Lemma 33.16.2. The set of dotted arrows making (33.16.1.1) commute has a canonical $\kappa (x)$-vector space structure.

Proof. Set $\kappa = \kappa (x)$. Observe that we have a pushout in the category of schemes

$\mathop{\mathrm{Spec}}(\kappa [\epsilon ]) \amalg _{\mathop{\mathrm{Spec}}(\kappa )} \mathop{\mathrm{Spec}}(\kappa [\epsilon ]) = \mathop{\mathrm{Spec}}(\kappa [\epsilon _1, \epsilon _2])$

where $\kappa [\epsilon _1, \epsilon _2]$ is the $\kappa$-algebra with basis $1, \epsilon _1, \epsilon _2$ and $\epsilon _1^2 = \epsilon _1\epsilon _2 = \epsilon _2^2 = 0$. This follows immediately from the corresponding result for rings and the description of morphisms from spectra of local rings to schemes in Schemes, Lemma 26.13.1. Given two arrows $\theta _1, \theta _2 : \mathop{\mathrm{Spec}}(\kappa [\epsilon ]) \to X$ we can consider the morphism

$\theta _1 + \theta _2 : \mathop{\mathrm{Spec}}(\kappa [\epsilon ]) \to \mathop{\mathrm{Spec}}(\kappa [\epsilon _1, \epsilon _2]) \xrightarrow {\theta _1, \theta _2} X$

where the first arrow is given by $\epsilon _ i \mapsto \epsilon$. On the other hand, given $\lambda \in \kappa$ there is a self map of $\mathop{\mathrm{Spec}}(\kappa [\epsilon ])$ corresponding to the $\kappa$-algebra endomorphism of $\kappa [\epsilon ]$ which sends $\epsilon$ to $\lambda \epsilon$. Precomposing $\theta : \mathop{\mathrm{Spec}}(\kappa [\epsilon ]) \to X$ by this selfmap gives $\lambda \theta$. The reader can verify the axioms of a vector space by verifying the existence of suitable commutative diagrams of schemes. We omit the details. (An alternative proof would be to express everything in terms of local rings and then verify the vector space axioms on the level of ring maps.) $\square$

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