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Positivity of Serre's Intersection Multiplicity
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Abstract
We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- maximusrufus, bocowgill, mo271 (GitHub), Add reference note: claimed proof of intersectionMultiplicity_pos (unrefereed; status unchanged) (google-deepmind/formal-conjectures PR #7215, opened 2026-10-12, open). A docstring-only PR that records Theorem 1.1 as an unrefereed claim and leaves
intersectionMultiplicity_posas research open; in a 2026-10-19 reply the PR author says they reviewed the proof in detail and found no errors, but that it should stay labelled "claimed, unrefereed".
Formalization
- unaoya (GitHub), SerrePositivity (GitHub repository, accessed 2026-10-21). An early-stage independent Lean 4 project to formalize this paper, with Theorem 1.1 stated with
sorryand the paper's Theorems 3.1 and 4.7 planned as named axioms; asorry-free core step derives $\chi>0$ only from an assumed input bundle. - mo271 (GitHub), Wikipedia/SerreMultiplicityConjectures: the positivity conjecture is true (google-deepmind/formal-conjectures issue #6911, 2026-10-07, open, label "solution found"). Proposes marking
intersectionMultiplicity_posandintersectionMultiplicity_pos_quotientas solved on the basis of this paper; the author says the comparison was prepared with AI assistance and should be checked against the Lean statements. - j2d9w5xtjn-png, feat: add Serre's multiplicity conjectures (google-deepmind/formal-conjectures PR #5382, merged 2026-09-23). This independent, AI-assisted contribution defines intersection multiplicity and states the module and prime-quotient positivity conjectures, without formalizing the manuscript's proof. [AI-generated per source]
- Formal Conjectures authors, FormalConjectures/Wikipedia/SerreMultiplicityConjectures.lean (google-deepmind/formal-conjectures, accessed 2026-10-21). Lean statements of Serre's conjectures, all with
sorryproofs; the general positivity statements that Theorem 1.1 addresses are still tagged research open, and the equicharacteristic and unramified cases are tagged solved.
Related prior literature mentioned by others
- Andrei Suslin and Vladimir Voevodsky, Singular homology of abstract algebraic varieties (Inventiones Mathematicae 123, 61–94, 1996). Ying Xie's survey (§7.7, §9.1.5) identifies its pseudo-Galois covering and orbit argument as an uncited antecedent of the manuscript's supported rational descent construction; no Suslin–Voevodsky work appears in the paper's bibliography.
- Editan, Copilot and Gemini, Θ：セールの正値性予想の構造的始原による完結：交叉多重度による代数幾何学的存在証明 (note.com, 2025-12-09). The manuscript's introductory footnote and Xie's survey mention this earlier assertion of the prime-quotient positivity conjecture as a claim, rather than an established theorem. [AI-generated per source]
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics, Section 9.1.5 "193 · Serre's positivity conjecture" (arXiv:2610.17090 [math.HO], 2026-10-12). An AI-assisted survey of the whole release whose section on this paper extracts three "instruments": supported rational descent, the rank quotient with supported asymptotic complexes, and the two Galois-norm comparisons. Its caveats say the Verdier-quotient identification (perfect modules over a connective algebra, base change, spectral sequences on idempotent summands, ultralimits) needs checking and that the argument depends on 2022–2026 results. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 193
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positivity-of-Serres-Intersection-Multiplicity-September-23-2026,
author = {{OpenAI}},
title = {{Positivity of Serre's Intersection Multiplicity}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026/paper.pdf}{OAI:Positivity-of-Serres-Intersection-Multiplicity-September-23-2026}},
year = {2026}
}Family 193: Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity $\chi^R(M,N)$ for nonzero finitely generated modules over any regular local ring, provided $M\otimes_R N$ has finite length and $\dim M+\dim N=\dim R$. This resolves the positivity conjecture, including ramified mixed characteristic.
This is the only paper in this family.