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43.14 Intersection multiplicities using Tor formula

A basic fact we will use frequently is that given sheaves of modules $\mathcal{F}$, $\mathcal{G}$ on a ringed space $(X, \mathcal{O}_ X)$ and a point $x \in X$ we have

\[ \text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x = \text{Tor}_ p^{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x) \]

as $\mathcal{O}_{X, x}$-modules. This can be seen in several ways from our construction of derived tensor products in Cohomology, Section 20.26, for example it follows from Cohomology, Lemma 20.26.4. Moreover, if $X$ is a scheme and $\mathcal{F}$ and $\mathcal{G}$ are quasi-coherent, then the modules $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ are quasi-coherent too, see Derived Categories of Schemes, Lemma 36.3.9. More important for our purposes is the following result.

Lemma 43.14.1. Let $X$ be a locally Noetherian scheme.

  1. If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_ X$-modules, then $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is too.

  2. If $L$ and $K$ are in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, then so is $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K$.

Proof. Let us explain how to prove (1) in a more elementary way and part (2) using previously developed general theory.

Proof of (1). Since formation of $\text{Tor}$ commutes with localization we may assume $X$ is affine. Hence $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$ and $\mathcal{F}$, $\mathcal{G}$ correspond to finite $A$-modules $M$ and $N$ (Cohomology of Schemes, Lemma 30.9.1). By Derived Categories of Schemes, Lemma 36.3.9 we may compute the $\text{Tor}$'s by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_ X$-module. Since the modules $\text{Tor}_ p^ A(M, N)$ are finite by Algebra, Lemma 10.75.7 we conclude.

By Derived Categories of Schemes, Lemma 36.10.3 the assumption is equivalent to asking $L$ and $K$ to be (locally) pseudo-coherent. Then $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K$ is pseudo-coherent by Cohomology, Lemma 20.44.5. $\square$

Lemma 43.14.2. Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-module $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_ p^{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{ 0, \ldots , \dim (X)\} $.

Proof. The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma 43.14.1. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_ p$ for $p > \dim (X)$ can be seen as follows: the local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim (X)$ (Algebra, Lemma 10.116.1), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim (X)$ (Algebra, Lemma 10.110.8) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma 15.65.19). $\square$

Let $X$ be a nonsingular variety and $W, V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$. Assume $V$ and $W$ intersect properly. In this case Lemma 43.13.4 tells us all irreducible components of $V \cap W$ have dimension equal to $r + s - \dim (X)$. The sheaves $\text{Tor}_ j^{\mathcal{O}_ X}(\mathcal{O}_ W, \mathcal{O}_ V)$ are coherent, supported on $V \cap W$, and zero if $j < 0$ or $j > \dim (X)$ (Lemma 43.14.2). We define the intersection product as

\[ W \cdot V = \sum \nolimits _ i (-1)^ i [\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ W, \mathcal{O}_ V)]_{r + s - \dim (X)}. \]

We stress that this makes sense only because of our assumption that $V$ and $W$ intersect properly. This fact will necessitate a moving lemma in order to define the intersection product in general.

With this notation, the cycle $V \cdot W$ is a formal linear combination $\sum e_ Z Z$ of the irreducible components $Z$ of the intersection $V \cap W$. The integers $e_ Z$ are called the intersection multiplicities

\[ e_ Z = e(X, V \cdot W, Z) = \sum \nolimits _ i (-1)^ i \text{length}_{\mathcal{O}_{X, Z}} \text{Tor}_ i^{\mathcal{O}_{X, Z}}(\mathcal{O}_{W, Z}, \mathcal{O}_{V, Z}) \]

where $\mathcal{O}_{X, Z}$, resp. $\mathcal{O}_{W, Z}$, resp. $\mathcal{O}_{V, Z}$ denotes the local ring of $X$, resp. $W$, resp. $V$ at the generic point of $Z$. These alternating sums of lengths of $\text{Tor}$'s satisfy many good properties, as we will see later on.

In the case of transversal intersections, the intersection number is $1$.

Lemma 43.14.3. Let $X$ be a nonsingular variety. Let $V, W \subset X$ be closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ and assume that the multiplicity (in the sense of Section 43.4) of $Z$ in the closed subscheme $V \cap W$ is $1$. Then $e(X, V \cdot W, Z) = 1$ and $V$ and $W$ are smooth in a general point of $Z$.

Proof. Let $(A, \mathfrak m, \kappa ) = (\mathcal{O}_{X, \xi }, \mathfrak m_\xi , \kappa (\xi ))$ where $\xi \in Z$ is the generic point. Then $\dim (A) = \dim (X) - \dim (Z)$, see Varieties, Lemma 33.20.3. Let $I, J \subset A$ cut out the trace of $V$ and $W$ in $\mathop{\mathrm{Spec}}(A)$. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$. Then $\dim _\kappa \overline{I} \leq \dim (X) - \dim (V)$ with equality if and only if $A/I$ is regular (this follows from the lemma cited above and the definition of regular rings, see Algebra, Definition 10.60.10 and the discussion preceding it). Similarly for $\overline{J}$. If the multiplicity is $1$, then $\text{length}_ A(A/I + J) = 1$, hence $I + J = \mathfrak m$, hence $\overline{I} + \overline{J} = \mathfrak m/\mathfrak m^2$. Then we get equality everywhere (because the intersection is proper). Hence we find $f_1, \ldots , f_ a \in I$ and $g_1, \ldots g_ b \in J$ such that $\overline{f}_1, \ldots , \overline{g}_ b$ is a basis for $\mathfrak m/\mathfrak m^2$. Then $f_1, \ldots , g_ b$ is a regular system of parameters and a regular sequence (Algebra, Lemma 10.106.3). The same lemma shows $A/(f_1, \ldots , f_ a)$ is a regular local ring of dimension $\dim (X) - \dim (V)$, hence $A/(f_1, \ldots , f_ a) \to A/I$ is an isomorphism (if the kernel is nonzero, then the dimension of $A/I$ is strictly less, see Algebra, Lemmas 10.106.2 and 10.60.13). We conclude $I = (f_1, \ldots , f_ a)$ and $J = (g_1, \ldots , g_ b)$ by symmetry. Thus the Koszul complex $K_\bullet (A, f_1, \ldots , f_ a)$ on $f_1, \ldots , f_ a$ is a resolution of $A/I$, see More on Algebra, Lemma 15.30.2. Hence

\begin{align*} \text{Tor}_ p^ A(A/I, A/J) & = H_ p(K_\bullet (A, f_1, \ldots , f_ a) \otimes _ A A/J) \\ & = H_ p(K_\bullet (A/J, f_1 \bmod J, \ldots , f_ a \bmod J)) \end{align*}

Since we've seen above that $f_1 \bmod J, \ldots , f_ a \bmod J$ is a regular system of parameters in the regular local ring $A/J$ we conclude that there is only one cohomology group, namely $H_0 = A/(I + J) = \kappa $. This finishes the proof. $\square$

Example 43.14.4. In this example we show that it is necessary to use the higher tors in the formula for the intersection multiplicities above. Let $X$ be a nonsingular variety of dimension $4$. Let $p \in X$ be a closed point. Let $V, W \subset X$ be closed subvarieties in $X$. Assume that there is an isomorphism

\[ \mathcal{O}_{X, p}^\wedge \cong \mathbf{C}[[x, y, z, w]] \]

such that the ideal of $V$ is $(xz, xw, yz, yw)$ and the ideal of $W$ is $(x - z, y - w)$. Then a computation shows that

\[ \text{length}\ \mathbf{C}[[x, y, z, w]]/ (xz, xw, yz, yw, x - z, y - w) = 3 \]

On the other hand, the multiplicity $e(X, V \cdot W, p) = 2$ as can be seen from the fact that formal locally $V$ is the union of two smooth planes $x = y = 0$ and $z = w = 0$ at $p$, each of which has intersection multiplicity $1$ with the plane $x - z = y - w = 0$ (Lemma 43.14.3). To make an actual example, take a general morphism $f : \mathbf{P}^2 \to \mathbf{P}^4$ given by $5$ homogeneous polynomials of degree $> 1$. The image $V \subset \mathbf{P}^4 = X$ will have singularities of the type described above, because there will be $p_1, p_2 \in \mathbf{P}^2$ with $f(p_1) = f(p_2)$. To find $W$ take a general plane passing through such a point.

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