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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.

Manuscripts in this family · 1

Positivity of Serre's Intersection Multiplicity

Sep 23, 2026·paper.pdf

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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