# Divisors in algebraic geometry

*1958–59*

#### Translator’s note

*This page is a translation into English of the following:*

Seshadri, C. S. “Diviseurs en géométrie algébrique.” *Séminaire Claude Chevalley* **4** (1958–59), Talk no. 4. `http://www.numdam.org/item/SCC_1958-1959__4__A4_0`

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

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

# 1 Preliminaries

If *torsion element* if there exists some non-zero *torsion module* (resp. *torsion-free module*) if every element of

Let

A sheaf *variety* *torsion sheaf* (resp. *torsion-free sheaf*) if, for every

If

*Proof*. The uniqueness is trivial.
The exists is a consequence of the fact that, if

If ^{1}

If

*Proof*. This is a trivial consequence of the fact that, if

If

*Proof*. Let

Let *rank of {\mathscr{F}}*, and we can then consider

Under the same hypotheses as in Proposition 3, there exists a coherent sheaf

*Proof*. The proof is immediate.

If

Let

*Proof*. We can reduce to the case where

Let

*Proof*. By Proposition 3 and the hypothesis, there exists a coherent sheaf

# 2 Dévissage theorem

Let *left exact in {\mathcal{C}}* if

^{2}

- every subobject of an object of
{\mathcal{C}}' is in{\mathcal{C}}' ; - for every exact sequence
0\to{\mathscr{A}}'\to{\mathscr{A}}\to{\mathscr{A}}''\to0 in{\mathcal{C}} , the object{\mathscr{A}} is in{\mathcal{C}}' if the other two objects are in{\mathcal{C}}' .

Let

Let

*Proof*. The proof works by induction on the dimension of

Now assume that we have proven the theorem for all dimensions

We will now prove that, if

Suppose that

Suppose again that

Now let

We know that

If

*Proof*. We take

# 3 Divisors (Generalities)

Let

A *divisor* *definition function of D at x*.
More generally, a function

*definition function of*D in an open subset U if, for all

A divisor *positive* if, for each

Since *group of divisors on X*.
The composition law in this group is written additively, and the identity element in this group is thus called the

*zero divisor*, and is denoted by

If *principal divisors*, and form a subgroup of the group of divisors on *group of classes of divisors on X*.
Two divisors

We can define, in an analogous way, an *additive divisor* on a variety *multiplicative divisors*, or simply *divisors*).
The additive divisors form an abelian group, and even a vector space over

Let

The support of a divisor

*Proof*. The latter claim is trivial.
For the former, we prove that the set

If

*Proof*. If

If

Suppose that *associated cycle* of the divisor

Let

*Proof*. The proof is trivial.

If

*Proof*. It suffices to show that, for every hypersurface

Proposition 10 is not necessarily true if

# Bibliography

*Bull. Soc. Math. France*. (1958).

*Fondements de la Géométrie algébrique*. Paris, Secrétariat mathématique, 1958.

*Propriétés analytiques des localités*. 1955-56.

**8**.

*Sur les faisceaux algébriques et les faisceaux analytiques cohérents*. 1956-57.

**9**.

*Ann. Math.*

**61**(1955), 197–279.

*J. Math. Pures Et Appl.*

**36**(1957), 1–16.

*Fibre spaces in algebraic geometry*. University of Chicago, 1955.

*[Trans.] The condition that*↩︎{\mathscr{F}}\neq0 is unnecessary, but we include it here since it is in the original. Note that the zero sheaf is indeed a torsion-free sheaf, otherwise any coherent torsion sheaf{\mathscr{F}} provides a counterexample to this corollary.The axioms here that define a left-exact subcategory are slightly stronger than those of Grothendieck [4].↩︎