definition, not fitted to any data, and Theorem 5.4 is proved by an induction in which Lemma 5.12 provides lower-dimensional fibers; the fiber inclusions such as SL_{m+1,s_2-1} x A^d and W'_{s'} x A^d are asserted rather than fully verified, and the codomain in Lemma 5.12 appears to be mis-specified (the image lies in sum X_alpha Y_alpha = 1), but these are correctness or exposition gaps, not circular reductions. The paper cites no prior work by its own authors, and Remark 5.16 explicitly states which cases remain open, further indicating that no result is being assumed through a self-citation chain. The suspicious 'one checks' smoothness and fibration claims would affect the validity of Theorem 5.4 if false, but they do not make the derivation equivalent to its own inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The central claims rest on standard results from algebraic number theory, arithmetic geometry, and the Brauer–Manin formalism. No free parameters are fitted; the only new construct is the stratified Brauer–Manin obstruction.

assumptions (8)

standard math Chebotarev density theorem
Used in Lemma 3.2 to conclude a polynomial splits over k when it splits modulo almost all primes.
standard math Strong approximation for number fields and affine spaces
Used in Proposition 3.5 to adjust eigenbases globally, and in the good-fibration arguments of Section 5; the paper cites [CTS21, Theorem 13.1.1].
standard math Unimodular completion theorem (Reiner)
Used in Lemma 3.1(1) to construct an SL_n(R) matrix with prescribed primitive first column over a PID.
standard math Brauer–Manin equality for connected linear algebraic groups (Demarche)
Used in Lemma 2.2 to handle the GL_n factors in Theorem 4.2.
standard math Product and disjoint-union formulas for Brauer–Manin sets
Used in Lemmas 2.1 and 2.5 to decompose the Brauer–Manin set of products and finite disjoint unions; cites [SZ14], [Lv20], [CTS21].
standard math Strong approximation for complements of codimension ≥2 closed subschemes
Used in Lemma 2.3 and Corollary 5.7 to get X(k)=X(A_•) for such open subschemes; cites [HW16, Lemma 1.8].
standard math Miracle flatness
Used in Corollary 5.7 to assert flatness (and hence smoothness) of a morphism with constant fiber dimension onto a regular base.
standard math Spreading out / existence of integral models
Used in Definition 5.5 to choose models for fibrations; cites [Poo23, Theorem 3.2.1].

invented entities (1)

Stratified Brauer–Manin obstruction
purpose: To explain failures of the local-global principle for triangularization varieties by taking the union of Brauer–Manin sets of dominant irreducible components.
Introduced in Definition 5.1. The conjecture that it is the only obstruction is established only in special cases in this paper, so the predictive handle is internal.

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Pith. "Pith review of Local-global principle for triangularizability and diagonalizability of matrices." pith.science (2026). https://pith.science/paper/BJBRC7D4

@misc{pith2026251115827,
  author       = {Pith},
  title        = {Pith review of: Local-global principle for triangularizability and diagonalizability of matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJBRC7D4}},
  note         = {Machine review of arXiv:2511.15827}
}

abstract

Given a number field $k$ with the ring of integers $\mathcal{O}_k$ and a matrix $M\in \mathrm{M}_{n}(\mathcal{O}_k)$. We prove that if $\mathcal{O}_k$ is a principal ideal domain, the local-global principle for triangularizability and diagonalizability of $M$ holds. To explain the possible failures of the local-global principle, we prove that the stratified Brauer--Manin obstruction is the only obstruction to the local-global principle for triangularizability and diagonalizability of $M$ in some special cases.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On Bounds of Extension Degrees for Similarity of Integral Matrices over Number Fields
math.NT 2026-06 unverdicted novelty 7.0 of 10

No uniform bound exists for the extension degree making integral matrices similar over number fields after local similarity everywhere, but a bound depending on a separable characteristic polynomial is provided.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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Brian Conrad, Weil and Grothendieck approaches to adelic points , L’Enseignement Mathématique 58 (2012), no. 1/2, 61--97

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Skorobogatov, The Brauer--Grothendieck group , Ergebnisse der Mathematik und ihrer Grenzgebiete

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Chang Lv, A note on the Brauer group and the Brauer--Manin set of a product , Bulletin of the London Mathematical Society 52 (2020), 932--941

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