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