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

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

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.

Comments (2)

Comment #7197 by Anonymous on

The display equation doesn't compile on the web version, says "\nolimits is allowed only on operators''. It does seem to compile correctly in the pdf file, however.

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