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 $X\times Y$?



First infinitesimal neighborhood of diagonal on abelian variety
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 $Y \times X$.


