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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In Positivity of Serre’s Intersection Multiplicity, 23 September 2026
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Let (R,\m) be a commutative Noetherian regular local ring of dimension d, and let M,N be nonzero finitely generated R-modules. If \length_R(M\otimes_R N)<\infty and \dim_R M+\dim_R N=d, then
In Positivity of Serre’s Intersection Multiplicity, 23 September 2026
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Let (A,\m) be a commutative Noetherian regular local ring with residue field k, and let P,Q be prime ideals such that A/(P+Q) has finite length and \dim(A/P)+\dim(A/Q)=\dim A. Write e_\m for Hilbert–Samuel multiplicity and put
where all associated graded rings use the maximal-ideal filtrations. If \dim T\leq1, then
with equality if and only if \dim T=0.
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mo271, Wikipedia/SerreMultiplicityConjectures: the positivity conjecture is true (issue 6911), 7 October 2026. Source
The release manuscript proves Serre’s positivity conjecture including ramified mixed characteristic.
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Why this passage is shown
Theorem 1.1
In Positivity of Serre’s Intersection Multiplicity, 23 September 2026
Why this passage is shown
Corollary 1.2 (Finite strict-transform intersection)
In Positivity of Serre’s Intersection Multiplicity, 23 September 2026