The Stacks project

43.1 Introduction

In this chapter we construct the intersection product on the Chow groups modulo rational equivalence on a nonsingular projective variety over an algebraically closed field. Our tools are Serre's Tor formula (see [Chapter V, Serre_algebre_locale]), reduction to the diagonal, and the moving lemma.

We first recall cycles and how to construct proper pushforward and flat pullback of cycles. Next, we introduce rational equivalence of cycles which gives us the Chow groups $\mathop{\mathrm{CH}}\nolimits _*(X)$. Proper pushforward and flat pullback factor through rational equivalence to give operations on Chow groups. This takes up Sections 43.3, 43.4, 43.5, 43.6, 43.7, 43.8, 43.9, 43.10, and 43.11. For proofs we mostly refer to the chapter on Chow homology where these results have been proven in the setting of schemes locally of finite type over a universally catenary Noetherian base, see Chow Homology, Section 42.7 ff.

Since we work on a nonsingular projective $X$ any irreducible component of the intersection $V \cap W$ of two irreducible closed subvarieties has dimension at least $\dim (V) + \dim (W) - \dim (X)$. We say $V$ and $W$ intersect properly if equality holds for every irreducible component $Z$. In this case we define the intersection multiplicity $e_ Z = e(X, V \cdot W, Z)$ by the formula

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

We need to do a little bit of commutative algebra to show that these intersection multiplicities agree with intuition in simple cases, namely, that sometimes

\[ e_ Z = \text{length}_{\mathcal{O}_{X, Z}} \mathcal{O}_{V \cap W, Z}, \]

in other words, only $\text{Tor}_0$ contributes. This happens when $V$ and $W$ are Cohen-Macaulay in the generic point of $Z$ or when $W$ is cut out by a regular sequence in $\mathcal{O}_{X, Z}$ which also defines a regular sequence on $\mathcal{O}_{V, Z}$. However, Example 43.14.4 shows that higher tors are necessary in general. Moreover, there is a relationship with the Samuel multiplicity. These matters are discussed in Sections 43.13, 43.14, 43.15, 43.16, and 43.17.

Reduction to the diagonal is the statement that we can intersect $V$ and $W$ by intersecting $V \times W$ with the diagonal in $X \times X$. This innocuous statement, which is clear on the level of scheme theoretic intersections, reduces an intersection of a general pair of closed subschemes, to the case where one of the two is locally cut out by a regular sequence. We use this, following Serre, to obtain positivity of intersection multiplicities. Moreover, reduction to the diagonal leads to additivity of intersection multiplicities, associativity, and a projection formula. This can be found in Sections 43.18, 43.19, 43.20, 43.21, and 43.22.

Finally, we come to the moving lemmas and applications. There are two parts to the moving lemma. The first is that given closed subvarieties

\[ Z \subset X \subset \mathbf{P}^ N \]

with $X$ nonsingular, we can find a subvariety $C \subset \mathbf{P}^ N$ intersecting $X$ properly such that

\[ C \cdot X = [Z] + \sum m_ j [Z_ j] \]

and such that the other components $Z_ j$ are “more general” than $Z$. The second part is that one can move $C \subset \mathbf{P}^ N$ over a rational curve to a subvariety in general position with respect to any given list of subvarieties. Combined these results imply that it suffices to define the intersection product of cycles on $X$ which intersect properly which was done above. Of course this only leads to an intersection product on $\mathop{\mathrm{CH}}\nolimits _*(X)$ if one can show, as we do in the text, that these products pass through rational equivalence. This and some applications are discussed in Sections 43.23, 43.24, 43.25, 43.26, 43.27, and 43.28.

Comments (5)

Comment #1378 by jojo on

At the end of the second paragraph there seems to be an extra "ff".

Comment #1380 by jojo on

Ah ! The stacks project really is a amazing source of information :) Thanks a lot.

Comment #2339 by oliver on

Why is the ground field in this chapter assumed to be algebraically closed?

Comment #2341 by on

@#2339: Mainly because in the proof of the moving lemma it is nice to have many rational points available. Burt Totaro pointed out to me that it is not necessary to do so. In the moving lemma one chooses always a rational point in a projective space or a grassmanian. Hence only finite fields cause trouble. In this case you can solve the problem after an extension of degree and after an extension of degree . Then you can combine the cycles you get to win. But you do have to be very careful with the order in which you do things so you don't get stuck so it isn't completely "free" to make the corresponding changes. Thanks for the comment!

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