43.19 Reduction to the diagonal
Let $X$ be a nonsingular variety. We will use $\Delta $ to denote either the diagonal morphism $\Delta : X \to X \times X$ or the image $\Delta \subset X \times X$. Reduction to the diagonal is the statement that intersection products on $X$ can be reduced to intersection products of exterior products with the diagonal on $X \times X$.
Lemma 43.19.1. Let $X$ be a nonsingular variety.
If $\mathcal{F}$ and $\mathcal{G}$ are coherent $\mathcal{O}_ X$-modules, then there are canonical isomorphisms
\[ \text{Tor}_ i^{\mathcal{O}_{X \times X}}(\mathcal{O}_\Delta , \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{G}) = \Delta _*\text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \]
If $K$ and $M$ are in $D_\mathit{QCoh}(\mathcal{O}_ X)$, then there is a canonical isomorphism
\[ L\Delta ^* \left( L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_2^*M \right) = K \otimes _{\mathcal{O}_ X}^\mathbf {L} M \]
in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and a canonical isomorphism
\[ \mathcal{O}_\Delta \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times X}}^\mathbf {L} L\text{pr}_2^*M = \Delta _*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]
in $D_\mathit{QCoh}(X \times X)$.
Proof.
Let us explain how to prove (1) in a more elementary way and part (2) using more general theory. As (2) implies (1) the reader can skip the proof of (1).
Proof of (1). Choose an affine open $\mathop{\mathrm{Spec}}(A) \subset X$. Then $A$ is a Noetherian $\mathbf{C}$-algebra 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 $\text{Tor}_ i$ over $\mathcal{O}_ X$ by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_ X$-module. For the $\text{Tor}_ i$ over $\mathcal{O}_{X \times X}$ we compute the tor of $A$ and $M \otimes _\mathbf {C} N$ over $A \otimes _\mathbf {C} A$ and then take the associated $\mathcal{O}_{X \times X}$-module. Hence on this affine patch we have to prove that
\[ \text{Tor}_ i^{A \otimes _\mathbf {C} A}(A, M \otimes _\mathbf {C} N) = \text{Tor}_ i^ A(M, N) \]
To see this choose resolutions $F_\bullet \to M$ and $G_\bullet \to M$ by finite free $A$-modules (Algebra, Lemma 10.71.1). Note that $\text{Tot}(F_\bullet \otimes _\mathbf {C} G_\bullet )$ is a resolution of $M \otimes _\mathbf {C} N$ as it computes Tor groups over $\mathbf{C}$! Of course the terms of $F_\bullet \otimes _\mathbf {C} G_\bullet $ are finite free $A \otimes _\mathbf {C} A$-modules. Hence the left hand side of the displayed equation is the module
\[ H_ i(A \otimes _{A \otimes _\mathbf {C} A} \text{Tot}(F_\bullet \otimes _\mathbf {C} G_\bullet )) \]
and the right hand side is the module
\[ H_ i(\text{Tot}(F_\bullet \otimes _ A G_\bullet )) \]
Since $A \otimes _{A \otimes _\mathbf {C} A} (F_ p \otimes _\mathbf {C} G_ q) = F_ p \otimes _ A G_ q$ we see that these modules are equal. This defines an isomorphism over the affine open $\mathop{\mathrm{Spec}}(A) \times \mathop{\mathrm{Spec}}(A)$ (which is good enough for the application to equality of intersection numbers). We omit the proof that these isomorphisms glue.
Proof of (2). The second statement follows from the first by the projection formula as stated in Derived Categories of Schemes, Lemma 36.22.1. To see the first, represent $K$ and $M$ by K-flat complexes $\mathcal{K}^\bullet $ and $\mathcal{M}^\bullet $. Since pullback and tensor product preserve K-flat complexes (Cohomology, Lemmas 20.26.5 and 20.26.8) we see that it suffices to show
\[ \Delta ^*\text{Tot}( \text{pr}_1^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}^\bullet ) = \text{Tot}( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{M}^\bullet ) \]
Thus it suffices to see that there are canonical isomorphisms
\[ \Delta ^*(\text{pr}_1^*\mathcal{K} \otimes _{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}) \longrightarrow \mathcal{K} \otimes _{\mathcal{O}_ X} \mathcal{M} \]
whenever $\mathcal{K}$ and $\mathcal{M}$ are $\mathcal{O}_ X$-modules (not necessarily quasi-coherent or flat). We omit the details.
$\square$
Lemma 43.19.2. Let $X$ be a nonsingular variety. Let $\alpha $, resp. $\beta $ be an $r$-cycle, resp. $s$-cycle on $X$. Assume $\alpha $ and $\beta $ intersect properly. Then
$\alpha \times \beta $ and $[\Delta ]$ intersect properly
we have $\Delta _*(\alpha \cdot \beta ) = [\Delta ] \cdot \alpha \times \beta $ as cycles on $X \times X$,
if $X$ is proper, then $\text{pr}_{1, *}([\Delta ] \cdot \alpha \times \beta ) = \alpha \cdot \beta $, where $pr_1 : X\times X \to X$ is the projection.
Proof.
By linearity it suffices to prove this when $\alpha = [V]$ and $\beta = [W]$ for some closed subvarieties $V \subset X$ and $W \subset Y$ which intersect properly. Recall that $V \times W$ is a closed subvariety of dimension $r + s$. Observe that scheme theoretically we have $V \cap W = \Delta ^{-1}(V \times W)$ as well as $\Delta (V \cap W) = \Delta \cap V \times W$. This proves (1).
Proof of (2). Let $Z \subset V \cap W$ be an irreducible component with generic point $\xi $. We have to show that the coefficient of $Z$ in $\alpha \cdot \beta $ is the same as the coefficient of $\Delta (Z)$ in $[\Delta ] \cdot \alpha \times \beta $. The first is given by the integer
\[ \sum (-1)^ i \text{length}_{\mathcal{O}_{X, \xi }} \text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, \mathcal{O}_ W)_\xi \]
and the second by the integer
\[ \sum (-1)^ i \text{length}_{\mathcal{O}_{X \times Y, \Delta (\xi )}} \text{Tor}_ i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta , \mathcal{O}_{V \times W})_{\Delta (\xi )} \]
However, by Lemma 43.19.1 we have
\[ \text{Tor}_ i^{\mathcal{O}_ X}(\mathcal{O}_ V, \mathcal{O}_ W)_\xi \cong \text{Tor}_ i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta , \mathcal{O}_{V \times W})_{\Delta (\xi )} \]
as $\mathcal{O}_{X \times X, \Delta (\xi )}$-modules. Thus equality of lengths (by Algebra, Lemma 10.52.5 to be precise).
Part (2) implies (3) because $\text{pr}_{1, *} \circ \Delta _* = \text{id}$ by Lemma 43.6.2.
$\square$
reference
Proposition 43.19.3. Let $X$ be a nonsingular variety. Let $V \subset X$ and $W \subset Y$ be closed subvarieties which intersect properly. Let $Z \subset V \cap W$ be an irreducible component. Then $e(X, V \cdot W, Z) > 0$.
Proof.
By Lemma 43.19.2 we have
\[ e(X, V \cdot W, Z) = e(X \times X, \Delta \cdot V \times W, \Delta (Z)) \]
Since $\Delta : X \to X \times X$ is a regular immersion (see Lemma 43.13.3), we see that $e(X \times X, \Delta \cdot V \times W, \Delta (Z))$ is a positive integer by Lemma 43.16.3.
$\square$
The following is a key lemma in the development of the theory as is done in this chapter. Essentially, this lemma tells us that the intersection numbers have a suitable additivity property.
reference
Lemma 43.19.4. Let $X$ be a nonsingular variety. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ with $\dim (\text{Supp}(\mathcal{F})) \leq r$, $\dim (\text{Supp}(\mathcal{G})) \leq s$, and $\dim (\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G}) ) \leq r + s - \dim X$. In this case $[\mathcal{F}]_ r$ and $[\mathcal{G}]_ s$ intersect properly and
\[ [\mathcal{F}]_ r \cdot [\mathcal{G}]_ s = \sum (-1)^ p [\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})]_{r + s - \dim (X)}. \]
Proof.
The statement that $[\mathcal{F}]_ r$ and $[\mathcal{G}]_ s$ intersect properly is immediate. Since we are proving an equality of cycles we may work locally on $X$. (Observe that the formation of the intersection product of cycles, the formation of $\text{Tor}$-sheaves, and forming the cycle associated to a coherent sheaf, each commute with restriction to open subschemes.) Thus we may and do assume that $X$ is affine.
Denote
\[ RHS(\mathcal{F}, \mathcal{G}) = [\mathcal{F}]_ r \cdot [\mathcal{G}]_ s \quad \text{and}\quad LHS(\mathcal{F}, \mathcal{G}) = \sum (-1)^ p [\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})]_{r + s - \dim (X)} \]
Consider a short exact sequence
\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]
of coherent sheaves on $X$ with $\text{Supp}(\mathcal{F}_ i) \subset \text{Supp}(\mathcal{F})$, then both $LHS(\mathcal{F}_ i, \mathcal{G})$ and $RHS(\mathcal{F}_ i, \mathcal{G})$ are defined for $i = 1, 2, 3$ and we have
\[ RHS(\mathcal{F}_2, \mathcal{G}) = RHS(\mathcal{F}_1, \mathcal{G}) + RHS(\mathcal{F}_3, \mathcal{G}) \]
and similarly for LHS. Namely, the support condition guarantees that everything is defined, the short exact sequence and additivity of lengths gives
\[ [\mathcal{F}_2]_ r = [\mathcal{F}_1]_ r + [\mathcal{F}_3]_ r \]
(Chow Homology, Lemma 42.10.4) which implies additivity for RHS. The long exact sequence of $\text{Tor}$s
\[ \ldots \to \text{Tor}_1(\mathcal{F}_3, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_1, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_2, \mathcal{G}) \to \text{Tor}_0(\mathcal{F}_3, \mathcal{G}) \to 0 \]
and additivity of lengths as before implies additivity for LHS.
By Algebra, Lemma 10.62.1 and the fact that $X$ is affine, we can find a filtration of $\mathcal{F}$ whose graded pieces are structure sheaves of closed subvarieties of $\text{Supp}(\mathcal{F})$. The additivity shown in the previous paragraph, implies that it suffices to prove $LHS = RHS$ with $\mathcal{F}$ replaced by $\mathcal{O}_ V$ where $V \subset \text{Supp}(\mathcal{F})$. By symmetry we can do the same for $\mathcal{G}$. This reduces us to proving that
\[ LHS(\mathcal{O}_ V, \mathcal{O}_ W) = RHS(\mathcal{O}_ V, \mathcal{O}_ W) \]
where $W \subset \text{Supp}(\mathcal{G})$ is a closed subvariety. If $\dim (V) = r$ and $\dim (W) = s$, then this equality is the definition of $V \cdot W$. On the other hand, if $\dim (V) < r$ or $\dim (W) < s$, i.e., $[V]_ r = 0$ or $[W]_ s = 0$, then we have to prove that $RHS(\mathcal{O}_ V, \mathcal{O}_ W) = 0$ 1.
Let $Z \subset V \cap W$ be an irreducible component of dimension $r + s - \dim (X)$. This is the maximal dimension of a component and it suffices to show that the coefficient of $Z$ in $RHS$ is zero. Let $\xi \in Z$ be the generic point. Write $A = \mathcal{O}_{X, \xi }$, $B = \mathcal{O}_{X \times X, \Delta (\xi )}$, and $C = \mathcal{O}_{V \times W, \Delta (\xi )}$. By Lemma 43.19.1 we have
\[ \text{coeff of }Z\text{ in } RHS(\mathcal{O}_ V, \mathcal{O}_ W) = \sum (-1)^ i \text{length}_ B \text{Tor}_ i^ B(A, C) \]
Since $\dim (V) < r$ or $\dim (W) < s$ we have $\dim (V \times W) < r + s$ which implies $\dim (C) < \dim (X)$ (small detail omitted). Moreover, the kernel $I$ of $B \to A$ is generated by a regular sequence of length $\dim (X)$ (Lemma 43.13.3). Hence vanishing by Lemma 43.16.2 because the Hilbert function of $C$ with respect to $I$ has degree $\dim (C) < n$ by Algebra, Proposition 10.60.9.
$\square$
The arguments that prove Lemma 43.19.4 and Proposition 43.19.3 can be leveraged (as is done in Serre's text) to show that (1) and (2) are true if $A$ contains a field. Currently, conjecture (1) is known in general and it is known that $\chi (M, N) \geq 0$ in general (Gabber). Positivity is, as far as we know, still an open problem.
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