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.
released 2026-09-23 | 4 theorems · 17 lemmas · 25 proofs · 14,887 words |
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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.