I have a question about a step in the proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
In the Theorem 1 where we have to show that an abelian variety is commutative the author says in the red tagged line that it suffice to show that
$$\dim T \le \dim A$$
where $T$ is the locus if $(x, yxy^{-1}) \in A \times A$.
Following step isn't clear:
Why completeness of $A$ implies that the point $(e,a) \in T \cap (e \times A)$ has a preimage $(e,b)$ in $A \times A$ under the map $(x,y) \mapsto (x, yxy^{-1})$?
I'm working with wiki's definition of completeness.
