Picard group

The Picard group is a term from the mathematical sub-areas of commutative algebra and algebraic geometry . It is an important invariant of commutative rings with ones and schemes . It is named after the mathematician Émile Picard .

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

The Picard group of rings

definition

If a module is above a ring , it is called projective of rank 1 if it is projective and local of rank 1, i.e. if the following applies to all prime ideals of : ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle M_{p}\cong R_{p}}$

Are and projective of rank 1, then too ${\displaystyle M}$ ${\displaystyle N}$

${\displaystyle M\otimes N}$

and the dual module

${\displaystyle M^{\vee }:=Hom_{R}(M,R)}$

The following applies:

${\displaystyle M\otimes M^{\vee }\cong R}$

and

${\displaystyle M\otimes R\cong M}$

The isomorphism classes of projective modules of rank 1 over a ring therefore form a group. This is known as the Picard group . ${\displaystyle R}$

properties

Pic as functor

A ring homomorphism

${\displaystyle f\colon A\to B}$

induces a group homomorphism

${\displaystyle \mathrm {Pic} (f)\colon \mathrm {Pic} (A)\to \mathrm {Pic} (B)}$

${\displaystyle \mathrm {Pic} (f)\colon P\mapsto P\otimes _{A}B}$

because through becomes an algebra. If a projective module is of rank 1 above , then is ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle A}$

${\displaystyle P\otimes _{A}B}$

a projective module from the rank above . ${\displaystyle 1}$ ${\displaystyle B}$

${\displaystyle \mathrm {Pic} }$is a covariant functor .

The Picard group and the ideal class group

The following is a multiplicative set without a zero divisor . (A set is multiplicative if and .) An ideal is a submodule of that has an element such that ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle 1\in S}$ ${\displaystyle r,s\in S\Rightarrow rs\in S}$ ${\displaystyle S}$ ${\displaystyle A}$ ${\displaystyle I}$ ${\displaystyle S^{-1}}$ ${\displaystyle s\in S}$

${\displaystyle sA\subset M\subset s^{-1}I}$

Denote

${\displaystyle \mathrm {Inv} (A,S)}$

the set of invertible S-ideals of and ${\displaystyle A}$

${\displaystyle \mathrm {H} (A,S):=\{I\in \mathrm {Inv} (A,S)|\exists x\in S^{-1}AI=Ax\}}$

the set of invertible main ideals.

${\displaystyle \mathrm {Inv} (A,S)/\mathrm {H} (A,S)}$

is referred to as the - ideal class group. ${\displaystyle S}$

There is an exact sequence:

${\displaystyle 0\longrightarrow \mathrm {H} (A,S)\longrightarrow \mathrm {Inv} (A,S)\longrightarrow \mathrm {Pic} (A)\longrightarrow \mathrm {Pic} (S^{-1}A)}$

So in order to represent the Picard group as an ideal class group, a multiplicative set without zero divisors must be found, so that

${\displaystyle \mathrm {Pic} (S^{-1}A)=0}$

is.

When one of the following conditions is true:

• ${\displaystyle A}$ is an integrity ring and ${\displaystyle S=A\backslash \{0\}}$
• ${\displaystyle A}$is a reduced ring that has only a finite number of minimal prime ideals and ${\displaystyle \{p_{1},\dots ,p_{n}\}}$

${\displaystyle S=A\backslash \bigcup _{i=1}^{n}p_{i}}$

• ${\displaystyle A}$ is noetherian and

${\displaystyle S=A\backslash \bigcup _{p\in Ass(A)}p}$

Then the Picard group of is equal to the ideal class group of . ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle A}$

The Picard group of a scheme

definition

The definition of rings can be applied to limited spaces , especially to schemes.

An invertible sheaf of a small space is a locally free module sheaf of rank 1.

If and invertible sheaves are in a small space, then there is also an invertible sheaf. There is also an invertible sheaf ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {L}}\otimes {\mathcal {M}}}$

${\displaystyle {\mathcal {L}}^{\vee }={\mathcal {H}}{\mathit {om}}({\mathcal {L}},{\mathcal {O}}_{X})}$

so that

${\displaystyle {\mathcal {L}}^{\vee }\otimes {\mathcal {L}}\cong {\mathcal {O}}_{X}}$

The following also applies:

${\displaystyle {\mathcal {L}}\otimes {\mathcal {O}}_{X}\cong {\mathcal {L}}}$

The Picard group of a small space, especially of a scheme, is the group of the isomorphism class of invertible sheaves with the tensor product as a link.

properties

The Picard group is isomorphic to the first cohomology group :

${\displaystyle \mathrm {H} ^{1}(X,{\mathcal {O}}_{X}^{*})}$

example

Is

${\displaystyle X=\mathbf {P_{k}^{n}} }$

the projective space to a body , so

${\displaystyle \mathrm {Pic} (X)\cong \mathbb {Z} }$

literature

• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9