Picard group

The Picardgruppe is a term used in the mathematical sciences in commutative algebra and algebraic geometry. It is an important invariant of commutative rings with unity and schemes. It is named after the mathematician Émile Picard.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

  • 2.1 Definition
  • 2.2 Features
  • 2.3 Example

The Picardgruppe of rings

Definition

If a module over a ring, it is called projective of rank 1 if it is projective and is locally of rank 1, ie, if valid for all prime ideals of:

Are and projective of rank 1, then

And the dual module

The following applies:

And

Therefore, the isomorphism classes of projective modules of rank 1 over a ring form a group. This is referred to as Picardgruppe.

Properties

Pic as a functor

A ring homomorphism

Induces a group homomorphism

Because by becoming a - algebra. Is a projective module of rank 1 over, so is

A projective module of rank over.

Is a covariant functor.

The Picardgruppe and the ideal class group

Below is a multiplicative quantity without zero divisors. ( A lot is multiplicative if and. ) A - ideal is a -submodule of, for which there is an element, so

Denote

The set of invertible ideals of S- and

The set of invertible principal ideals.

Is referred to as the - ideal class group.

There exists an exact sequence:

So in order to represent the Picardgruppe than ideal class group, a multiplicative quantity without zero divisors must be found so that

Is.

If any of the following conditions is met:

  • Is an integral domain and
  • Is a reduced ring, which has only finitely many minimal prime ideals and
  • Is noetherian and

Then the Picardgruppe of the equal - ideal class group of.

The Picardgruppe a schema

Definition

The definition for rings can be transferred to ringed spaces, and in particular schemes.

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

Are invertible sheaves and on a ringed space, then is also an invertible sheaf. Also, there is an invertible sheaf

So that

In addition:

The Picardgruppe a ringed space, in particular a scheme is the group of invertible sheaves of Isomorphismenklasse with the tensor product as a link.

Properties

The Picardgruppe is isomorphic to the first cohomology group:

Example

Is

The projective space over a field, then

Fractional ideal Robin Hartshorne
648916
de