Example 53.15.3 (Squishing a tangent vector). Let $k$ be an algebraically closed field. Let $f : X' \to X$ be a morphism of algebraic $k$-schemes. We say $X$ is obtained by squishing the tangent vector $\vartheta$ in $X'$ if the following are true:

1. $\vartheta : \mathop{\mathrm{Spec}}(k[\epsilon ]) \to X'$ is a closed immersion over $k$ such that $f \circ \vartheta$ factors through a point $x \in X(k)$,

2. $f$ is finite and $f^{-1}(X \setminus \{ x\} ) \to X \setminus \{ x\}$ is an isomorphism,

3. there is a short exact sequence

$0 \to \mathcal{O}_ X \to f_*\mathcal{O}_{X'} \xrightarrow {\vartheta } x_*k \to 0$

where arrow on the right sends a local section $h$ of $f_*\mathcal{O}_{X'}$ to the coefficient of $\epsilon$ in $\vartheta ^\sharp (h) \in k[\epsilon ]$.

If this is the case, then there also is a short exact sequence

$0 \to \mathcal{O}_ X^* \to f_*\mathcal{O}_{X'}^* \xrightarrow {\vartheta } x_*k \to 0$

where arrow on the right sends a local section $h$ of $f_*\mathcal{O}_{X'}^*$ to $\text{d}\log (\vartheta ^\sharp (h))$ where $\text{d}\log : k[\epsilon ]^* \to k$ is the homomorphism of abelian groups sending $a + b\epsilon$ to $b/a \in k$.

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