View source for Divisor (algebraic geometry)

A generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer [[#References|[1]]] under the name of  "ideal divisorideal divisor"  in his studies on cyclotomic fields.

The theory of divisors for an integral commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337001.png" /> with a unit element consists in constructing a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337002.png" /> from the multiplicative semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337003.png" /> of non-zero elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337004.png" /> into some semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337005.png" /> with unique factorization, the elements of which are known as (integral) divisors of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337006.png" />. The theory of divisors makes it possible to reduce a series of problems connected with prime factorization in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337007.png" />, where this factorization may be not unique, to the problem of prime factorization in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337008.png" />. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d0337009.png" /> of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370010.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370011.png" /> and is called the principal divisor of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370012.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370013.png" /> is divisible by the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370015.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370017.png" />.

More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370018.png" /> be a free Abelian semi-group with a unit element, the free generators of which are known as prime divisors, and let a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370019.png" /> be given. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370020.png" /> defines a theory of divisors of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370021.png" /> if it satisfies the following conditions.

1) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370022.png" /> the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370023.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370025.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370026.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370028.png" />.

2) For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370029.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370030.png" /></td> </tr></table>

is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370031.png" />.

3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370032.png" /> and if, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370034.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370035.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370036.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370038.png" />.

If a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370039.png" /> exists, it is uniquely determined, up to an isomorphism, by the conditions just listed. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370040.png" /> coincides with the group of unit elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370041.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370042.png" /> are called positive divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370043.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370044.png" /> be the field of quotients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370045.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370046.png" /> be the free Abelian group generated by the set of prime divisors. Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370048.png" />, it is possible to define a principal divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370049.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370050.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370052.png" />. The elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370053.png" /> are known as fractional divisors (or, simply, divisors) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370054.png" /> (or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370055.png" />). Any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370056.png" /> may be written in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370057.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370058.png" /> is a prime divisor. In additive notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370059.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370061.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370062.png" /> is a discrete valuation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370063.png" />, and is known as the essential valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370064.png" />. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370065.png" /> is extended to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370067.png" />, contained in the exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370068.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370069.png" /> is the group of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370070.png" />, while the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370071.png" /> is called the divisor class group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370072.png" /> (or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370073.png" />). Two divisors which belong to the same equivalence class by the subgroup of principal divisors are called equivalent (in algebraic geometry, where a large number of other divisor equivalences are considered, this equivalence is known as linear).

The theory of divisors is valid for any [[Dedekind ring|Dedekind ring]], in particular for rings of integral elements in algebraic number fields, and the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370074.png" /> are in one-to-one correspondence with the non-zero ideals of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370075.png" /> (to the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370076.png" /> corresponds the ideal of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370077.png" /> that are divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370078.png" />). This is why, in a Dedekind ring, the group of divisors is also called the group of ideals, while the divisor class group is called the ideal class group.

The divisor class group of an algebraic number field is finite, and many problems in algebraic number theory involve the computation of its order (the number of classes) and structure [[#References|[2]]].

More generally, the theory of divisors is valid for Krull rings (cf. [[Krull ring|Krull ring]], [[#References|[11]]]). In such a case the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370079.png" /> is played by the semi-group of divisorial ideals (cf. [[Divisorial ideal|Divisorial ideal]]) of the ring, while the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370080.png" /> is played by the group of fractional divisorial ideals.

The concept of a Weil divisor is a generalization of the concept of a fractional divisorial ideal of a commutative ring to algebraic varieties or analytic spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370081.png" />. The name Weil divisor is given to integral formal finite linear combinations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370082.png" /> of irreducible closed subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370084.png" /> of codimension 1. A Weil divisor is called positive, or effective, if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370085.png" />. All Weil divisors form a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370086.png" /> (the group of Weil divisors). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370087.png" /> is a smooth algebraic variety, the concept of a Weil divisor coincides with that of an [[Algebraic cycle|algebraic cycle]] of codimension 1.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370088.png" /> is a Noetherian Krull ring, each prime divisorial ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370089.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370090.png" /> defines a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370091.png" /> of codimension 1 in the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370092.png" />, while each divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370093.png" /> may thus be identified with the Weil divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370094.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370095.png" /> be a normal scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370096.png" /> be a rational (meromorphic in the analytic case) function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370097.png" />. A principal Weil divisor is defined canonically:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370098.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d03370099.png" /> is the value of the discrete valuation of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700100.png" /> of the subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700101.png" /> on the representative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700103.png" />. If

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700104.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700106.png" />, the Weil divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700107.png" /> is known as the divisor of the zeros, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700108.png" /> is known as the divisor of the poles of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700109.png" />. The set of principal Weil divisors is a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700110.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700111.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700112.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700113.png" /> and is known as the divisor class group of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700114.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700115.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700116.png" /> is a Noetherian Krull ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700117.png" /> coincides with the divisor class group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700118.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700119.png" /> be an algebraic function field. A divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700120.png" /> is sometimes defined as a formal integral combination of discrete valuations of rank 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700121.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700122.png" /> is a field of algebraic functions in one variable, each such divisor may be identified with the Weil divisor of its complete non-singular model.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700123.png" /> be a regular scheme or a complex variety and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700124.png" /> be a Weil divisor. For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700125.png" /> there exists an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700126.png" /> such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700127.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700128.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700129.png" /></td> </tr></table>

is the principal divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700130.png" /> for a certain meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700131.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700132.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700133.png" /> is uniquely defined, up to an invertible function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700134.png" />, and is known as the local equation of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700135.png" /> in the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700136.png" />, while the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700137.png" /> defines a section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700138.png" />. In general, a Cartier divisor on a ringed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700139.png" /> is defined as a global section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700140.png" /> of germs of divisors. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700141.png" /> denotes the sheaf of germs of meromorphic (or rational) functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700142.png" />, i.e. the sheaf which brings into correspondence each open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700143.png" /> with the total quotient ring of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700144.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700145.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700146.png" /> are the sheaves of invertible elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700148.png" />, respectively. A Cartier divisor may be defined by a selection of local equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700149.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700150.png" /> is an open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700151.png" />, and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700152.png" /> should be a section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700153.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700154.png" />. In particular, a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700155.png" /> defines a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700156.png" /> known as a principal divisor. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700157.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700158.png" /> is called the support of the divisor. The Cartier divisors form an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700159.png" />, while the principal divisors form a subgroup of it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700160.png" />. Each divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700161.png" /> defines an invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700162.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700163.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700164.png" /> is represented by the local equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700165.png" /> on the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700166.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700167.png" /></td> </tr></table>

The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700168.png" /> is a homomorphism of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700169.png" /> into the [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700170.png" />. This homomorphism is included in the exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700171.png" /></td> </tr></table>

which is obtained from the exact sequence of sheaves

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700172.png" /></td> </tr></table>

Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700173.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700174.png" /> is a principal divisor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700175.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700176.png" /> are said to be linearly equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700177.png" /> is a quasi-projective algebraic variety or a complex [[Stein space|Stein space]], the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700178.png" /> is surjective and induces an isomorphism of the group of classes of linearly equivalent divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700179.png" /> onto the Picard group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700180.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700181.png" /> is a [[Complex space|complex space]], the problem arises as to when a given divisor is a principal divisor; this is the so-called second Cousin problem (cf. [[Cousin problems|Cousin problems]]). For example, the divisor class group on a complex Stein space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700182.png" /> is trivial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700183.png" />.

A divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700184.png" /> is said to be effective (or positive) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700185.png" />. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700186.png" /> is a sheaf of ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700187.png" />; the support of a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700188.png" /> with structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700189.png" /> forms a subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700190.png" />, which is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700191.png" />.

For a normal Noetherian scheme or a [[Normal analytic space|normal analytic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700192.png" /> there is a natural homomorphism:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700193.png" /></td> </tr></table>

mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700194.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700195.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700196.png" /> and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700197.png" /> is a local equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700198.png" /> in the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700199.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700200.png" /> is the discrete valuation corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700201.png" /> [[#References|[3]]]. The homomorphism cyc is injective and maps effective divisors to effective cycles; cyc is bijective if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700202.png" /> is locally factorial (e.g. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700203.png" /> is a non-singular scheme or an analytic manifold). If cyc is bijective, Weil and Cartier divisors coincide.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700204.png" /> be a morphism of schemes which is flat in codimension 1. Then, for any Cartier or Weil divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700205.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700206.png" /> the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700207.png" /> is defined; also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700208.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700209.png" /> is a homomorphism of groups which maps principal divisors to principal ones, and thus defines a homomorphism of groups

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700210.png" /></td> </tr></table>

(respectively,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700211.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700212.png" /> is an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700213.png" /> whose codimension of the complement is at least 2 and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700214.png" /> is the imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700215.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700216.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700217.png" /> is an isomorphism, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700218.png" /> is an isomorphism if the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700219.png" /> is locally factorial.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700220.png" /> be a smooth projective variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700221.png" />. Any divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700222.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700223.png" /> defines a homology class

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700224.png" /></td> </tr></table>

The cohomology class which is Poincaré dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700225.png" /> is identical with the [[Chern class|Chern class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700226.png" /> of the invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700227.png" />. Thus there appears a homological equivalence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700228.png" />. There exists a theory of intersections of divisors [[#References|[7]]], leading to the concept of algebraic equivalence of divisors (cf. [[Algebraic cycle|Algebraic cycle]]). The group

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700229.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700230.png" /> denotes the group of divisors which are algebraically equivalent to zero, is naturally provided with the structure of an Abelian variety (the [[Picard variety|Picard variety]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700231.png" /> is a curve, it is also called the [[Jacobi variety|Jacobi variety]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700232.png" />). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700233.png" />, known as the Néron–Severi group, has a finite number of generators. The last two facts also apply to algebraic varieties over an arbitrary field.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700234.png" /> is a one-dimensional complex manifold (a [[Riemann surface|Riemann surface]]), a divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700235.png" /> is a finite linear combination

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700236.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700237.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700238.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700239.png" /> is called the degree of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700240.png" />. For a compact Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700241.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700242.png" /> the group of divisor classes of degree zero is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700243.png" />-dimensional Abelian variety and is identical with the [[Picard variety|Picard variety]] (or with the [[Jacobi variety|Jacobi variety]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700244.png" /> is a meromorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700245.png" />, a principal divisor is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700246.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700247.png" /> are the zeros and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700248.png" /> are the poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700249.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700250.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700251.png" /> are their multiplicities. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700252.png" />, i.e. a principal divisor has degree 0. A divisor of degree 0 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700253.png" /> is principal if and only if there exists a singular one-dimensional chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700254.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700255.png" /></td> </tr></table>

for all holomorphic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700256.png" /> of degree 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033700/d033700257.png" /> (Abel's theorem). See also [[Abelian differential|Abelian differential]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.E. Kummer,   "Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexe Zahlen in ihre Primfaktoren"  ''J. Reine Angew. Math.'' , '''35'''  (1847)  pp. 327–367</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Weil,   "Introduction à l'Aeetude des variétés kahlériennes" , Hermann  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Cartier,   "Questions de rationalité des diviseurs en géometrie algébrique"  ''Bull. Soc. Math. France'' , '''86'''  (1958)  pp. 177–251</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Grothendieck,   "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  C. Chevalley,   "Introduction to the theory of algebraic functions of one variable" , Amer. Math. Soc.  (1951)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  S.S. Chern,   "Complex manifolds without potential theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience)  (1978)  pp. 178; 674; 179; 349; 525; 532; 535; 632; 743</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>