Lemma 43.16.1. Let X be a nonsingular variety and W, V \subset X closed subvarieties which intersect properly. Let Z be an irreducible component of V \cap W with generic point \xi . Assume that \mathcal{O}_{W, \xi } and \mathcal{O}_{V, \xi } are Cohen-Macaulay. Then
e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }}(\mathcal{O}_{V \cap W, \xi })
where V \cap W is the scheme theoretic intersection. In particular, if both V and W are Cohen-Macaulay, then V \cdot W = [V \cap W]_{\dim (V) + \dim (W) - \dim (X)}.
Proof.
Set A = \mathcal{O}_{X, \xi }, B = \mathcal{O}_{V, \xi }, and C = \mathcal{O}_{W, \xi }. By Auslander-Buchsbaum (Algebra, Proposition 10.111.1) we can find a finite free resolution F_\bullet \to B of length
\text{depth}(A) - \text{depth}(B) = \dim (A) - \dim (B) = \dim (C)
First equality as A and B are Cohen-Macaulay and the second as V and W intersect properly. Then F_\bullet \otimes _ A C is a complex of finite free modules representing B \otimes _ A^\mathbf {L} C hence has cohomology modules with support in \{ \mathfrak m_ A\} . By the Acyclicity lemma (Algebra, Lemma 10.102.8) which applies as C is Cohen-Macaulay we conclude that F_\bullet \otimes _ A C has nonzero cohomology only in degree 0. This finishes the proof.
\square
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