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.

Manuscript

Positivity of Serre's Intersection Multiplicity

September 23, 2026 · source · cite as \cite{OAI:Positivity-of-Serres-Intersection-Multiplicity-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.

Result 193 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.