## 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.

 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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