OpenAI research collection · 1 manuscript · Source checked 2026-10-19
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.
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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.
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