The Stacks project

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

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.