Let $X$ be a scheme over a field $k$, and let $x \in X$. One definition of the tangent space $T_x(X)$ of $X$ at $x$ is the $\kappa(x)$-vector space of morphisms $\textrm{Spec}(\kappa(x)[\epsilon]) \rightarrow X$ satisfying a certain commutative diagram, see here http://stacks.math.columbia.edu/tag/0B28.

Now, suppose $x$ is a $k$-rational point, i.e. $\kappa(x) = k$. We can think of $x$ as a morphism of schemes $x:\textrm{Spec}(k) \rightarrow X$ with image $x$. Let $\psi: \textrm{Spec}(k) \rightarrow \textrm{Spec}(k[\epsilon])$ be the morphism of schemes corresponding to evaluation at zero. Unraveling the definition of the tangent space, we see that we can describe it as the set of morphisms of $k$-schemes $h: \textrm{Spec}(k[\epsilon]) \rightarrow G$ such that $h \circ \psi = x$.

Suppose further that $X$ is a group scheme over $k$, and that $x$ is the identity section. Let's change notation to $G = X, e = x$. As $G$ is a group scheme over $k$, $G(k[\epsilon]) = \textrm{Hom}_{k-\textrm{sch}}(\textrm{Spec}(k[\epsilon],G)$ acquires the structure of a group, as does $G(k)$.

The description of the tangent space of $G$ at $e$ in the previous paragraph tells us that it is exactly the kernel of the group homomorphism

$$G(k[\epsilon]) \rightarrow G(k)$$

In particular, $T_g(G)$ acquires a group structure as a normal subgroup of $G(k[\epsilon])$! Is this group structure of any significance, or is it a useless observation? I was wondering, for example, whether this group structure would have anything to do with the vector space structure on $T_x(X)$, or with a Lie algebra structure.

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    $\begingroup$ It's the underlying additive group of the Lie algebra. $\endgroup$ – Qiaochu Yuan Jun 11 '17 at 21:27
  • $\begingroup$ That's interesting..so this kernel is an abelian group, no matter what $G(k[\epsilon])$ is. $\endgroup$ – D_S Jun 11 '17 at 21:31
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    $\begingroup$ It is a useful exercise to work out the kernel for more general Artin local rings. (It will be nilpotent.) $\endgroup$ – Moishe Kohan Jun 11 '17 at 22:37

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