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Serre's positivity conjecture for intersection multiplicities

Let (R,m)(R,\mathfrak m) be a regular local ring of dimension dd and M,NM,N finitely generated RR-modules with ℓ(M⊗RN)<∞\ell(M\otimes_R N)<\infty. Serre defined the intersection multiplicity χR(M,N)=∑i(−1)i ℓ(ToriR(M,N))\chi^R(M,N)=\sum_{i}(-1)^i\,\ell(\mathrm{Tor}_i^R(M,N)) and conjectured that it vanishes when dim⁡M+dim⁡N<d\dim M+\dim N<d, is nonnegative, and is strictly positive when dim⁡M+dim⁡N=d\dim M+\dim N=d. Serre proved these in equicharacteristic and in the unramified case; vanishing was proved by Roberts and by Gillet-Soule and nonnegativity by Gabber, while positivity stayed open in ramified mixed characteristic, with special cases by Skalit and by KC-Soto Levins. Is χR(M,N)>0\chi^R(M,N)>0 for all nonzero M,NM,N over every regular local ring with ℓ(M⊗RN)<∞\ell(M\otimes_R N)<\infty and dim⁡M+dim⁡N=dim⁡R\dim M+\dim N=\dim R?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Commutative algebra; intersection multiplicities over regular local rings
Posed by
J.-P. Serre, Local Algebra, Chapter V (cited by the manuscript in the 2000 English translation, Springer Monographs in Mathematics)
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: if (R,m)(R,\mathfrak m) is a commutative Noetherian regular local ring of dimension dd and M,NM,N are nonzero finitely generated modules with ℓ(M⊗RN)<∞\ell(M\otimes_R N)<\infty and dim⁡M+dim⁡N=d\dim M+\dim N=d, then χR(M,N)>0\chi^R(M,N)>0. This covers ramified mixed characteristic and needs no lift to an unramified ring. Consequences recorded: the Kurano-Roberts symbolic-power containment P∩Q(r)⊆mr+1P\cap Q^{(r)}\subseteq\mathfrak m^{r+1} in ramified mixed characteristic, and a strict-transform inequality χA(A/P,A/Q)≥e(A/P)e(A/Q)\chi^A(A/P,A/Q)\ge e(A/P)e(A/Q) with an equality criterion. It does not address Serre's other questions beyond positivity (vanishing and nonnegativity were already known), and gives no explicit lower bound beyond χ≥1\chi\ge1.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against Serre's positivity conjecture as the manuscript states it (Chapter V of Local Algebra). The theorem states the full conjecture for every commutative Noetherian regular local ring, with no restriction on characteristic, ramification or liftability. The proof (normalized lengths from perfectoid methods, rational K-theory descent, and an asymptotic rank quotient) was not refereed. No Lean formalization. The manuscript itself records, in a footnote, a public post by Editan dated 9 December 2025 and credited to Editan, Copilot and Gemini that asserts the prime-quotient form of the same conjecture; the manuscript treats it as a same-scope claim, not an established theorem.

Sources

Changelog1 change

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