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Fundamental SESs in algebraic geometry

The divisor class group embeds into the Picard group: \begin{align*} 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} \textrm{Cl}(\mathcal{O}_{X,x}) \to H^2(X, \mathcal{O}_X^{\ast}) \to 0. \end{align*}
The exponential exact sequence: \begin{align*} 0 \rightarrow 2 \pi i \mathbb{Z} \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^{\times}\rightarrow 0 \end{align*}
The Chow inclusion exact sequence: \begin{align*} Y \hookrightarrow X \hookleftarrow X\setminus Y \leadsto\quad \cdots \to {\operatorname{CH}}_k(Y) \to {\operatorname{CH}}_k(X) \twoheadrightarrow{\operatorname{CH}}_k(X\setminus Y) \end{align*}
The Cartier divisor exact sequences: \begin{align*} 0\to \mathcal{O}_{X}(-D) \to \mathcal{O}_{X} \to \mathcal{O}_{D} \to 0 \end{align*} and for algebraic differentials: \begin{align*} 0\to \Omega^{k}_{X}(-D)\to \Omega^{k}_{X}\to \Omega^{k}_{D} \to 0 \end{align*}
Set ${\mathbf{T}}_X = {}_{{\mathcal{O}}_X}{\mathsf{Mod}} ( \Omega_{X_{/ {k}} }, {\mathcal{O}}_X)$ and $\mathbf{N}_{Y / X} \coloneqq {}_{{\mathcal{O}}_Y}{\mathsf{Mod}} (I/I^2, {\mathcal{O}}_Y)$ for $Y \subseteq X$ with ideal sheaf $I$.
- If $X$ is smooth and $Y$ closed irreducible smooth, then there is a SES \begin{align*} 0 \longrightarrow I/I^2 \longrightarrow \Omega_{X / k} \otimes \mathcal{O}_Y \longrightarrow \Omega_{Y / k} \longrightarrow 0 \end{align*}
- Take duals to get \begin{align*} 0 \to {\mathbf{T}}_X \to {\mathbf{T}}_Y \to \mathbf{N}_{Y/X}\to 0 \end{align*}
The Euler sequence: for $Y=\operatorname{Spec}A$ affine and $X= {\mathbf{P}}^n_{/ {A}} $, there is a SES of ${\mathcal{O}}_X{\hbox{-}}$modules: \begin{align*} 0 \longrightarrow \Omega_{X / Y} \longrightarrow \mathcal{O}_X(-1)^{n+1} \longrightarrow \mathcal{O}_X \longrightarrow 0 \end{align*}