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.
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.
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 20.47.5.
\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 15.66.19).
\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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