193. Serre’s intersection-multiplicity conjecture

Serre’s intersection-multiplicity conjecture

1 paper

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.

Papers 1

Positivity of Serre's Intersection Multiplicity

OpenAI · September 23, 2026

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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@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}
}

Serre’s intersection-multiplicity conjecture

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Papers 1

Positivity of Serre's Intersection Multiplicity

Formal verification