What is a $k$-scheme isomorphism?

Question

In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $\pi:X=Spec(k[t])\to Spec(k)$ a scheme over $k$ and $F:X\to X$ and $Spec(k)\to Spec(k)$ the morphism with identity on the topological space and $p$-th power on the structure sheaf. It says the $k$-scheme $X_p$ given by $F\circ\pi:X\to Spec(k)$ is isomorphic to $\pi:X\to Spec(k)$, but it also says the corresponding ring homomorphism is defined by $k[t]\to k[t]$ $t\mapsto t^p$, which is not an isomorphism. I'm confused about the notion "isomorphism" here. Could anyone explain it?

Answer 1

The isomorphism refers to the isomorphism of schemes $X_p$ and $X$, but this isomorphism is not given by $F'$. I believe the isomorphism is given by $at^n \mapsto a^p t^n$ for $a \in k$ extended by linearity. Since $k$ is perfect, the inclusion $k^p \subset k$ is indeed equality.