The Stacks project

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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