Reference: Arithmetic Jet Spaces
Notes taken when I try to study the paper Arithmetic Jet Spaces
There are the following viewpoints for giving a derivation on a ring \(R\)
A map \(\varphi: R\to R\) such that \(\varphi(ab)=\varphi(a)b+a\varphi(b)\).
A map in \(\mathrm{Hom}_R(\Omega^1_R, R)\).
A map in \(\mathrm{Hom}_{\mathbf{Rings}}(R, R[\epsilon]/\epsilon^2)\), i.e. a map of schemes \(\mathrm{Spec}(R[\epsilon]/\epsilon^2)\to \mathrm{Spec}R\).
A vector field on \(\mathrm{Spec}R\).
One can construct a jet space for each \(n\), let \(D_n(R) = R[\epsilon]/\epsilon^{n+1}\).