This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
Previously we defined the intersection number of two divisors in §Riemann-Roch Theorem for Surfaces, ⁋Definition 1. This is of course a very interesting concept, and in this post we define the Chow group to generalize this notion to arbitrary varieties.
Chow Groups
In §Divisors, ⁋Definition 1, we defined a formal sum of codimension 1 closed irreducible subvarieties as a (Weil) divisor, and 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 of \(k\)-dimensional closed irreducible subvarieties of \(X\)
\[Z = \sum_{i} n_i V_i\]Here each \(V_i \subset X\) 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 definition, an algebraic \(k\)-cycle is close to homology. When we need to interpret this from the viewpoint of cohomology (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 by imposing a certain equivalence relation on these \(Z_k(X)\).
Definition 2 For a rational function \(f \in \mathbb{K}(Y)^\ast\) on a \((k+1)\)-dimensional closed irreducible subvariety \(Y \subset X\) of a variety \(X\), we define the principal cycle \(\divisor(f) \in Z_k(X)\) by the formula
\[\divisor(f) = \sum_{V \subset Y, \dim V = k} v_V(f) \cdot V\]Here \(v_V(f)\) is the valuation of \(f\) at \(V\).
Intuitively, this definition is merely §Divisors, ⁋Definition 3 repeated with \(Y\) as the ambient variety, and thus is a natural generalization of that definition. A somewhat subtle point is the 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 in this case normalization is somewhat more essential. Keeping this in mind, we make the following definition.
Definition 3 Two \(k\)-cycles \(Z_1, Z_2\) are rationally equivalent if there exist \((k+1)\)-dimensional closed irreducible subvarieties \(Y_j\) of \(X\) and rational functions \(f_j \in \mathbb{K}(Y_j)^\ast\) on them such that
\[Z_1 - Z_2 = \sum_j \divisor(f_j)\]We write this as \(Z_1 \sim_{\text{rat}} Z_2\).
That is, just as when defining the divisor class group, we regard two divisors as the same if they differ by a principal divisor. This equivalence relation can be thought of, in the same spirit as the intuition explained right after [Algebraic Varieties] §Divisors, ⁋Definition 9, as translating 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 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 The \(k\)-th Chow group \(\CH_k(X)\) is defined as the group of \(k\)-cycles modulo rational equivalence
\[\CH_k(X) = Z_k(X) / \sim_{\text{rat}}\]We define the codimension \(k\) Chow group as \(\CH^k(X) = \CH_{n-k}(X)\), and as mentioned above, it is mainly used when the cohomology convention is needed.
Functoriality
In algebraic topology, homology and cohomology are functorial for arbitrary continuous maps, but Chow groups are not. Chow groups have pushforward functoriality only for proper morphisms, and pullback functoriality only for flat morphisms.
First, a morphism \(f: X \to Y\) between two varieties being a proper morphism can roughly be regarded as the algebro-geometric analogue of a compact map. ([Schemes] §Valuation Rings, ⁋Definition 8) One should be somewhat careful: compactness does not work well in algebraic geometry, so we cannot simply import it directly. The intuition is that just as the fibers and image of a compact map do not leak off to infinity, the same holds for a proper morphism; in particular, what is important is that only finitely many additional coordinates are needed to describe this fiber. ([Schemes] §Properties of Scheme Morphisms, ⁋Example 15) The number of coordinates needed in this case is computed by the extension degree of function fields \([\mathbb{K}(V):\mathbb{K}(f(V))]\), which is defined when \(V\) and \(f(V)\) have the same dimension. For convenience, if we write
\[\deg(V/f(V))=\begin{cases}[\mathbb{K}(V):\mathbb{K}(f(V))]&\text{if $\dim f(V)=\dim V$,}\\ 0&\text{if $\dim f(V)<\dim V$}\end{cases}\]then the following holds.
Proposition 6 For a proper morphism \(f: X \to Y\), there exists a pushforward \(f_\ast: \CH_k(X) \to \CH_k(Y)\). In particular, for any subvariety \(V\subset X\),
\[f_\ast[V]=\deg(V/f(V))[f(V)]\]holds.
Thus, intuitively, if an algebraic cycle \([V]\) is mapped to \([f(V)]\) with overlap of degree \(d\) via a proper morphism \(f\), then \(f_\ast[V]\) captures precisely this degree.
Now we examine the pullback. Since this is closer to the cohomology convention than the homology convention, we consider the codimension \(k\) Chow group. The pullback \(f^\ast: \CH^k(Y)\rightarrow \CH^k(X)\) can be thought of intuitively as taking 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, when viewing each point of \(Y\) as a parameter, the structure of the fiber must not change abruptly as this parameter varies. A flat morphism is precisely the morphism reflecting this property, and in such cases we obtain the following proposition.
Proposition 7 For a flat morphism \(f: X \to Y\), there exists a pullback \(f^\ast: \CH^k(Y) \to \CH^k(X)\). For a subvariety \(V \subset Y\), we have \(f^\ast[V] = [f^{-1}(V)]\).
Computing Chow Groups
We have seen two kinds of functoriality so far, and using them together allows us to understand the structure of Chow groups better. For example, let \(Z \subset X\) be a closed subvariety and let \(U = X \setminus Z\). Then \(i: Z \hookrightarrow X\) is a closed embedding, hence a proper morphism, and therefore the pushforward \(i_\ast\) is defined. On the other hand, \(j: U \hookrightarrow X\) is an open embedding, hence a flat morphism, and therefore the pullback \(j^\ast\) is defined.
One point 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, since \(U\) has the same dimension as \(X\), restricting a \(k\)-dimensional cycle directly to \(U\) is naturally defined.
Proposition 8 (Localization Exact Sequence) If \(Z \subset X\) is a closed subvariety and \(U = X \setminus Z\), then the following exact sequence holds:
\[\operatorname{CH}_k(Z) \xrightarrow{i_\ast} \operatorname{CH}_k(X) \xrightarrow{j^\ast} \operatorname{CH}_k(U) \to 0\]Here \(i: Z \hookrightarrow X\) is a closed embedding and \(j: U \hookrightarrow X\) is an open embedding.
The reason this exact sequence holds is as follows. First, it is obvious that \(j^\ast\) is surjective. More importantly, \(\ker j^\ast = \im i_\ast\), meaning that cycles vanishing on \(U\) must be cycles supported along \(Z\), which is obvious since \(U\) is defined as \(X\setminus Z\).
The following example serves as a 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 homology of Euclidean space and projective space, showing that the Chow group we defined actually reflects geometric intuition well.
As a concrete illustration,
Example 10 To make the above example more intuitive, define a degree \(d\) morphism \(f: \mathbb{P}^1 \to \mathbb{P}^1\) 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 \(d\). Therefore, 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 \(d\)-fold over \(\mathbb{P}^1\), and the pushforward captures this.
By definition, the following proposition is almost obvious.
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; this can be formulated rigorously as follows.
Proposition 12 For a complex variety \(X\), the cycle class map
\[\cl: \CH_k(X) \to H^{\text{BM}}_{2k}(X, \mathbb{Z})\]exists. This is a map interpreting algebraic cycles topologically, and if \(X\) is smooth projective, it can be viewed as \(\cl: \CH^k(X) \to H^{2k}(X, \mathbb{Z})\) by Poincaré duality.
Here \(H^{\text{BM}}\) on the right-hand side is Borel-Moore homology; unlike singular homology, (in non-compact situations) a closed oriented submanifold can be viewed as a class in Borel-Moore homology. From this perspective, we can see that Borel-Moore homology is a slightly better analogue for our Chow group than singular cohomology. Also, since \(X\) is a complex variety, it is worth noting that the dimension on the right-hand side doubles to become \(2k\).
Chow Rings
We end this post by introducing the following proposition as motivation for introducing the intersection product.
Proposition 13 For a smooth variety \(X\), \(\CH^\ast(X) = \bigoplus_k \CH^k(X)\) forms a graded ring under the intersection product. (§Intersection Product)
This ring structure also matches the cohomology ring structure we already knew, just as in Proposition 12.
Example 14 ($\mathbb{P}^n$) \(\CH^\ast(\mathbb{P}^n) \cong \mathbb{Z}[H] / (H^{n+1})\)
Here \(H\) is the hyperplane class. \(H^k\) represents a \(k\)-codimensional linear subspace.
The intersection product of Proposition 13 will be introduced rigorously in the next post.
References
[Ful] W. Fulton, Intersection Theory, Springer, 1984.
[Har] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
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