The Stacks project

42.8 Cycles

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 $k \in \mathbf{Z}$.

  1. 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).

  2. A $k$-cycle on $X$ is a cycle

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

    where $n_ Z \not= 0 \Rightarrow \dim _\delta (Z) = k$.

  3. 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

\[ \alpha + \beta = \sum (n_ Z + m_ Z)[Z], \]

i.e., by adding the coefficients.

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:

  1. $K_0^ M(\kappa (x)) = \mathbf{Z}$ 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

  2. the direct sum on the right is over all points $x \in X$ with $\delta (x) = k$,

  3. 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 $).

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02QQ. Beware of the difference between the letter 'O' and the digit '0'.



View Section 42.8 as pdf