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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