Published paper
Abstract
The article systematizes the structure theory of linear algebraic groups. The selected downstream uses concern Jordan factors and the semisimple–unipotent direct-product decomposition of connected nilpotent algebraic groups.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statements:
- Theorem 8.4 · original article printed p. 47; exact statement cross-checked in Borel–Tits Corollary 8.5 and published-book proofsJordan semisimple and unipotent factors remain inside the algebraic group.
- Theorem 11.1 · theorem number and statement cross-checked in exact Ono 1958 Proposition 1.11 and Rosenlicht 1957 printed p. 31A connected nilpotent algebraic group is the direct product of its semisimple torus and unipotent subgroup, as used in Kazhdan–Margulis Lemma 3.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Target: Borel, Groupes linéaires algébriques (1956) · reviewed fallback: Ono, Sur la réduction modulo p des groupes linéaires algébriques (1958), exact Osaka University publication scan
Target: Armand Borel. Groupes linéaires algébriques. Annals of Mathematics, Second Series 64 (1956), no. 1, 20–82. Reviewed fallback: Takashi Ono. Sur la réduction modulo p des groupes linéaires algébriques. Osaka Mathematical Journal 10 (1958), 57–73.
The exact Borel 1956 publication scan is access-restricted at JSTOR, Internet Archive, and HathiTrust. Ono 1958 is an exact, openly accessible journal publication that reproduces the nilpotent direct-product statement and explicitly cites Borel Theorem 11.1; exact Borel–Tits and Rosenlicht consumer scans and published-book terminals are used separately for the Jordan-decomposition and independent-proof checks.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The two selected structural statements are correct and match the exact uses downstream: Jordan factors remain inside the algebraic group, and a connected nilpotent algebraic group is the direct product of its semisimple torus and unipotent subgroup. Exact consumer scans and published-book terminal proofs independently verify these claims. The exact 1956 publication scan is access-restricted, so no claim is made that its displayed wording was visually checked.
Theorem 8.4No issue found
Jordan factors remain in the algebraic group
Original article · Theorem 8.4, printed p. 47, locator independently reported by Couty–Esterle–Zarouf; Borel–Tits exact Numdam VOR · physical PDF p. 71 / printed p. 124 · Corollary 8.5, citation [1, th. 8.4]
For in a linear algebraic group, the multiplicative Jordan decomposition has semisimple and unipotent, with both factors again in the group. Over a perfect ground field, implies . Borel–Tits use exactly this rational form in characteristic zero. Chevalley's Théorie des groupes de Lie II, Theorem 18, and Borel's later Linear Algebraic Groups, Chapter I, Theorem 4.4, independently supply the required theorem.
Borel–Tits exact Numdam version of record ↗Theorem 11.1No issue found
A connected nilpotent algebraic group splits into semisimple and unipotent factors
Original article · Theorem 11.1, exact theorem locator printed in Ono 1958, Proposition 1.11, footnote 11; Ono exact Osaka University scan · physical PDF p. 5 / printed p. 60; Rosenlicht exact VOR · physical PDF p. 7 / printed p. 31
For a connected nilpotent algebraic group , the semisimple elements form a unique central torus , the unipotent elements form , and multiplication gives the direct product . Ono's Proposition 1.11 cites this statement specifically to Borel Theorem 11.1, and Rosenlicht states the same direct-product structure while citing Borel. Borel's later monograph, Chapter III, Theorem 10.6(3), independently closes the claim.
Ono 1958 exact Osaka University publication scan ↗Kazhdan–Margulis Lemma 3Dependency match verified
The nilpotent decomposition route matches the exact consumer
Kazhdan–Margulis exact MathNet VOR · physical PDF pp. 3–4 / printed pp. 149–150 · proof of Lemma 3
Kazhdan–Margulis first replace the connected nilpotent Lie subgroup by its Zariski closure and then write the resulting algebraic nilpotent group as . On the next page they repeat for the varying intersection subgroup. Their citation [10] to Borel–Tits is attached only to the later containment of in a positive-root unipotent subgroup. The direct-product sentence therefore has the separate Borel Theorem 11.1 route identified here.
Kazhdan–Margulis exact MathNet version of record ↗Source boundaryExact source identified · visual inspection unavailable
Exact article pagination is 20–82, but the scan is not accessible
Annals of Mathematics, Second Series 64 (1956), no. 1; DOI 10.2307/1969949; Internet Archive exact issue identifier sim_annals-of-mathematics_1956-07_64_1; HathiTrust item uc1.b3627311
The exact article occupies pp. 20–82, not the truncated 20–80 printed in the Borel–Tits bibliography. JSTOR returned an access challenge, Internet Archive marks the exact issue PDF and OCR private and returned HTTP 403, and HathiTrust classifies the exact volume as limited search-only. The correctness findings therefore rely on exact consumer scans and independent book-terminal proofs, not on an unperformed inspection of the 1956 page images.
02Proofs4 reported findingsFurther proof verification needed
The selected statements have independent, proof-complete routes to published monographs, but the completeness and citation ancestry of Borel's own 1956 proofs cannot be evaluated without the access-restricted publication scan. No non-book source has been established as a genuine proof-critical parent of the selected arguments.
Exact 1956 proof textNot independently verified
The displayed proofs are not available for direct audit
JSTOR stable item 1969949; Internet Archive exact issue sim_annals-of-mathematics_1956-07_64_1; HathiTrust exact volume uc1.b3627311
The exact publication PDF, OCR, and page images were not lawfully retrievable from any identified repository in this environment. Consequently, line-by-line proof completeness, internal cross-references, and the exact bibliography attached to Theorems 8.4 and 11.1 remain unverified. This source limitation does not count as evidence of an error.
Theorem 8.4 terminal routeIndependent proof verified at book terminals
Jordan decomposition closes at published books
Claude Chevalley, Théorie des groupes de Lie II (1951), Theorem 18, p. 184; Armand Borel, Linear Algebraic Groups, second edition (1991), Chapter I, Theorem 4.4
The semisimple and unipotent parts are polynomial expressions in an element on finite-dimensional stable subspaces. Applied to the coordinate-ring action, these polynomials preserve the defining ideal of the algebraic subgroup, so both factors remain in the subgroup; over a perfect field the same polynomial construction preserves -rationality. This is the exact mechanism required by Borel–Tits Corollary 8.5.
Theorem 11.1 terminal routeIndependent proof verified at book terminal
The nilpotent direct product closes at Borel's monograph
Armand Borel, Linear Algebraic Groups, second edition (1991), Chapter III, Theorem 10.6(3)
In a connected nilpotent algebraic group the semisimple parts form a closed subgroup, the unipotent parts form the unipotent subgroup, and the two mutually normalize, intersect trivially, and therefore commute. Every geometric point has a Jordan decomposition, so multiplication identifies the group with the direct product of these factors. This supplies the exact factorization used at the start of Kazhdan–Margulis Lemma 3.
Historical citation boundaryPotential antecedent recorded · proof-critical status not independently verified
Kolchin 1948 is not promoted to a proof-critical parent
Couty–Esterle–Zarouf, historical discussion, physical PDF p. 15; reference [35]
The accessible historical source says that Borel cites E. R. Kolchin's 1948 Annals article together with Chevalley's 1951 book for the characteristic-zero antecedent to Theorem 8.4. Because Borel's exact proof paragraph is unavailable, the Kolchin article cannot be classified as a genuine imported proof step. Chevalley's book already proves the full characteristic-zero result required by the inspected consumer, so no unresolved non-book branch is necessary for independent closure.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.