Section 42.8 (02QQ): Cycles—The Stacks project

Since we are not assuming our schemes are quasi-compact we have to be a little careful when defining cycles. We have to allow infinite sums because a rational function may have infinitely many poles for example. In any case, if

$X$ is quasi-compact then a cycle is a finite sum as usual.

Definition 42.8.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . Let $k \in \mathbf{Z}$ .

A cycle on $X$ is a formal sum

$\alpha = \sum n_ Z [Z]$

where the sum is over integral closed subschemes $Z \subset X$ , each $n_ Z \in \mathbf{Z}$ , and the collection $\{ Z; n_ Z \not= 0\}$ is locally finite (Topology, Definition 5.28.4).
A $k$ -cycle on $X$ is a cycle

$\alpha = \sum n_ Z [Z]$

where $n_ Z \not= 0 \Rightarrow \dim _\delta (Z) = k$ .
The abelian group of all $k$ -cycles on $X$ is denoted $Z_ k(X)$ .

In other words, a

$k$ -cycle on

$X$ is a locally finite formal

$\mathbf{Z}$ -linear combination of integral closed subschemes of

$\delta$ -dimension

$k$ . Addition of

$k$ -cycles

$\alpha = \sum n_ Z[Z]$ and

$\beta = \sum m_ Z[Z]$ is given by

Remark 42.8.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . Let $k \in \mathbf{Z}$ . Then we can write

$Z_ k(X) = \bigoplus \nolimits _{\delta (x) = k}' K_0^ M(\kappa (x)) \quad \subset \quad \bigoplus \nolimits _{\delta (x) = k} K_0^ M(\kappa (x))$

with the following notation and conventions:

$K_0^ M(\kappa (x)) = \mathbf{Z}$ is the degree $0$ part of the Milnor K-theory of the residue field $\kappa (x)$ of the point $x \in X$ (see Remark 42.6.4), and
the direct sum on the right is over all points $x \in X$ with $\delta (x) = k$ ,
the notation $\bigoplus '_ x$ signifies that we consider the subgroup consisting of locally finite elements; namely, elements $\sum _ x n_ x$ such that for every quasi-compact open $U \subset X$ the set of $x \in U$ with $n_ x \not= 0$ is finite.

Definition 42.8.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . The support of a cycle $\alpha = \sum n_ Z [Z]$ on $X$ is

$\text{Supp}(\alpha ) = \bigcup \nolimits _{n_ Z \not= 0} Z \subset X$

Since the collection

$\{ Z; n_ Z \not= 0\}$ is locally finite we see that

$\text{Supp}(\alpha )$ is a closed subset of

$X$ . If

$\alpha$ is a

$k$ -cycle, then every irreducible component

$Z$ of

$\text{Supp}(\alpha )$ has

$\delta$ -dimension

$k$ .

Definition 42.8.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . A cycle $\alpha$ on $X$ is effective if it can be written as $\alpha =\sum n_ Z [Z]$ with $n_ Z \geq 0$ for all $Z$ .

The set of all effective cycles is a monoid because the sum of two effective cycles is effective, but it is not a group (unless

$X = \emptyset$ ).

The Stacks project

42.8 Cycles

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