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.

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