First infinitesimal neighborhood of diagonal on abelian variety

Question

Consider an abelian variety $X$ over a field and denote by $Z$ the first infinitesimal neighborhood of the diagonal coming with natural projections

$p_1: Z \rightarrow X$, $p_2: Z \rightarrow X$.

Let $Y$ be the first infinitesimal neighborhood of zero in $X$.

Then why has one an isomorphism

$X\times Y \rightarrow Z$, such that $p_1$ corresponds to the first projection of $X\times Y$ and $p_2$ corresponds to the addition on $X\times Y$?

Answer 1

Consider an automorphism $X\times X \to X\times X$, $(x_1,x_2) \mapsto (x_1-x_2,x_2)$. It identifies the diagonal with $0 \times X$, and hence the infinitesimal neighborhood of the diagonal with the infinitesimal neighborhood of $0\times X$ which is $Y \times X$.