---
date: 2022-09-05 00:38
modification date: Monday 5th September 2022 00:38:43
title: "Fundamental SESs in algebraic geometry"
aliases: [Fundamental SESs in algebraic geometry]
---

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# Fundamental SESs in algebraic geometry

- The [divisor class group](Unsorted/divisor%20class%20group.md) embeds into the [Picard group](Unsorted/class%20group.md):
$$
0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus_{x \in \textrm{Sing}(X)} \textrm{Cl}(\mathcal{O}_{X,x}) \to H^2(X, \mathcal{O}_X^{\ast}) \to 0.
$$

- The exponential exact sequence:
$$
0 \rightarrow 2 \pi i \mathbb{Z} \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X\units \rightarrow 0
$$

- The Chow inclusion exact sequence:
$$
Y \injects X \injectsfrom X\sm Y \leadsto\quad \cdots \to \CH_k(Y) \to \CH_k(X) \surjects \CH_k(X\sm Y)
$$

- The [Cartier divisor](Unsorted/Weil%20divisor.md) exact sequences:
$$
0\to \mathcal{O}_{X}(-D) \to \mathcal{O}_{X} \to \mathcal{O}_{D} \to 0
$$
and for [algebraic differentials](Unsorted/algebraic%20de%20Rham%20cohomology.md): 
$$
0\to \Omega^{k}_{X}(-D)\to \Omega^{k}_{X}\to \Omega^{k}_{D} \to 0
$$

- Set $\T_X = \mods{\OO_X}( \Omega_{X\slice k}, \OO_X)$ and $\mathbf{N}_{Y / X} \da \mods{\OO_Y}(I/I^2, \OO_Y)$ for $Y \subseteq X$ with ideal sheaf $I$.
	- If $X$ is smooth and $Y$ closed irreducible smooth, then there is a SES
$$
0 \longrightarrow I/I^2 \longrightarrow \Omega_{X / k} \otimes \mathcal{O}_Y \longrightarrow \Omega_{Y / k} \longrightarrow 0
$$
	- Take duals to get
$$
0 \to \T_X \to \T_Y \to \mathbf{N}_{Y/X}\to 0
$$
- The Euler sequence: for $Y=\spec A$ affine and $X= \PP^n\slice A$, there is a SES of $\OO_X\dash$modules:
$$
0 \longrightarrow \Omega_{X / Y} \longrightarrow \mathcal{O}_X(-1)^{n+1} \longrightarrow \mathcal{O}_X \longrightarrow 0
$$