Remark: 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.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.