Published paper

Some Basic Theorems on Algebraic Groups

Maxwell Rosenlicht

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Publication metadata

Journal citation: American Journal of Mathematics 78(2), 401–443 (1956)
Publication date: 1956-04-01
DOI: 10.2307/2372523 ↗

Abstract

The article proves quotient and orbit-space results for algebraic group actions. Theorem 5 and its corollary give the quotient-action descent used by Rosenlicht's later fixed-point argument; a separate Kazhdan–Margulis citation to this paper does not match any theorem in it.

Role in dependence graphs

Proof-critical source

The Dirichlet spectrum

This paper is included only for the following marked statements:

Claim attributed by Kazhdan–Margulis · complete exact article scan, physical PDF pp. 181–223 / printed pp. 401–443; Kazhdan–Margulis citation on physical PDF p. 4 / printed p. 150Alleged local semicontinuity of dimensions in the ordered-torus family.
Theorem 5 and corollary · exact issue scan physical PDF pp. 196–197 / printed pp. 416–417Constructs the quotient action by a normal subgroup and descends it to the ground field for Rosenlicht's 1957 fixed-point induction.

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Audit summary

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Exact issue-level publication scan · American Journal of Mathematics 78 (1956), no. 2, 401–443

Maxwell Rosenlicht. Some Basic Theorems on Algebraic Groups. American Journal of Mathematics 78 (1956), no. 2, 401–443.

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Generated August 27, 2026

01Statements4 reported findingsNo statement issues found

The dimension-semicontinuity claim imported by Kazhdan–Margulis is mathematically correct in the stated algebraic-family setting, but it is not stated as a theorem or proposition in Rosenlicht's cited article. Theorem 5 and its corollary, which Rosenlicht 1957 uses for quotient-action descent, are correctly stated in the exact 1956 publication.

Kazhdan–Margulis imported claimCorrect · source attribution not direct

Dimensions in the algebraic family are generically minimal

Kazhdan–Margulis exact MathNet English VOR, physical PDF p. 4 / printed p. 150, proof of Lemma 3, citation [11]

For an algebraic family, upper semicontinuity of fiber dimension gives, for each $x_0$ , a Zariski-open neighborhood $U$ on which $d(x)\leq d(x_0)$ . On the irreducible ordered-torus parameter variety this makes the minimum locus open and dense. Kazhdan–Margulis calls the displayed inequality lower semicontinuity; in modern terminology for integer-valued functions it is usually called upper semicontinuity.

Exact Rosenlicht source checkSource scope not independently verified

No matching semicontinuity theorem occurs in the cited article

Exact publication scan, complete article on physical PDF pp. 181–223 / printed pp. 401–443

The complete article contains no statement of semicontinuity for dimensions of an algebraic family of subgroups. Its nearest dimension result, Proposition 1 on printed pp. 409–411, concerns inverse images under one equivariant map between homogeneous spaces and proves the fixed formula $\dim W_1=\dim V_1-\dim V_2+\dim W_2$ ; that is not the Kazhdan–Margulis family claim.

Theorem 5No issue found

A normal subgroup quotient acts on the generic orbit variety

Exact issue-level publication scan, physical PDF p. 196 / printed p. 416, Theorem 5

If an algebraic group $G$ acts on a variety $V$ and $H$ is a normal algebraic subgroup, Rosenlicht proves that $G/H$ acts rationally on the variety $W$ of generic $H$ -orbits by $gH(Hv)=H(gv)$ . He also identifies the variety of generic $(G/H)$ -orbits on $W$ birationally with the variety of generic $G$ -orbits on $V$ . Normality is exactly the hypothesis needed for the displayed rule to be independent of representatives.

Theorem 5 corollaryNo issue found

The quotient action and orbit-space comparison descend to the stated field

Exact issue-level publication scan, physical PDF pp. 196–197 / printed pp. 416–417, corollary to Theorem 5

When the groups, varieties, operations, orbit varieties, and natural rational maps occurring in Theorem 5 are all defined over $k$ , the corollary correctly concludes that both the induced $G/H$ -action on $W$ and the natural birational equivalence of the two generic orbit varieties are defined over $k$ . This is the exact descent input cited in Rosenlicht 1957's fixed-point proof.

02Proofs7 reported findingsFurther proof verification needed

Rosenlicht 1956 does not supply a proof of the specific semicontinuity claim cited by Kazhdan–Margulis, so that citation route remains unverified. Independently, the printed proof of Theorem 5 and its corollary correctly establishes the quotient-action descent used by Rosenlicht 1957; its only genuine external article parent on the selected route is Weil 1955, which is audited separately.

Direct citation routeNot independently verified from cited source

The cited paper does not prove the selected claim

Rosenlicht exact publication scan, printed pp. 401–443; Kazhdan–Margulis physical PDF p. 4 / printed p. 150

An exhaustive text check of the article and visual inspection of its relevant theorem pages found no theorem matching the dimension-semicontinuity assertion. Therefore Rosenlicht [11] cannot by itself be recorded as a verified proof parent for this step.

Nearest result: Proposition 1Correct and complete · not the imported theorem

The homogeneous-space inverse-image formula is correct but does not fill the gap

Exact publication scan, physical PDF pp. 189–191 / printed pp. 409–411

Transitivity makes every fiber have the same dimension, so the graph-intersection calculation yields $\dim W_1=\dim V_1-\dim V_2+\dim W_2$ and surjectivity on each component. The coefficient and separability clauses use Weil's intersection-cycle theorems. This fixed homogeneous-space calculation does not imply semicontinuity for a varying family of subgroup intersections.

Recursive source routeDependency scope verified with source substitution

The selected claim closes at an existing book terminal

Canonical terminal substitution already recorded for Kazhdan–Margulis Lemma 3

The standard upper-semicontinuity theorem for algebraic fiber dimensions supplies the exact local inequality needed by Kazhdan–Margulis and is already represented by Hartshorne's Algebraic Geometry as a published-book terminal. Rosenlicht Proposition 1 itself reaches only Weil's Foundations of Algebraic Geometry, also a published-book terminal. No additional non-book proof-critical parent is exposed.

Theorem 5 proofNo issue found in the reviewed proof

Invariant function fields define the quotient action and identify the two orbit fields

Exact issue-level publication scan, physical PDF p. 196 / printed p. 416, complete proof of Theorem 5

Writing $θ:G\dashrightarrow G/H$ and $τ:V\dashrightarrow W$ for the natural rational maps, the proof first shows that $τ(gv)$ is rational over $k(g,τ(v))$ and then, using normality of $H$ , over $k(θ(g),τ(v))$ . These field identities give rational maps from every component of $(G/H)\times W$ to $W$ . Independent generic points verify the group law, and equality of the $(G/H)$ -invariant subfield of $k(W)$ with the $G$ -invariant subfield of $k(V)$ gives the claimed birational orbit-space identification.

Corollary proofNo issue found in the reviewed proof

The field-of-definition conclusion is already encoded in the generic-point construction

Exact issue-level publication scan, physical PDF pp. 196–197 / printed pp. 416–417, proof and corollary to Theorem 5

Every rational function and generic-point identity used to construct the quotient action is taken over the common field $k$ . The same invariant-subfield identity defines the orbit-space birational equivalence over $k$ . Thus the corollary adds no unsupported descent step.

Historical parent for quotient existenceRequired historical article branch represented

The exact non-book parent is the separately audited Weil regularization theorem

Exact issue-level publication scan, physical PDF pp. 193–194 / printed pp. 413–414, proof of Theorem 4; Weil 1955, printed p. 375, unnumbered theorem, especially part (ii)

Theorem 5 uses the quotient group $G/H$ supplied by Theorem 4 in the same paper. Rosenlicht's direct proof of Theorem 4 invokes Weil's result that a prehomogeneous rational action has a birational homogeneous-space model. That exact 1955 article theorem is represented at ordinary-audits/weil-algebraic-transformation-groups.json. The separate Rosenlicht 1955 citation used in Theorem 2 proves separable generation of an invariant-field extension; that separability assertion is not used in Theorem 5's quotient-action or invariant-field argument and is therefore not a proof-critical parent of this selected claim.

Rosenlicht 1957 consumerDependency match verified

The later fixed-point proof uses precisely the checked descent corollary

Rosenlicht 1957 exact Springer VOR, physical PDF p. 12 / printed p. 36, proof of the fixed-point lemma, citation [5, Theorem 5]

After a normal subgroup $G_1$ fixes the reduced variety, the 1957 proof needs the quotient $G/G_1$ to act on that variety over $k$ . Theorem 5 and its corollary give exactly this induced quotient action and field-of-definition conclusion.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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