2024-10-03

Last time

We considered the surjection

for an ideal . If is a Noetherian domain, then the above map is an isomorphism if and only if is a domain.

We also calculated that

It follows that to show

we just need that is prime. Equivalently, since is a UFD, we just need to show that is irreducible. This follows from Gauss’s Lemma: consider . This is degree in and .

Definition (Homogeneous ideals)

An ideal is homogeneous if any of the equivalent conditions hold:

  1. if and only if the th graded component for all .
  2. is generated by homogeneous elements.

Examples 1

  1. Let be a graded ring homomorphism, i.e., . Then is homogeneous.
  2. (Homogenisation) Consider the ideal . The ideal is not homogeneous, while is graded by total degree. But let’s think of these as not being graded: is certainly not graded. We now add an extra variable and map , . Then we get . Clearing denominators, we find , which is homogeneous! It turns out we can even recover what we started with. To do this, we localise at : . Here, is a -graded ring, and we pick off the degree- part. In fact, , and for our ideal, , which is what we started with.
  3. (Prime ideals) A homogeneous ideal is prime if and only if for all homogeneous, implies or . (We only have to check this for homogeneous elements.)
  4. If is a homogeneous ideal, then where . The quotient is again graded with .

Definition (Proj)

We define the set

(Recall that is the irrelevant ideal consisting of elements of degree .)

Examples 2

  1. . We know that
    The homogeneous ideals are just and , and contains . So . We have “cut down” the dimension by .

    is meant to be “lines through the origin”.

  2. . We have
    The homogeneous ones are , , and where is homogeneous and irreducible in . But if we have a homogeneous polynomial in two variables over , it must factor: we can turn into a polynomial in one variable by dividing out by a power of one of the variables. More precisely, we have
    for some . Now factors as over , and therefore so does :
    By irreducibility of , for not both zero. So
    The ideals are precisely the lines with , .

Definition (Vanishing locus)

If is a homogeneous ideal, we define the vanishing locus to be

These define the closed subsets for a topology on . If is homogeneous, we define the distinguished open

Lemma

A base of opens for a topology on is given by the for homogeneous. Moreover, .

Key construction

Let be homogeneous of degree . Define

where means to localise at , and then take the degree part of the resulting -graded ring. To see the above equality , we first note that we certainly have . Conversely, if , then for some . Now write the graded decomposition

This means that

Another property

We also have (actual equalities)

Example 3

Let , we have . It follows that

These are the charts that appeared when we did projective space. Also, for gluing, we have

Lemma (Distinguished opens)

Let be homogeneous. There is a natural homeomorphism

Moreover, if , then there exists a homogeneous such that the following diagram commutes:

where is the restriction of to , and comes from applying the universal property of localisation at to .

Remark

The purpose of this lemma is to show that the map is natural with localisation.

Lemma

Let and be homogeneous and belong to . Then if and only if for some (exactly the same as the affine case: nullstellensatz for prime ideals). In particular, if is a finitely generated ideal, then is quasi-compact.

Structure sheaf on

Let be homogeneous. We define

where (this only depends on , not ). If then and we get a naturally induced map . Hence, is a (pre)sheaf of rings on a base of opens of . One can verify that really does define a sheaf of rings on .

Theorem

Let be an -graded ring. Then is a scheme with affine cover given by for homogeneous.

Quasi-coherent sheaves on Proj

Let be a graded ring. A graded -module is a -graded abelian group

with an action of such that .

Examples 4

  1. If and is a graded -module, then we can make a “twist”:
  2. So we have the graded -modules for all .

Quasi-coherent sheaf of modules

Just as in the affine case, we can define for each homogeneous

Examples 5

Let be a homogeneous ideal. Then we get a quasi-coherent sheaf of ideals . Hence, there is an induced closed immersion of schemes.

The converse doesn’t always hold. It holds for projective space.

Definition (Serre twisting sheaf)

Let . We define , called the Serre twisting sheaf. If is a quasi-coherent sheaf on , we define , called a Serre twist of .

Example 6

If then we have for homogeneous

Actually,

Another thing

We have an isomorphism , from which we get -module homomorphisms

However, this is not an isomorphism in general.