The Stacks project

43.12 The short exact sequence for an open

Let $X$ be a variety and let $U \subset X$ be an open subvariety. Let $X \setminus U = \bigcup Z_ i$ be the decomposition into irreducible components1. Then for each $k \geq 0$ there exists a commutative diagram

\[ \xymatrix{ \bigoplus Z_ k(Z_ i) \ar[r] \ar[d] & Z_ k(X) \ar[r] \ar[d] & Z_ k(U) \ar[d] \ar[r] & 0 \\ \bigoplus \mathop{\mathrm{CH}}\nolimits _ k(Z_ i) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r] & \mathop{\mathrm{CH}}\nolimits _ k(U) \ar[r] & 0 } \]

with exact rows. Here the vertical arrows are the canonical quotient maps. The left horizontal arrows are given by proper pushforward along the closed immersions $Z_ i \to X$. The right horizontal arrows are given by flat pullback along the open immersion $j : U \to X$. Since we have seen that these maps factor through rational equivalence we obtain the commutativity of the squares. The top row is exact simply because every subvariety of $X$ is either contained in some $Z_ i$ or has irreducible intersection with $U$. The bottom row is exact because every principal divisor $\text{div}_ W(f)$ on $U$ is the restriction of a principal divisor on $X$. More precisely, if $W \subset U$ is a $(k + 1)$-dimensional closed subvariety and $f \in \mathbf{C}(W)^*$, then denote $\overline{W}$ the closure of $W$ in $X$. Then $W \subset \overline{W}$ is an open immersion, so $\mathbf{C}(W) = \mathbf{C}(\overline{W})$ and we may think of $f$ as a nonconstant rational function on $\overline{W}$. Then clearly

\[ j^*\text{div}_{\overline{W}}(f) = \text{div}_ W(f) \]

in $Z_ k(X)$. The exactness of the lower row follows easily from this. For details see Chow Homology, Lemma 42.19.3.

[1] Since in this chapter we only consider Chow groups of varieties, we are prohibited from taking $Z_ k(X \setminus U)$ and $\mathop{\mathrm{CH}}\nolimits _ k(X \setminus U)$, hence the approach using the varieties $Z_ i$.

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