OpenAI · 193-01 · 2026-09-23 · Algebra

Positivity of Serre's Intersection Multiplicity

preprints/Positivity-of-Serres-Intersection-Multiplicity-September-23-2026

Family 193. Serre's intersection-multiplicity conjecture.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves strict positivity of Serre's intersection multiplicity $χ^{R} (M, N)$ 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.

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.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.