Published paper
Abstract
The article regularizes rational group laws and prehomogeneous rational actions into algebraic groups and homogeneous spaces. Its final theorem is the source used in Rosenlicht's quotient and fixed-point arguments.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statements:
- Final unnumbered theorem, clause (ii) · exact issue scan physical PDF p. 183 / printed p. 375Replaces a prehomogeneous rational action by a birationally equivalent homogeneous-space model over the ground field.
- Proposition 7 and Appendix input · Proposition 3 proof printed pp. 363–364; Appendix Proposition 7 printed pp. 380–381, citation on p. 381The open-set and specialization construction used in the descent argument imports Weil 1954 Lemma 7 directly, independently of the Lang–Weil route.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact digitized issue scan · American Journal of Mathematics 77(2) (1955), 355–391
André Weil. On Algebraic Groups of Transformations. American Journal of Mathematics 77(2) (1955), 355–391.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The exact publication states the homogeneous-space regularization theorem used by Rosenlicht, with the required ground-field descent and uniqueness. Its proof's Appendix Proposition 7 has the correct closed-fiber statement imported directly from Weil 1954, Lemma 7. No incorrect selected statement was found.
Theorem (ii)No issue found
A pre-homogeneous space has a homogeneous-space model over the same field
Exact issue scan · physical PDF p. 183 / printed p. 375 · unnumbered theorem, clause (ii)
Given a pre-homogeneous space for a pre-group over , Weil asserts the existence over of a birationally equivalent homogeneous space for the regularized group , unique up to isomorphism. This is the precise existence and descent statement Rosenlicht invokes.
Theorem (i)No issue found
The acting pre-group is regularized over the ground field
Exact issue scan · physical PDF p. 183 / printed p. 375 · unnumbered theorem, clause (i)
Clause (i) constructs a group over birationally equivalent to and determines it uniquely up to isomorphism. When Rosenlicht begins with an actual algebraic group, uniqueness identifies the regularized object with that group, so clause (ii) has exactly the acting group needed in the consumer.
Rosenlicht dependency matchDependency match verified
The 1957 fixed-point proof uses exactly theorem (ii)
Rosenlicht exact Springer VOR · physical PDF p. 12 / printed p. 36 · fixed-point lemma proof, citation [8]
Rosenlicht obtains a prehomogeneous rational quotient and asks for a birational homogeneous-space model over . Weil's theorem (ii) supplies exactly that replacement; no stronger regularity or completeness assertion is imported from the citation.
Appendix Proposition 7No issue found
Full-fiber containment is a closed condition on the base
Exact issue scan · physical PDF pp. 188–189 / printed pp. 380–381 · Appendix Proposition 7
For a -closed subset , the set of such that is -closed. Weil reduces this to the fiber-dimension jump locus in the complete projective closure of a representative of , exactly as covered by Weil 1954, Lemma 7.
02Proofs7 reported findingsNo proof issues found
The selected existence and uniqueness proof is complete. The article constructs affine chunks, glues their translates into regular objects, descends the construction to the ground field, and uses two separately represented article branches: Weil 1954, Lemma 7, for closed-fiber openness arguments and Lang–Weil 1954 for the finite-field point. No unresolved non-book proof parent remains.
Chunk constructionNo issue found in the reviewed proof
Proposition 4 gives an affine model carrying the rational law
Exact issue scan · physical PDF p. 173 / printed p. 365 · Proposition 4
After shrinking around a generic point, Weil obtains an affine chunk on which the product, inverse, and action identities hold wherever both sides are defined. The construction preserves the field and gives the local data used by the gluing argument.
Regular gluingNo issue found in the reviewed proof
Translates of the chunk glue to a regular group and homogeneous space
Exact issue scan · physical PDF pp. 176–177 / printed pp. 368–369 · beginning of Section III
The overlap maps are the rational translations supplied by the chunk identities and are biregular on the relevant open pieces. Their cocycle compatibility is exactly associativity, so the glued abstract variety carries everywhere-regular group operations and the glued orbit space is homogeneous.
Descent and Proposition 9No issue found in the reviewed proof
A rational separating cycle descends the glued model
Exact issue scan · physical PDF pp. 177–182 / printed pp. 369–374 · descent construction and Proposition 9
Weil encodes finitely many translated affine pieces by Chow coordinates, takes all ground-field conjugates, and uses a rational zero-cycle satisfying condition (S) to prevent unintended stabilizers. The resulting conjugation-compatible charts and transition maps descend the group and space from the auxiliary extension to .
Direct Lemma 7 branchRequired historical article branch represented
Appendix Proposition 7 is exactly supplied by Weil 1954
Exact issue scan · proof of Proposition 3 on printed pp. 363–364 ('k-open by Prop. 7' on p. 364); Appendix Proposition 7 on pp. 380–381 (explicit Lemma 7 citation on p. 381); later use on pp. 386–387
Appendix Proposition 7 explicitly invokes Lemma 7 of Weil's 1954 Mathematische Annalen paper. At the threshold , Lemma 7 says precisely that the base points whose fiber contains the whole complete fiber form a closed locus. This direct branch is represented at ordinary-audits/weil-equivalence-criteria-algebraic-geometry.json.
Finite-field branchRequired historical article branch represented
Lang–Weil supplies the required open-set point
Exact issue scan · physical PDF p. 182 / printed p. 374 · final paragraph of Proposition 9 proof
For finite , the proof needs a point of over an extension of prime degree larger than . The Lang–Weil consequence gives such a point over every sufficiently large extension, so the extra primality and size conditions can be imposed. This branch is represented at ordinary-audits/lang-weil-number-points-finite-fields.json.
UniquenessNo issue found in the reviewed proof
Everywhere-regular translations make the comparison global
Exact issue scan · physical PDF p. 183 / printed p. 375 · paragraph after the theorem
The birational comparison is biregular near one base point. Translating source and target by arbitrary group elements transports that neighborhood across every orbit point, proving the asserted uniqueness for homogeneous spaces.
Dependency closureClosed with represented article branch
The original article branch is retained rather than replaced
Provenance graph · Weil 1955 to Lang–Weil 1954; remaining citations to Weil's published books
The direct Weil 1954 Lemma 7 input and the finite-field Lang–Weil input are genuine historical article dependencies and each has its own strict ordinary node. The remaining proof-critical external inputs terminate in Weil's Foundations of Algebraic Geometry and Variétés abéliennes et courbes algébriques.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.