Proposition 1.8 in Fulton's book on Intersection Theory is as follows:
Proposition 1.8: Let $Y$ be a closed subscheme of a scheme $X$, and let $U$ be $X - Y$. Let $i: Y\to X$ and $j: U\to X$ be the inclusions. Then the sequence $$ A_kY\xrightarrow{i_*} A_kX \xrightarrow{j^*} A_kU \to 0$$ is exact for all $k$.
I think that exactness at the left may fail because some subvariety $W$ of $Y$ may fail to be rationally equivalent in $Y$ itself, while being so in $X$, due to the presence of $k+1$-dimensional subvariety $W'$ and an $r$ and $div(r)$ coming from $X$ but not from $Y$. However, my questions are as follows:
Is the sequence exact on the left when we replace $A$ by $Z$? More precisely, is $$ 0\to Z_kY\xrightarrow{i_*} Z_kX \xrightarrow{j^*} Z_kU \to 0$$ exact? I get the feeling this is obvious, or else I'm missing something.
What are some examples of failure of $i_*$ not being injective (in the original sequence at least, if not in the other)?
Thanks.
$Z_k Y \to Z_k X$ is injective, because both groups are freely generated, and $Z_k Y$ is generated by a subset of the generators of $Z_k X$.
Hartshorne, Algebraic geometry, Example II 6.5.2 considers $X = \operatorname{Spec} k[x,y,z] / (xy - z^2)$, the affine quadric cone in $\mathbb A^3$. If $Y = \{y=z=0\}$ is a line, $A_1 Y = \mathbb Z[Y]$ and he proves that $2[Y] = 0$ in $A_1 X$.