The Stacks project

Remark 42.27.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$. The complex in Remark 42.27.2 and the presentation of $\mathop{\mathrm{CH}}\nolimits _ k(X)$ in Remark 42.19.2 suggests that we can define a first higher Chow group

\[ \mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1) = H_1(\text{the complex of Remark 0GU9}) \]

We use the supscript ${}^ M$ to distinguish our notation from the higher chow groups defined in the literature, e.g., in the papers by Spencer Bloch ([Bloch] and [Bloch-moving]). Let $U \subset X$ be open with complement $Y \subset X$ (viewed as reduced closed subscheme). Then we find a split short exact sequence

\[ 0 \to \bigoplus \nolimits _{y \in Y, \delta (y) = k + i}' K_ i^ M(\kappa (y)) \to \bigoplus \nolimits _{x \in X, \delta (x) = k + i}' K_ i^ M(\kappa (x)) \to \bigoplus \nolimits _{u \in U, \delta (u) = k + i}' K_ i^ M(\kappa (u)) \to 0 \]

for $i = 2, 1, 0$ compatible with the boundary maps in the complexes of Remark 42.27.2. Applying the snake lemma (see Homology, Lemma 12.13.6) we obtain a six term exact sequence

\[ \mathop{\mathrm{CH}}\nolimits ^ M_ k(Y, 1) \to \mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1) \to \mathop{\mathrm{CH}}\nolimits ^ M_ k(U, 1) \to \mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(U) \to 0 \]

extending the canonical exact sequence of Lemma 42.19.3. With some work, one may also define flat pullback and proper pushforward for the first higher chow group $\mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1)$. We will return to this later (insert future reference here).

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