Published paper
Abstract
The article establishes ground-field triangularization, split-solvability, and rational fixed-point results for algebraic groups. These are used in Borel–Tits's construction of rational positive-root subgroups.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statements:
- Corollary 1 · exact Springer VOR physical PDF p. 9 / printed p. 33Triangularizes a unipotent matrix group over the ground field when every component is defined there.
- Corollary 2 · exact Springer VOR physical PDF pp. 10–11 / printed pp. 34–35Gives split solvability over a perfect ground field, including the connected unipotent case.
- Rational fixed-point lemma · exact Springer VOR physical PDF pp. 11–12 / printed pp. 35–36A split connected solvable group acting on a complete variety with a rational point has a rational fixed point.
- Proposition 7 · exact Springer VOR physical PDF p. 12 / printed p. 36Triangularizes a connected split solvable matrix group over the ground field.
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Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact Springer version of record · Annali di Matematica Pura ed Applicata 43 (1957), 25–50
Maxwell Rosenlicht. Some rationality questions on algebraic groups. Annali di Matematica Pura ed Applicata 43 (1957), 25–50.
Open audited source ↗01Statements6 reported findingsNo statement issues found
The exact version of record states the split-solvability, triangularization, and rational fixed-point results used by Borel–Tits. Their hypotheses match the later uses, and no incorrect selected statement was found. The exact pagination places Proposition 7 on printed p. 36 rather than within pp. 33–35.
Corollary 1No issue found
A unipotent matrix group with rational components is triangularizable over the ground field
Exact Springer version of record · physical PDF p. 9 / printed p. 33 · Corollary 1
If every component of the matrix group is defined over and every element of is unipotent, Rosenlicht concludes that is triangularizable over . The component hypothesis is the descent input: after triangularization over an algebraic closure, one can use generic points over of all components in the argument of Proposition 5. This is the result cited in Borel–Tits Section 8.1.
Corollary 2No issue found
The characteristic-root condition gives split solvability over a perfect field
Exact Springer version of record · physical PDF pp. 10–11 / printed pp. 34–35 · Corollary 2 and proof
For a connected matrix group over a perfect field satisfying Proposition 5's characteristic-root condition, the corollary produces a normal chain of connected normal subgroups defined over , with every successive quotient isomorphic over to or . This is precisely Rosenlicht's definition that is a field of definition for solvability, and it supplies the perfect-field splitness input used in Borel–Tits.
Fixed-point lemmaNo issue found
A split solvable action on a complete variety has a rational fixed point
Exact Springer version of record · physical PDF pp. 11–12 / printed pp. 35–36 · lemma preceding Proposition 7
Let be connected and solvable, let be a field of definition for its solvability, and suppose acts regularly over on a complete variety . If has a -point, the lemma gives a -point fixed by all of . This exactly matches the statement cited in Borel–Tits Section 2.7(i).
Proposition 7No issue found
A connected split solvable matrix group is triangularizable over the ground field
Exact Springer version of record · physical PDF p. 12 / printed p. 36 · Proposition 7
If is a field of definition for the solvability of the connected solvable matrix group , Proposition 7 asserts that is triangularizable over . Borel–Tits Section 8.1 and the proof of Theorem 8.2 use this statement to obtain a rational invariant flag.
Borel–Tits dependency matchDependency match verified
The later consumer uses exactly the checked Rosenlicht results
Borel–Tits exact Numdam VOR · physical PDF p. 14 / printed p. 67, Section 2.7(i); physical PDF pp. 69–70 / printed pp. 122–123, Section 8.1 and proof of Theorem 8.2
Section 2.7(i) imports the rational fixed-point lemma. Section 8.1 imports triangularization of split solvable and unipotent groups and perfect-field splitness. Theorem 8.2 then uses a rational stable flag, a rational fixed point in a complete orbit closure, and splitness of connected unipotent groups. Each use falls within the hypotheses of the selected Rosenlicht statement.
Locator clarificationSource locator corrected
Proposition 7 is on printed p. 36
Exact Springer version of record · physical PDF pp. 9–12 / printed pp. 33–36
Corollary 1 is on printed p. 33, Corollary 2 is on p. 34 with related discussion on p. 35, and the fixed-point lemma runs from pp. 35–36. Proposition 7 begins and ends on p. 36. This corrects the compact page description of the citation cluster and does not identify a mathematical defect.
02Proofs8 reported findingsNo proof issues found
The selected proofs are correct and complete at the level required for their Borel–Tits use. Corollaries 1–2 give valid descent and normal-series arguments; the fixed-point lemma closes its induction through a one-dimensional quotient; and Proposition 7 correctly applies the lemma to the flag variety. All three historical article branches are represented by separate ordinary-audit nodes; the modern book references are supplementary cross-checks, not substitutes for those branches.
Corollary 1 proofNo issue found in the reviewed proof
The generic-point descent handles every component
Exact Springer version of record · physical PDF p. 9 / printed p. 33 · Corollary 1 proof, with Proposition 5 on printed pp. 31–33
Kolchin triangularization supplies a common flag over the algebraic closure. Because each component is defined over , Rosenlicht can choose its generic point over and repeat Proposition 5's eigenvector descent with the finite set of component-generic points. Every element then preserves the descended flag. The abbreviated reference to the pertinent part of Proposition 5 omits no new case.
Corollary 2 normal chainNo issue found in the reviewed proof
Intersecting the standard triangular chain gives normal one-dimensional factors
Exact Springer version of record · physical PDF p. 10 / printed p. 34 · first paragraph of the proof of Corollary 2
After Proposition 5 triangularizes over , Rosenlicht intersects with a prime-field-defined normal chain in the full triangular group and takes identity components. The resulting connected subgroups are normal in and their dimensions drop by at most one. Perfectness ensures that these identity components are defined over ; after repetitions are removed, each nontrivial quotient is one-dimensional.
Corollary 2 quotient typesNo issue found in the reviewed proof
The toric and unipotent factors descend to the claimed standard groups
Exact Springer version of record · physical PDF p. 10 / printed p. 34 · second paragraph of the proof of Corollary 2
Above the unipotent radical, the one-dimensional quotient embeds over in a split diagonal torus and is therefore . Below it, a one-dimensional unipotent quotient becomes after scalar extension. Rosenlicht's genus-zero completion argument uses the identity point and perfectness to make the unique point at infinity rational over , after which the group law is the additive one. This supplies every factor required by the definition of split solvability.
Fixed-point lemma inductionNo issue found in the reviewed proof
The reduction through or yields a rational boundary fixed point
Exact Springer version of record · physical PDF pp. 11–12 / printed pp. 35–36 · complete proof of the fixed-point lemma
Induction first produces a -point fixed by a connected normal subgroup with quotient isomorphic to or . After reduction to the closure of the generic orbit, the quotient's orbit map extends from its standard open subset of the projective line to the complete variety. If the boundary is nonempty, it consists of one or two -points and connectedness makes each fixed; if the boundary is empty, commutativity and transitivity make the original rational point fixed. The alternatives exhaust the one-dimensional quotient case.
Proposition 7 proofNo issue found in the reviewed proof
A rational fixed flag gives the required triangular basis
Exact Springer version of record · physical PDF p. 12 / printed p. 36 · complete proof of Proposition 7
The full flag variety of the underlying vector space is complete, defined over the prime field, and has a -point. The fixed-point lemma therefore supplies a -fixed full flag rational over . Choosing a -basis adapted to that flag simultaneously puts every element of in triangular form.
Historical parent: Rosenlicht 1956Required historical article branch represented
The quotient-action descent has an exact earlier-article locator
Rosenlicht, Some Basic Theorems on Algebraic Groups · exact issue scan physical PDF pp. 196–197 / printed pp. 416–417 · Theorem 5 and Corollary; cited in the 1957 lemma on printed p. 36
Theorem 5 constructs the action on the variety of -orbits for a normal algebraic subgroup , and its corollary records descent to the field of definition. This precise branch is represented at ordinary-audits/rosenlicht-basic-theorems-algebraic-groups.json.
Historical parents: Borel and WeilRequired historical article branches represented
The Borel 1956 and Weil 1955 branches have separate audit nodes
Exact Rosenlicht 1957 VOR · printed p. 33, citation [1, Chapter IV], and printed pp. 35–36, citations [1] and [8]
Corollary 1 invokes Borel's 1956 presentation of Kolchin triangularization, and the fixed-point argument follows Borel while using Weil's 1955 homogeneous-space regularization. These genuine historical parents are represented at ordinary-audits/borel-linear-algebraic-groups-1956.json and ordinary-audits/weil-algebraic-transformation-groups.json.
Supplementary canonical cross-checkSupplementary published-book terminal
A published monograph independently confirms the consumer-critical conclusions
T. A. Springer, Linear Algebraic Groups, Second Edition, 1998 · Proposition 14.1.2 on printed pp. 237–238, Proposition 14.1.4 and Remark 14.1.5 on p. 238, Theorem 14.1.7 on p. 239, and Corollary 14.3.10 on p. 247
Springer proves the ground-field triangularization criterion, triangularization of connected split solvable groups, the rational fixed-point theorem for projective varieties, and splitness of connected unipotent groups over perfect fields. This is an independent canonical cross-check. It does not replace the Borel 1956, Rosenlicht 1956, or Weil 1955 branches of Rosenlicht's original proof.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.