I have a basic question concerning dual numbers and tangent vectors.
If I have a scheme $S$ over a field $k$ and a closed $k-$rational point $s\in S$, then one knows that to give a tangent vector in $s$ corresponds to a $k-$morphism $D\rightarrow S$, sending the unique point of $D$ to $s$, where $D$ denotes the scheme of dual numbers.
My question is simply: the morphism from $D$ to $S$ which you get is not a closed immersion, isn't it? Of course, the image is a closed point, namely $s$, but I don't see if the morphism on sheaves is surjective.
Thank you!