Difference between revisions of "Divisor (algebraic geometry)"
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− | 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 | + | 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" />. | 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" />. | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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) {{MR|0195803}} {{ZBL|0145.04902}} </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 {{MR|0106223}} {{ZBL|0091.33501}} </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) {{MR|0238860}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </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) {{MR|0042164}} {{ZBL|0045.32301}} </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) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) {{MR|0360549}} {{ZBL|0279.13001}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) {{MR|0092855}} {{ZBL|0078.06602}} </TD></TR></table> |
Line 122: | Line 122: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </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 {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR></table> |
Revision as of 21:51, 30 March 2012
A generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer [1] under the name of "ideal divisorideal divisor" in his studies on cyclotomic fields.
The theory of divisors for an integral commutative ring with a unit element consists in constructing a homomorphism
from the multiplicative semi-group
of non-zero elements of
into some semi-group
with unique factorization, the elements of which are known as (integral) divisors of the ring
. The theory of divisors makes it possible to reduce a series of problems connected with prime factorization in
, where this factorization may be not unique, to the problem of prime factorization in
. The image
of an element
is denoted by
and is called the principal divisor of the element
. One says that
is divisible by the divisor
if
divides
in
.
More exactly, let be a free Abelian semi-group with a unit element, the free generators of which are known as prime divisors, and let a homomorphism
be given. The homomorphism
defines a theory of divisors of the ring
if it satisfies the following conditions.
1) For the element
divides
in
if and only if
divides
in
.
2) For any ,
![]() |
is an ideal of .
3) If and if, for any
,
is divisible by
if and only if
is divisible by
, then
.
If a homomorphism exists, it is uniquely determined, up to an isomorphism, by the conditions just listed. The kernel
coincides with the group of unit elements of
. The elements of
are called positive divisors of
. Let
be the field of quotients of
, and let
be the free Abelian group generated by the set of prime divisors. Then for any
,
, it is possible to define a principal divisor
. If
where
, then
. The elements of the group
are known as fractional divisors (or, simply, divisors) of
(or of
). Any divisor
may be written in the form
![]() |
where is a prime divisor. In additive notation:
. If
and
, the mapping
is a discrete valuation on
, and is known as the essential valuation of
. The homomorphism
is extended to a homomorphism
, where
, contained in the exact sequence
![]() |
Here is the group of invertible elements of
, while the group
is called the divisor class group of
(or of
). 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, in particular for rings of integral elements in algebraic number fields, and the elements of are in one-to-one correspondence with the non-zero ideals of the ring
(to the divisor
corresponds the ideal of all elements of
that are divisible by
). 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 [2].
More generally, the theory of divisors is valid for Krull rings (cf. Krull ring, [11]). In such a case the role of is played by the semi-group of divisorial ideals (cf. Divisorial ideal) of the ring, while the part of
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 . The name Weil divisor is given to integral formal finite linear combinations
of irreducible closed subspaces
in
of codimension 1. A Weil divisor is called positive, or effective, if all
. All Weil divisors form a group
(the group of Weil divisors). If
is a smooth algebraic variety, the concept of a Weil divisor coincides with that of an algebraic cycle of codimension 1.
If is a Noetherian Krull ring, each prime divisorial ideal
in
defines a subspace
of codimension 1 in the scheme
, while each divisor
may thus be identified with the Weil divisor
.
Let be a normal scheme and let
be a rational (meromorphic in the analytic case) function on
. A principal Weil divisor is defined canonically:
![]() |
Here is the value of the discrete valuation of the ring
of the subvariety
on the representative of
in
. If
![]() |
where and
, the Weil divisor
is known as the divisor of the zeros, while
is known as the divisor of the poles of the function
. The set of principal Weil divisors is a subgroup
of the group
. The quotient group
is denoted by
and is known as the divisor class group of the scheme
. If
, where
is a Noetherian Krull ring,
coincides with the divisor class group of the ring
.
Let be an algebraic function field. A divisor of
is sometimes defined as a formal integral combination of discrete valuations of rank 1 of
. If
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 be a regular scheme or a complex variety and let
be a Weil divisor. For any point
there exists an open neighbourhood
such that the restriction of
on
,
![]() |
is the principal divisor for a certain meromorphic function
on
. The function
is uniquely defined, up to an invertible function on
, and is known as the local equation of the divisor
in the neighbourhood
, while the correspondence
defines a section of the sheaf
. In general, a Cartier divisor on a ringed space
is defined as a global section of the sheaf
of germs of divisors. Here
denotes the sheaf of germs of meromorphic (or rational) functions on
, i.e. the sheaf which brings into correspondence each open
with the total quotient ring of the ring
, while
and
are the sheaves of invertible elements in
and
, respectively. A Cartier divisor may be defined by a selection of local equations
![]() |
where is an open covering of
, and the functions
should be a section of the sheaf
over
. In particular, a meromorphic function
defines a divisor
known as a principal divisor. The set of
such that
is called the support of the divisor. The Cartier divisors form an Abelian group
, while the principal divisors form a subgroup of it,
. Each divisor
defines an invertible sheaf
contained in
: If
is represented by the local equations
on the covering
, then
![]() |
The correspondence is a homomorphism of the group
into the Picard group
. This homomorphism is included in the exact sequence
![]() |
which is obtained from the exact sequence of sheaves
![]() |
Thus, . If
is a principal divisor,
and
are said to be linearly equivalent. If
is a quasi-projective algebraic variety or a complex Stein space, the homomorphism
is surjective and induces an isomorphism of the group of classes of linearly equivalent divisors
onto the Picard group
.
If is a 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). For example, the divisor class group on a complex Stein space
is trivial if and only if
.
A divisor is said to be effective (or positive) if
. In such a case
is a sheaf of ideals in
; the support of a divisor
with structure sheaf
forms a subspace in
, which is also denoted by
.
For a normal Noetherian scheme or a normal analytic space there is a natural homomorphism:
![]() |
mapping into
, where
and where
is a local equation of
in the neighbourhood
, while
is the discrete valuation corresponding to
[3]. The homomorphism cyc is injective and maps effective divisors to effective cycles; cyc is bijective if and only if
is locally factorial (e.g. when
is a non-singular scheme or an analytic manifold). If cyc is bijective, Weil and Cartier divisors coincide.
Let be a morphism of schemes which is flat in codimension 1. Then, for any Cartier or Weil divisor
on
the inverse image
is defined; also,
. The mapping
is a homomorphism of groups which maps principal divisors to principal ones, and thus defines a homomorphism of groups
![]() |
(respectively,
![]() |
If is an open set in
whose codimension of the complement is at least 2 and if
is the imbedding of
into
, then
is an isomorphism, while
is an isomorphism if the scheme
is locally factorial.
Let be a smooth projective variety over
. Any divisor
on
defines a homology class
![]() |
The cohomology class which is Poincaré dual to is identical with the Chern class
of the invertible sheaf
. Thus there appears a homological equivalence on
. There exists a theory of intersections of divisors [7], leading to the concept of algebraic equivalence of divisors (cf. Algebraic cycle). The group
![]() |
where denotes the group of divisors which are algebraically equivalent to zero, is naturally provided with the structure of an Abelian variety (the Picard variety; if
is a curve, it is also called the Jacobi variety of
). The group
, 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 is a one-dimensional complex manifold (a Riemann surface), a divisor on
is a finite linear combination
![]() |
where ,
. The number
is called the degree of the divisor
. For a compact Riemann surface
of genus
the group of divisor classes of degree zero is a
-dimensional Abelian variety and is identical with the Picard variety (or with the Jacobi variety). If
is a meromorphic function on
, a principal divisor is
![]() |
where are the zeros and
are the poles of
and
,
are their multiplicities. Then
, i.e. a principal divisor has degree 0. A divisor of degree 0 on
is principal if and only if there exists a singular one-dimensional chain
such that
![]() |
for all holomorphic forms of degree 1 on
(Abel's theorem). See also Abelian differential.
References
[1] | 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 |
[2] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902 |
[3] | A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958) |
[4] | P. Cartier, "Questions de rationalité des diviseurs en géometrie algébrique" Bull. Soc. Math. France , 86 (1958) pp. 177–251 MR0106223 Zbl 0091.33501 |
[5] | A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967) MR0238860 Zbl 0144.19904 Zbl 0135.39701 Zbl 0136.15901 |
[6] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |
[7] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
[8] | C. Chevalley, "Introduction to the theory of algebraic functions of one variable" , Amer. Math. Soc. (1951) MR0042164 Zbl 0045.32301 |
[9] | R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601 |
[10] | S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004 |
[11] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) MR0360549 Zbl 0279.13001 |
[12] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) MR0092855 Zbl 0078.06602 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
[a2] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 178; 674; 179; 349; 525; 532; 535; 632; 743 MR0507725 Zbl 0408.14001 |
[a3] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004 |
Divisor (algebraic geometry). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divisor_(algebraic_geometry)&oldid=18967