I'm confused about the precise definition of an inner automorphism of an algebraic group. Here is what Milne says in his book on algebraic groups:


Let $k$ be a field, let $\overline{k}$ be an algebraic closure, and let $G$ be an algebraic group over $k$. Let $Z$ (or $Z(G)$) denote the center of $G$. Then an automorphism of $G$ is said to be inner if it becomes of the form inn$(g)$ over $\overline{k}$, where $g \in G(\overline{k})$. The action of $G$ on itself by inner automorphisms, $(x,y) \mapsto xyx^{-1}$, is invariant under $Z(G) \times e$ acting by translation, and so it factors through $G/Z(G)$. The inner automorphisms of $G$ are exactly the automorphisms of $G$ defined by elements of $G/Z(k)$.


I don't understand why we don't simply say that the inner automorphisms of $G$ can be identified with $G/Z$. I think Milne is identifying $G/Z(k)$ with a subset of the underlying topological space $G/Z$.



Inner automorphisms of algebraic groups
The inner automorphisms of $G$ form an abstract group, whereas $G/Z$ is an algebraic group (i.e., group scheme of finite type over the field $k$), so you can't say that one is equal to the other --- they are different types of objects. By $(G/Z)(k)$ Milne means the group of $k$-rational points of $G/Z$, which is an abstract group. Each element of $(G/Z)(k)$ defines an automorphism of $G$, and these are exactly the inner automorphisms


Here is the passage in Milne's book 3.51: Let $G$ be an algebraic group over $k$. An automorphism of $G$ is
inner if it becomes of the form $ inn(g)$ over $k^{\mathrm{a}}$. Later
(17.63), we shall see that the group $\mathcal{I}(k)$ of inner
automorphisms of $G$ is equal to $\bar{G}(k)$, where $\bar{G}$ is the quotient
of $G$ by its centre. For example, for $t\in k^{\times}$, the automorphism
$\left(
\begin{smallmatrix}
a & b\\
c & d
\end{smallmatrix}
\right) \mapsto\left(
\begin{smallmatrix}
a & tb\\
t^{-1}c & d
\end{smallmatrix}
\right) $ of $SL\_{2}$ is inner because it becomes the inner automorphism
defined by $diag(\sqrt{t},\sqrt{t}\,^{-1})$ over $k^{\mathrm{a}}$. It is also
the inner automorphism defined by the element $diag(t,1)$ of $PGL\_{2}(k)$.


