The Lie algebra of a group scheme

Let $k$ be a field and let $G$ be a group scheme of finite type over $k$ (Group schemes of finite type over a field) with identity point $e=e_G\in G(k)$, the image of the unit section $e:\mathrm{Spec}k\to G$ under $G(\mathrm{Spec}k)=G(k)$. Let ${\mathfrak{m}}_e=\ker (\epsilon :{\mathcal{O}}_{G,e}\to k)$ be the maximal ideal of the local ring of $G$ at $e$ (The residue field at a point of an affine scheme) and let $T_{G\mathrm{/}k,e}={\mathrm{Hom}}_k({\mathfrak{m}}_e\mathrm{/}{\mathfrak{m}}_e^2,k)$ be the tangent space of $G$ over $k$ at $e$ (Relative cotangent and tangent spaces), the dual of the cotangent space; since $e$ is a $k$-rational point, the classical description ${\mathrm{Hom}}_k({\mathfrak{m}}_e\mathrm{/}{\mathfrak{m}}_e^2,k)$ of the tangent space is the identification of Cotangent space at a rational point. The Lie algebra of $G$ is

$\mathrm{Lie}(G):=T_{G\mathrm{/}k,e},$

written $\mathfrak{g}=\mathrm{Lie}(G)$. By Tangent vectors as dual-number points the tangent space is canonically the set of $k$-morphisms $\tau :\mathrm{Spec}k[\epsilon ]\mathrm{/}({\epsilon }^2)\to G$ whose composite with $\epsilon \mapsto 0$ is $e$ (The affine scheme of dual numbers), that is,

$\mathrm{Lie}(G)\cong \ker (G(k[\epsilon ])\to G(k)),$

the kernel of the reduction map induced by $\epsilon \mapsto 0$. For a commutative $k$-algebra $R$ one writes $\mathrm{Lie}(G)(R):=\ker (G(R[\epsilon ])\to G(R))$ with $R[\epsilon ]=R{\otimes }_{k}k[\epsilon ]$, and the elements are written $e^{\epsilon X}$.

The vector-space structure on $\mathrm{Lie}(G)$ over $k$ (Vector space over a field), the naturality of these identifications, and their agreement with the cotangent description are proved in The tangent space at the identity is a vector space, and Lie is a functor ↗, which is the well-definedness statement for this definition. No affineness, reducedness, smoothness, or characteristic hypothesis is imposed, and $G$ need not be affine.