Indeed, by part (3) and the fact that $n+m-c\geq n$, we have $\ker(\beta)\subseteq\ker(\alpha)$, while the reverse inclusion follows immediately from the factorization of $\beta$ through $\alpha$. Hence $\ker(\beta)=\ker(\alpha)$.

On the other hand, I am not sure how part (4) directly gives this under the assumption $m\geq c$, since the relevant statement in part (4) seems to require an index at least $2c$.

In the sentence

“Since $j$ is an immersion, so is $(j,j_{d-d_0})$,”

should “$j$” be “$j_{d-d_0}$” ?

“assume we have affine opens $U,V⊂S_{0}$ such that $U∩V$ is affine too.”

Should this be $U_{0},V_{0} \subset S_{0}?$ In the next sentence, $U,V \subset S$ are defined as the inverse images, and the subsequent notation also uses $U_{0},V_{0}$.

$x'$ generalizes (from) $x$

$x'$ specializes to $x$