## 42.42 Cycles of given codimension

In some cases there is a second grading on the abelian group of all cycles given by codimension.

Lemma 42.42.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Write $\delta = \delta _{X/S}$ as in Section 42.7. The following are equivalent

1. There exists a decomposition $X = \coprod _{n \in \mathbf{Z}} X_ n$ into open and closed subschemes such that $\delta (\xi ) = n$ whenever $\xi \in X_ n$ is a generic point of an irreducible component of $X_ n$.

2. For all $x \in X$ there exists an open neighbourhood $U \subset X$ of $x$ and an integer $n$ such that $\delta (\xi ) = n$ whenever $\xi \in U$ is a generic point of an irreducible component of $U$.

3. For all $x \in X$ there exists an integer $n_ x$ such that $\delta (\xi ) = n_ x$ for any generic point $\xi$ of an irreducible component of $X$ containing $x$.

The conditions are satisfied if $X$ is either normal or Cohen-Macaulay1.

Proof. It is clear that (1) $\Rightarrow$ (2) $\Rightarrow$ (3). Conversely, if (3) holds, then we set $X_ n = \{ x \in X \mid n_ x = n\}$ and we get a decomposition as in (1). Namely, $X_ n$ is open because given $x$ the union of the irreducible components of $X$ passing through $x$ minus the union of the irreducible components of $X$ not passing through $x$ is an open neighbourhood of $x$. If $X$ is normal, then $X$ is a disjoint union of integral schemes (Properties, Lemma 28.7.7) and hence the properties hold. If $X$ is Cohen-Macaulay, then $\delta ' : X \to \mathbf{Z}$, $x \mapsto -\dim (\mathcal{O}_{X, x})$ is a dimension function on $X$ (see Example 42.7.4). Since $\delta - \delta '$ is locally constant (Topology, Lemma 5.20.3) and since $\delta '(\xi ) = 0$ for every generic point $\xi$ of $X$ we see that (2) holds. $\square$

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ satisfying the equivalent conditions of Lemma 42.42.1. For an integral closed subscheme $Z \subset X$ we have the codimension $\text{codim}(Z, X)$ of $Z$ in $X$, see Topology, Definition 5.11.1. We define a codimension $p$-cycle to be a cycle $\alpha = \sum n_ Z[Z]$ on $X$ such that $n_ Z \not= 0 \Rightarrow \text{codim}(Z, X) = p$. The abelian group of all codimension $p$-cycles is denoted $Z^ p(X)$. Let $X = \coprod X_ n$ be the decomposition given in Lemma 42.42.1 part (1). Recalling that our cycles are defined as locally finite sums, it is clear that

$Z^ p(X) = \prod \nolimits _ n Z_{n - p}(X_ n)$

Moreover, we see that $\prod _ p Z^ p(X) = \prod _ k Z_ k(X)$. We could now define rational equivalence of codimension $p$ cycles on $X$ in exactly the same manner as before and in fact we could redevelop the whole theory from scratch for cycles of a given codimension for $X$ as in Lemma 42.42.1. However, instead we simply define the Chow group of codimension $p$-cycles as

$\mathop{\mathrm{CH}}\nolimits ^ p(X) = \prod \nolimits _ n \mathop{\mathrm{CH}}\nolimits _{n - p}(X_ n)$

As before we have $\prod _ p \mathop{\mathrm{CH}}\nolimits ^ p(X) = \prod _ k \mathop{\mathrm{CH}}\nolimits _ k(X)$. If $X$ is quasi-compact, then the product in the formula is finite (and hence is a direct sum) and we have $\bigoplus _ p \mathop{\mathrm{CH}}\nolimits ^ p(X) = \bigoplus _ k \mathop{\mathrm{CH}}\nolimits _ k(X)$. If $X$ is quasi-compact and finite dimensional, then only a finite number of these groups is nonzero.

Many of the constructions and results for Chow groups proved above have natural counterparts for the Chow groups $\mathop{\mathrm{CH}}\nolimits ^*(X)$. Each of these is shown by decomposing the relevant schemes into “equidimensional” pieces as in Lemma 42.42.1 and applying the results already proved for the factors in the product decomposition given above. Let us list some of them.

1. If $f : X \to Y$ is a flat morphism of schemes locally of finite type over $S$ and $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 then flat pullback determines a map

$f^* : \mathop{\mathrm{CH}}\nolimits ^ p(Y) \to \mathop{\mathrm{CH}}\nolimits ^ p(X)$
2. If $f : X \to Y$ is a morphism of schemes locally of finite type over $S$ and $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 let us say $f$ has codimension $r \in \mathbf{Z}$ if for all pairs of irreducible components $Z \subset X$, $W \subset Y$ with $f(Z) \subset W$ we have $\dim _\delta (W) - \dim _\delta (Z) = r$.

3. If $f : X \to Y$ is a proper morphism of schemes locally of finite type over $S$ and $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 and $f$ has codimension $r$, then proper pushforward is a map

$f_* : \mathop{\mathrm{CH}}\nolimits ^ p(X) \to \mathop{\mathrm{CH}}\nolimits ^{p + r}(Y)$
4. If $f : X \to Y$ is a morphism of schemes locally of finite type over $S$ and $X$ and $Y$ satisfy the equivalent conditions of Lemma 42.42.1 and $f$ has codimension $r$ and $c \in A^ q(X \to Y)$, then $c$ induces maps

$c \cap - : \mathop{\mathrm{CH}}\nolimits ^ p(Y) \to \mathop{\mathrm{CH}}\nolimits ^{p + q - r}(X)$
5. If $X$ is a scheme locally of finite type over $S$ satisfying the equivalent conditions of Lemma 42.42.1 and $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then

$c_1(\mathcal{L}) \cap - : \mathop{\mathrm{CH}}\nolimits ^ p(X) \to \mathop{\mathrm{CH}}\nolimits ^{p + 1}(X)$
6. If $X$ is a scheme locally of finite type over $S$ satisfying the equivalent conditions of Lemma 42.42.1 and $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module, then

$c_ i(\mathcal{E}) \cap - : \mathop{\mathrm{CH}}\nolimits ^ p(X) \to \mathop{\mathrm{CH}}\nolimits ^{p + i}(X)$

Warning: the property for a morphism to have codimension $r$ is not preserved by base change.

Remark 42.42.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ satisfying the equivalent conditions of Lemma 42.42.1. Let $X = \coprod X_ n$ be the decomposition into open and closed subschemes such that every irreducible component of $X_ n$ has $\delta$-dimension $n$. In this situation we sometimes set

$[X] = \sum \nolimits _ n [X_ n]_ n \in \mathop{\mathrm{CH}}\nolimits ^0(X)$

This class is a kind of “fundamental class” of $X$ in Chow theory.

[1] In fact, it suffices if $X$ is $(S_2)$. Compare with Local Cohomology, Lemma 51.3.2.

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