The Stacks project

43.16 Computing intersection multiplicities

In this section we discuss some cases where the intersection multiplicities can be computed by different means. Here is a first example.

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. $\square$

Lemma 43.16.2. Let $A$ be a Noetherian local ring. Let $I = (f_1, \ldots , f_ r)$ be an ideal generated by a regular sequence. Let $M$ be a finite $A$-module. Assume that $\dim (\text{Supp}(M/IM)) = 0$. Then

\[ e_ I(M, r) = \sum (-1)^ i\text{length}_ A(\text{Tor}_ i^ A(A/I, M)) \]

Here $e_ I(M, r)$ is as in Remark 43.15.6.

Proof. Since $f_1, \ldots , f_ r$ is a regular sequence the Koszul complex $K_\bullet (f_1, \ldots , f_ r)$ is a resolution of $A/I$ over $A$, see More on Algebra, Lemma 15.30.7. Thus the right hand side is equal to

\[ \sum (-1)^ i\text{length}_ A H_ i(K_\bullet (f_1, \ldots , f_ r) \otimes _ A M) \]

Now the result follows immediately from Theorem 43.15.5 if $I$ is an ideal of definition. In general, we replace $A$ by $\overline{A} = A/\text{Ann}(M)$ and $f_1, \ldots , f_ r$ by $\overline{f}_1, \ldots , \overline{f}_ r$ which is allowed because

\[ K_\bullet (f_1, \ldots , f_ r) \otimes _ A M = K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r) \otimes _{\overline{A}} M \]

Since $e_ I(M, r) = e_{\overline{I}}(M, r)$ where $\overline{I} = (\overline{f}_1, \ldots , \overline{f}_ r) \subset \overline{A}$ is an ideal of definition the result follows from Theorem 43.15.5 in this case as well. $\square$

Lemma 43.16.3. Let $X$ be a nonsingular variety. Let $W,V \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ with generic point $\xi $. Suppose the ideal of $V$ in $\mathcal{O}_{X, \xi }$ is cut out by a regular sequence $f_1, \ldots , f_ c \in \mathcal{O}_{X, \xi }$. Then $e(X, V\cdot W, Z)$ is equal to $c!$ times the leading coefficient in the Hilbert polynomial

\[ t \mapsto \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{W, \xi }/(f_1, \ldots , f_ c)^ t,\quad t \gg 0. \]

In particular, this coefficient is $> 0$.

Proof. The equality

\[ e(X, V\cdot W, Z) = e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c) \]

follows from the more general Lemma 43.16.2. To see that $e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c)$ is $> 0$ or equivalently that $e_{(f_1, \ldots , f_ c)}(\mathcal{O}_{W, \xi }, c)$ is the leading coefficient of the Hilbert polynomial it suffices to show that the dimension of $\mathcal{O}_{W, \xi }$ is $c$, because the degree of the Hilbert polynomial is equal to the dimension by Algebra, Proposition 10.60.9. Say $\dim (V) = r$, $\dim (W) = s$, and $\dim (X) = n$. Then $\dim (Z) = r + s - n$ as the intersection is proper. Thus the transcendence degree of $\kappa (\xi )$ over $\mathbf{C}$ is $r + s - n$, see Algebra, Lemma 10.116.1. We have $r + c = n$ because $V$ is cut out by a regular sequence in a neighbourhood of $\xi $, see Divisors, Lemma 31.20.8 and then Lemma 43.13.2 applies (for example). Thus

\[ \dim (\mathcal{O}_{W, \xi }) = s - (r + s - n) = s - ((n - c) + s - n) = c \]

the first equality by Algebra, Lemma 10.116.3. $\square$

Lemma 43.16.4. In Lemma 43.16.3 assume that $c = 1$, i.e., $V$ is an effective Cartier divisor. Then

\[ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1\mathcal{O}_{W, \xi }). \]

Proof. In this case the image of $f_1$ in $\mathcal{O}_{W, \xi }$ is nonzero by properness of intersection, hence a nonzerodivisor divisor. Moreover, $\mathcal{O}_{W, \xi }$ is a Noetherian local domain of dimension $1$. Thus

\[ \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1^ t\mathcal{O}_{W, \xi }) = t \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/f_1\mathcal{O}_{W, \xi }) \]

for all $t \geq 1$, see Algebra, Lemma 10.121.1. This proves the lemma. $\square$

Lemma 43.16.5. In Lemma 43.16.3 assume that the local ring $\mathcal{O}_{W, \xi }$ is Cohen-Macaulay. Then we have

\[ e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/ f_1\mathcal{O}_{W, \xi } + \ldots + f_ c\mathcal{O}_{W, \xi }). \]

Proof. This follows immediately from Lemma 43.16.1. Alternatively, we can deduce it from Lemma 43.16.3. Namely, by Algebra, Lemma 10.104.2 we see that $f_1, \ldots , f_ c$ is a regular sequence in $\mathcal{O}_{W, \xi }$. Then Algebra, Lemma 10.69.2 shows that $f_1, \ldots , f_ c$ is a quasi-regular sequence. This easily implies the length of $\mathcal{O}_{W, \xi }/(f_1, \ldots , f_ c)^ t$ is

\[ {c + t \choose c} \text{length}_{\mathcal{O}_{X, \xi }} (\mathcal{O}_{W, \xi }/ f_1\mathcal{O}_{W, \xi } + \ldots + f_ c\mathcal{O}_{W, \xi }). \]

Looking at the leading coefficient we conclude. $\square$

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