Divisors in algebraic geometry
C.S. Seshadri
Translator's note.
This text is one of a series* of translations of various papers into English. The translator takes full responsibility for any errors introduced in the passage from one language to another, and claims no rights to any of the mathematical content herein.
What follows is a translation of the French seminar talk:
SESHADRI, C. S. "Diviseurs en géométrie algébrique". Séminaire Claude Chevalley, Volume 4 (1958-1959), Talk no. 4. http://www.numdam.org/item/SCC_1958-1959_4__A4_0
Contents
1 Preliminaries 1
2 Dévissage theorem 3
3 Divisors (Generalities) 5
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In the first part of this talk, we will prove a theorem of Serre on complete varieties [6], following the methods of Grothendieck [4]. The second part is dedicated to generalities on divisors. In the literature, we often call the divisors studied here “locally principal” divisors.
The algebraic spaces considered here are defined over an algebraically closed field $K$. By “variety”, we mean an irreducible algebraic space. If $X$ is an algebraic space, we denote by $\mathcal{O}(X)$, $\mathcal{R}(X)$, etc. (or simply $\mathcal{O}$, $\mathcal{R}$, etc.) the structure sheaf, of regular functions, etc. on $X$ (to define $\mathcal{R}(X)$ we assume that $X$ is a variety). By “coherent sheaf” on $X$, we mean a coherent sheaf of $\mathcal{O}$-modules on $X$.
1 Preliminaries
[4, 5, 6]
If $M$ is a module over an integral ring $A$ (commutative and with 1), then we say that an element $m \in M$ is a torsion element if there exists some non-zero $a \in A$ such that $a \cdot m = 0$. We say that $M$ is a torsion module (resp. torsion-free module) if every element of $M$ is a torsion element (resp. if $M \neq 0$ and no non-zero element of $M$ is a torsion element). The torsion elements of $M$ form a torsion submodule of $M$ (denoted by $T(M)$); if $M \neq 0$, then