Let $X$ be a (reduced, affine) scheme and consider the closed embedding $i: X \cong X\times 0 \to X\times \mathbb A^1$.
Consider the pullback map $i^*: CH^k(X\times \mathbb A^1) \to CH^k(X)$. I know that Chow groups are invariant under taking products with the affine line so this map should be an isomorphism and indeed I can show that it is a surjection since the composition $X \to X\times \mathbb A^1 \to X$ is an isomorphism. Can we show directly that $i^*$ is injective?
I know the proof that proceeds by showing that $\pi^*: CH^k(X) \to CH^k(X\times \mathbb A^1)$ is a surjection (and hence isomorphism where $\pi: X\times \mathbb A^1 \to X$ is the projection.
I am only interested in the case where $X$ is affine in which case I think the problem looks like the following in the case where $k=1$ (codimension one):
Consider an ideal $I \subset A[t]$ and form the ideal $I_0 = \{f(0) : f(t) \in I\}$. Assume that the codimension of $I$ in $A[t]$ and $I_0$ in $A$ is both one (transversality + codimension one condition). We want to prove that if $I_0$ is principal, then so is $I$.
Is this the right translation to commutative algebra?
PS: While the above is the case of most interest, I would be interested in what one can say in general about pulling back cycles under closed subschemes.