Chow Groups

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Previously, we defined the intersection number of two divisors in §The Riemann–Roch Theorem for Surfaces, ⁋Definition 1. This is of course a very interesting notion, and in this post we define the Chow group in order to generalize this concept to arbitrary varieties.

Chow Groups

In §Divisors, ⁋Definition 1, we defined a (Weil) divisor as a formal sum of codimension 1 closed irreducible subvarieties, and we defined the divisor class group \(\Cl(X)\) by collecting these up to linear equivalence. Similarly, the Chow group is obtained by collecting formal sums of \(k\)-dimensional closed irreducible subvarieties up to rational equivalence.

Definition 1 An algebraic \(k\)-cycle of a variety \(X\) is a formal sum

\[Z = \sum_{i} n_i V_i\]

of \(k\)-dimensional closed irreducible subvarieties of \(X\). Here each \(V_i \subseteq X\) is a \(k\)-dimensional closed irreducible subvariety and \(n_i \in \mathbb{Z}\). We write \(Z_k(X)\) for the free abelian group generated by \(k\)-cycles.

By this definition, an algebraic \(k\)-cycle is close to homology. When there is occasion to interpret it from the cohomology point of view (via duality), we write a codimension \(k\) cycle as \(Z^k(X) = Z_{n-k}(X)\) (where \(n = \dim X\)). As mentioned above, the Chow group is obtained from these \(Z_k(X)\) by imposing a certain equivalence relation.

Definition 2 For a variety \(X\), a \((k+1)\)-dimensional closed irreducible subvariety \(Y \subseteq X\), and a rational function \(f \in \mathbb{K}(Y)^\times\) on it, we define the principal cycle \(\divisor(f) \in Z_k(X)\) by the formula

\[\divisor(f) = \sum_{V \subseteq Y, \dim V = k} v_V(f) \cdot V\]

where \(v_V(f)\) is the order of the zero or pole of \(f\) along \(V\).

Intuitively, this definition is nothing more than repeating §Divisors, ⁋Definition 3 with \(Y\) as the ambient variety, and thus it is a natural generalization of that definition. A somewhat subtle point is the issue of normality mentioned in the introduction of that post: even if \(X\) is a nice (say, normal) variety, an arbitrary subvariety of \(X\) need not inherit this property, so normalization enters the above definition a little more essentially.

That is, the order \(v_V(f)\) is defined via the normalization \(\nu: \widetilde{Y}\rightarrow Y\) of \(Y\). ([Commutative Algebra] §Regular Local Rings, ⁋Definition 9) Namely,

\[v_V(f) = \sum_{\nu(\widetilde{V})=V} [\mathbb{K}(\widetilde{V}):\mathbb{K}(V)]\cdot v_{\widetilde{V}}(\nu^\ast f)\]

where the sum is over the \(k\)-dimensional irreducible components \(\widetilde{V}\) of \(\nu^{-1}(V)\). Since \(\widetilde{Y}\) is normal, each \(v_{\widetilde{V}}\) is the valuation given by the local ring \(\mathcal{O}_{\widetilde{Y}, \eta_{\widetilde{V}}}\) at the generic point of \(\widetilde{V}\) (§Tangent Spaces and Smoothness, ⁋Definition 9), and if \(\mathcal{O}_{Y, \eta_V}\) itself is a discrete valuation ring then \(v_V(f)\) coincides with that valuation. Keeping this in mind, we make the following definition.

Definition 3 Two \(k\)-cycles \(Z_1, Z_2\) are said to be rationally equivalent if there exist \((k+1)\)-dimensional closed irreducible subvarieties \(Y_j\) of \(X\) and rational functions \(f_j \in \mathbb{K}(Y_j)^\times\) on them such that

\[Z_1 - Z_2 = \sum_j \divisor(f_j)\]

holds. We write this as \(Z_1 \sim_{\text{rat}} Z_2\).

That is, just as when defining the divisor class group, we regard two cycles as the same if they differ by a principal cycle. This equivalence relation can be thought of, in the same intuitive way explained right after §Divisors, ⁋Definition 9, as transporting the notion of homotopy into algebraic geometry.

Then the following proposition holds.

Proposition 4 Rational equivalence is an equivalence relation on \(Z_k(X)\).

The proof of this is almost a repetition of §Divisors, ⁋Proposition 8, so we omit it here. As a consequence of this proposition, we can finally make the following definition.

Definition 5 We define the \(k\)-th Chow group \(\CH_k(X)\) as the group of \(k\)-cycles modulo rational equivalence:

\[\CH_k(X) = Z_k(X) / \sim_{\text{rat}}\]

The codimension \(k\) Chow group is defined as \(\CH^k(X) = \CH_{n-k}(X)\), and as mentioned above, it is mainly used in situations where the cohomology convention is needed.

Functoriality

In algebraic topology, homology and cohomology are functorial for arbitrary continuous maps, but the Chow group is not. The Chow group has pushforward functoriality only for proper morphisms, and pullback functoriality only for flat morphisms.

First, a morphism \(f: X \rightarrow Y\) between two varieties being a proper morphism can roughly be described as the algebraic-geometric analogue of a compact map. ([Scheme Theory] §Valuation Rings, ⁋Definition 9) Something to be careful about is that compactness does not work well in algebraic geometry, so this cannot be transferred directly. The intuition is that, just as the fiber and image of a compact map do not leak off to infinity, a proper morphism also does not; what is particularly important is that only finitely many additional coordinates are needed to describe this fiber. ([Scheme Theory] §Properties of Scheme Morphisms, ⁋Example 16) On the other hand, if a subvariety \(V\subseteq X\) satisfies \(\dim f(V)=\dim V\), then \(V\) covers \(f(V)\) with finite multiplicity, and this multiplicity is the extension degree \([\mathbb{K}(V):\mathbb{K}(f(V))]\) of function fields. For convenience, writing

\[\deg(V/f(V))=\begin{cases}[\mathbb{K}(V):\mathbb{K}(f(V))]&\text{if $\dim f(V)=\dim V$,}\\ 0&\text{if $\dim f(V)<\dim V$}\end{cases}\]

the following holds.

Proposition 6 For a proper morphism \(f: X \rightarrow Y\), there exists a pushforward \(f_\ast: \CH_k(X) \rightarrow \CH_k(Y)\). In particular, for any subvariety \(V\subseteq X\),

\[f_\ast[V]=\deg(V/f(V))[f(V)]\]

holds.

That is, intuitively, if an algebraic cycle \([V]\) is mapped to \([f(V)]\) with multiplicity \(d\) via a proper morphism \(f\), then \(f_\ast[V]\) is exactly what captures this degree.

Now we examine pullback. This is closer to the cohomology convention than the homology convention, so we think in terms of the codimension \(k\) Chow group. The pullback \(f^\ast: \CH^k(Y)\rightarrow \CH^k(X)\) can be thought of intuitively as receiving a cycle on the target \(Y\) and stretching it in the fiber direction to give a cycle on the source. For this to be well defined, the dimension of the fiber over each point of \(Y\) must be constant, and moreover, as we vary the point of \(Y\) as a parameter, the structure of the fiber must not change abruptly. A flat morphism is precisely a morphism reflecting these properties, and in this case we obtain the following proposition.

Proposition 7 For a flat morphism \(f: X \rightarrow Y\), there exists a pullback \(f^\ast: \CH^k(Y) \rightarrow \CH^k(X)\). For a subvariety \(V \subseteq Y\), we have \(f^\ast[V] = [f^{-1}(V)]\).

Here \(f^{-1}(V)\) should be read not as the set-theoretic inverse image but as the subscheme cut out by the ideal \(\mathcal{I}\) generated by the ideal of \(V\) on \(X\), and \([f^{-1}(V)]\) is the cycle

\[[f^{-1}(V)] = \sum_i \length(\mathcal{O}_{X, \xi_i}/\mathcal{I}_{\xi_i})\cdot W_i\]

with multiplicities attached to its irreducible components \(W_i\). Here \(\xi_i\) is the generic point of \(W_i\) (§Tangent Spaces and Smoothness, ⁋Definition 9), and \(\length\) is the length of the local ring as a module over itself. ([Commutative Algebra] §The Jordan-Hölder Theorem, ⁋Definition 2) Since \(W_i\) is an irreducible component of the zero set of \(\mathcal{I}\), the ideal \(\mathcal{I}_{\xi_i}\) is primary for the maximal ideal of \(\mathcal{O}_{X,\xi_i}\), and hence this length is finite. Without this convention, one cannot capture the case where \(f^{-1}(V)\) is non-reduced. For example, for the \(f\) in Example 10 and the point \(p = [0:1]\), the set-theoretic inverse image \(f^{-1}(p)\) consists of the single point \(q = [0:1]\), but we must have \(f^\ast[p] = d\cdot[q]\).

Computing Chow Groups

We have examined two kinds of functoriality so far, and using them together allows us to understand the structure of Chow groups better. For example, let \(Z \subseteq X\) be a closed subvariety and let \(U = X \setminus Z\). Then \(i: Z \hookrightarrow X\) is a closed embedding, hence a proper morphism, and so the pushforward \(i_\ast\) is defined. On the other hand, \(j: U \hookrightarrow X\) is an open embedding, hence a flat morphism, and so the pullback \(j^\ast\) is defined.

One thing to note here is that the pullback \(j^\ast\) is originally a contravariant operation defined for the cohomology convention \(\CH^k\). However, in the case of an open embedding, \(U\) has the same dimension as \(X\), so restricting a \(k\)-dimensional cycle to \(U\) is naturally defined.

Proposition 8 (Localization Exact Sequence) If \(Z \subseteq X\) is a closed subvariety and \(U = X \setminus Z\), then the following exact sequence holds:

\[\CH_k(Z) \xrightarrow{i_\ast} \CH_k(X) \xrightarrow{j^\ast} \CH_k(U) \rightarrow 0\]

where \(i: Z \hookrightarrow X\) is the closed embedding and \(j: U \hookrightarrow X\) is the open embedding.

The reason this exact sequence holds is as follows. First, for a \(k\)-dimensional closed irreducible subvariety \(V\) of \(U\), taking its closure \(\overline{V}\) in \(X\) gives \(j^\ast[\overline{V}]=[V]\), so \(j^\ast\) is surjective. More important is that \(\ker j^\ast=\im i_\ast\), which means that cycles disappearing in \(U\) must be precisely those stacked along \(Z\). At the cycle level, it follows immediately from the definition \(U=X\setminus Z\) that the support of \(\alpha\) with \(j^\ast\alpha=0\) lies in \(Z\). On the other hand, at the class level, \(j^\ast\alpha\) need only be rationally equivalent to 0 on \(U\); taking the closures in \(X\) of the subvarieties of \(U\) giving this rational equivalence, the function field does not change so the same rational functions can be used, and subtracting this from \(\alpha\) yields a cycle rationally equivalent to \(\alpha\) whose support lies in \(Z\).

The following example is the basic starting point for computing various Chow groups.

Example 9 As the most basic example,

\[\CH_k(\mathbb{A}^n)=\begin{cases}\mathbb{Z}&\text{if $k=n$}\\0&\text{otherwise}\end{cases}\]

and

\[\CH_k(\mathbb{P}^n)=\mathbb{Z}\qquad\text{for all $0\leq k\leq n$}\]

hold. This agrees with the Borel–Moore homology of Euclidean space and projective space, showing that our defined Chow group actually reflects geometric intuition well. (Proposition 12)

In general, for an \(n\)-dimensional variety \(X\), we have \(\CH_n(X) \cong \mathbb{Z}\), and its generator is the class \([X]\) of \(X\) itself. This is because the only \(n\)-dimensional closed irreducible subvariety of \(X\) is \(X\) itself by §Dimension, ⁋Proposition 9, and there exists no \((n+1)\)-dimensional subvariety to define rational equivalence. The case \(k = n\) in the computation of Example 9 corresponds to this. On the other hand, for \(k < n\), the generator of \(\CH_k(\mathbb{P}^n)\) is the class \([\ell_k]\) of a \(k\)-dimensional linear subspace \(\ell_k = \mathbb{P}^k \subseteq \mathbb{P}^n\), and any \(k\)-dimensional closed irreducible subvariety \(V \subseteq \mathbb{P}^n\) satisfies \([V] = d[\ell_k]\) for some positive integer \(d\). This integer \(d\) is the number of intersection points of \(V\) with a general \((n-k)\)-dimensional linear subspace, and is called the degree of \(V\).

Example 10 To make the above example more concrete, define a degree \(d\) morphism \(f: \mathbb{P}^1 \rightarrow \mathbb{P}^1\) by

\[f([x:y]) = [x^d:y^d]\]

This is proper, and for the coordinate \(t = x/y\) on \(\mathbb{P}^1\) we have \(f^\ast(t) = t^d\), so the field extension \(\mathbb{K}(\mathbb{P}^1) \hookrightarrow \mathbb{K}(\mathbb{P}^1)\) is given by \(t \mapsto t^d\), and the extension degree in this case is \(d\). Hence by Proposition 6,

\[f_\ast[\mathbb{P}^1] = d \cdot [\mathbb{P}^1] \in \CH_1(\mathbb{P}^1) \cong \mathbb{Z}\]

holds. That is, \(\mathbb{P}^1\) is covered \(d\) times over \(\mathbb{P}^1\), and the pushforward captures this.

The \(n\)-dimensional closed irreducible subvarieties of \(X\) are only \(X\) itself, so rational equivalence of codimension 1 cycles is the same as linear equivalence, and adding §Divisors, ⁋Proposition 14 and §Line Bundles and Vector Bundles, ⁋Proposition 19 gives the following.

Proposition 11 For a smooth variety \(X\),

\[\CH^1(X) \cong \Cl(X) \cong \Pic(X)\]

holds.

Also, in Example 9 we saw that the cases of \(\mathbb{A}^n\) and \(\mathbb{P}^n\) match classical computations, and this can be formulated rigorously as follows.

Proposition 12 For a complex variety \(X\), there exists a cycle class map

\[\cl: \CH_k(X) \rightarrow H^{\text{BM}}_{2k}(X, \mathbb{Z})\]

This is a morphism interpreting algebraic cycles topologically, and if \(X\) is smooth projective then by Poincaré duality it can be viewed as \(\cl: \CH^k(X) \rightarrow H^{2k}(X, \mathbb{Z})\).

Here \(H^{\text{BM}}\) on the right-hand side is Borel–Moore homology, which unlike singular homology allows one to view a closed oriented submanifold (in the non-compact situation) as a class in Borel–Moore homology; from this point of view we see that Borel–Moore homology is a slightly better analogue of our Chow group than singular cohomology. Also, since \(X\) is a complex variety, the dimension on the right-hand side doubles to \(2k\), which is also worth noting.

Chow Ring

We close this post by introducing the following proposition as motivation for introducing the intersection product.

Proposition 13 For a smooth variety \(X\), the group \(\CH^\ast(X) = \bigoplus_k \CH^k(X)\) forms a graded ring under the intersection product. (§Intersection Product)

This ring structure, like Proposition 12, also matches the cohomology ring structure we already knew.

Example 14 (\(\mathbb{P}^n\)) \(\CH^\ast(\mathbb{P}^n) \cong \mathbb{Z}[H] / (H^{n+1})\)

Here \(H\) is the hyperplane class. The class \(H^k\) represents a \(k\)-codimensional linear subspace.

The intersection product of Proposition 13 will be introduced rigorously in the next post.

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References

[Ful] W. Fulton, Intersection Theory, Springer, 1984.
[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.