The Stacks project

Remark 42.19.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$. Let us show that we have a presentation

\[ \bigoplus \nolimits _{\delta (x) = k + 1}' K_1^ M(\kappa (x)) \xrightarrow {\partial } \bigoplus \nolimits _{\delta (x) = k}' K_0^ M(\kappa (x)) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \to 0 \]

Here we use the notation and conventions introduced in Remark 42.8.2 and in addition

  1. $K_1^ M(\kappa (x)) = \kappa (x)^*$ is the degree $1$ 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 differential $\partial $ is defined as follows: given an element $\xi = \sum _ x f_ x$ we denote $W_ x = \overline{x}$ the integral closed subscheme of $X$ with generic point $x$ and we set

    \[ \partial (\xi ) = \sum (W_ x \to X)_*\text{div}(f_ x) \]

    in $Z_ k(X)$ which makes sense as we have seen that the second term of the complex is equal to $Z_ k(X)$ by Remark 42.8.2.

The fact that we obtain a presentation of $\mathop{\mathrm{CH}}\nolimits _ k(X)$ follows immediately by comparing with Definition 42.19.1.

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