Serre's Intersection Multiplicity Conjecture

Serre introduced a homological definition of intersection multiplicity using Tor functors, providing a purely algebraic approach to intersection theory in algebraic geometry.

Definition

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $n$ , and let $\mathfrak{p}, \mathfrak{q}$ be prime ideals with $\operatorname{ht}(\mathfrak{p}) + \operatorname{ht}(\mathfrak{q}) = n$ (i.e., $V(\mathfrak{p})$ and $V(\mathfrak{q})$ meet in expected codimension). Serre's intersection multiplicity is $\chi(R/\mathfrak{p}, R/\mathfrak{q}) = \sum_{i=0}^{n} (-1)^i \ell_R(\operatorname{Tor}_i^R(R/\mathfrak{p}, R/\mathfrak{q}))$ where $\ell_R$ denotes length as an $R$ -module.

This alternating sum generalizes the classical intersection multiplicity from algebraic geometry and reduces to the expected geometric notion in the smooth case.

Serre's Conjectures

Theorem8.9Serre's Multiplicity Conjectures

Let $R$ be a regular local ring, $M$ and $N$ finitely generated $R$ -modules with $\ell(M \otimes_R N) < \infty$ . Then:

Dimension inequality: $\dim M + \dim N \leq \dim R$
Vanishing: If $\dim M + \dim N < \dim R$ , then $\chi(M, N) = 0$
Nonnegativity: $\chi(M, N) \geq 0$
Positivity: If $\dim M + \dim N = \dim R$ , then $\chi(M, N) > 0$

The dimension inequality (1) was proved by Serre. The vanishing conjecture (2) was proved by Roberts (1985) and independently by Gillet-Soule using $K$ -theory. The nonnegativity (3) is known in equal characteristic (Gabber, 1995) using de Jong's alterations. The positivity (4) remains open in mixed characteristic.

ExampleIntersection multiplicity in practice

In $R = k[[x, y]]$ , let $\mathfrak{p} = (y - x^2)$ and $\mathfrak{q} = (y)$ . Then $R/\mathfrak{p} \otimes R/\mathfrak{q} \cong k[[x]]/(x^2)$ , which has length $2$ . Higher Tors vanish since $R/\mathfrak{q}$ has projective dimension $1$ in the regular ring $R$ . So $\chi(R/\mathfrak{p}, R/\mathfrak{q}) = 2$ , correctly reflecting that the parabola $y = x^2$ and the line $y = 0$ meet with multiplicity $2$ at the origin.

RemarkComparison with geometric intersection theory

For smooth algebraic varieties over a field, Serre's definition agrees with the classical intersection product in the Chow ring. The advantage of the homological approach is that it works in complete generality (any regular local ring, including mixed characteristic) and does not require moving lemmas or transversality arguments.