Published paper
Abstract
The paper gives the Orlik–Solomon presentation of the cohomology algebra of a complex hyperplane-arrangement complement and the Möbius-function formula for its Poincaré polynomial. That formula is the finite-arrangement Betti-number input used in Lofano and Paolini's minimality argument.
Role in dependence graphs
Proof-critical source
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements
This paper is included only for the following marked statement:
- Theorem 5.2 and formula (1.4) · journal pp. 169 and 183–185Identifies hyperplane-complement cohomology with the Orlik–Solomon algebra and its Möbius Poincaré polynomial.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact journal scan · Inventiones Mathematicae 56 (1980), 167–189
Peter Orlik and Louis Solomon. Combinatorics and Topology of Complements of Hyperplanes. Inventiones Mathematicae 56 (1980), 167–189.
Open audited source ↗01Statements3 reported findingsCorrect
The algebra presentation, Möbius-basis theorem, Poincaré polynomial, cohomology comparison, and monodromy applications are correct in the complete version of record. No false or materially unsupported statement was found.
Theorems 2.6 and 3.5Correct
The combinatorial algebra has the stated basis and Hilbert series
Journal pages 171–179 · Theorems 2.6 and 3.5
The circuit-boundary ideal reduces every dependent monomial, and the resulting independent monomials indexed by the intersection lattice have the Möbius-number count claimed in formula (1.4). The degree signs agree with the characteristic-polynomial convention used later.
Theorem 5.2Correct
The Orlik–Solomon algebra equals complement cohomology
Journal pages 183–185 · Theorem 5.2 and proof
Logarithmic one-forms satisfy the circuit relations, so the combinatorial algebra maps to de Rham cohomology. Brieskorn's comparison on every intersection-flat summand gives injectivity and surjectivity. Consequently the complement Poincaré polynomial is the Möbius polynomial in formula (1.4), the exact fact used in the recursive audit chain.
Exact journal scan ↗Sections 6–7Correct
The local-system and monodromy consequences respect their stated scope
Journal pages 185–189 · Sections 6–7
The final applications specialize the established algebra and group action rather than adding an unproved general equivalence. The eigenspace decompositions and fixed-point traces are finite-dimensional and are used only under the arrangement hypotheses stated at the start of each section.
02Proofs2 reported findingsCorrect
The complete proof chain was checked. The deletion–restriction induction, Möbius inversion, decomposition by intersection flats, and de Rham comparison fit together without a missing case. The imported Brieskorn result is a published chapter and is treated as a terminal source.
Sections 2–4Correct
Circuit reduction and deletion–restriction establish the algebraic presentation
Journal pages 169–183 · Sections 2–4
Dependent sets are reduced by circuit boundaries, while the deletion and restriction maps preserve the grading and produce the required exact dimensions. Induction on the number of hyperplanes matches those dimensions to the Möbius recursion. No circular use of the later cohomology theorem occurs.
Proof of Theorem 5.2Correct
The flatwise comparison proves the global cohomology isomorphism
Journal pages 183–185 · proof of Theorem 5.2
Both the algebra and the logarithmic-form cohomology split by the same intersection-flat grading. Brieskorn's lemma identifies each summand, and their direct sum therefore identifies the full spaces. Multiplicativity follows from the wedge-product construction, completing the ring statement rather than only the vector-space dimension count.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.