Example 53.15.2 (Glueing points). 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 glueing $a$ and $b$ in $X'$ if the following are true:

1. $a, b \in X'(k)$ are distinct points which map to the same 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

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

where arrow on the right sends a local section $h$ of $f_*\mathcal{O}_{X'}$ to the difference $h(a) - h(b) \in k$.

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

$0 \to \mathcal{O}_ X^* \to f_*\mathcal{O}_{X'}^* \xrightarrow {ab^{-1}} x_*k^* \to 0$

where arrow on the right sends a local section $h$ of $f_*\mathcal{O}_{X'}^*$ to the multiplicative difference $h(a)h(b)^{-1} \in k^*$.

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