2024-09-17

Goal

Replace the conditions for quasi-compactness and quasi-separatedness, which are hard to check (since they quantify over very large sets) with conditions which are easier to check.

Corollary 1 (Quasi-compactness)

Let be a morphism of schemes. Then the following are equivalent:

  1. is quasi-compact.
  2. If is an affine open then has a finite cover by affine open subschemes.
  3. has an affine open cover such that each has a finite cover by affine open subschemes.

Corollary 2 (Quasi-separatedness)

Let be a scheme. Then the following are equivalent:

  1. is quasi-compact.
  2. is quasi-separated.
  3. The intersection of any two affine open subsets of has a finite cover by affines.

Proof

1. 2.

Let , be quasi-compact opens of . Then is quasi-compact open and the diagram

is Cartesian. If is quasi-compact then is quasi-compact. Therefore 1. 2..

Note that since is final, products and fibre products over are the same thing.

2. 3.

2. 3. follows from the last lecture. Since is quasi-separated, the intersection of any two affine opens is quasi-compact, which is the same as having a finite cover by affines by this proposition.

3. 1.

Let be quasi-compact open. The construction of the fibre product shows that has a basis of opens given by products of affines. So then for . Now, the preimage of under just :

We know that has a finite cover by affines by assumption, i.e., is quasi-compact. Since cover , cover . So has a finite cover by quasi-compact opens, and is therefore quasi-compact. So is quasi-compact.

Corollary 3 (Relative version)

Let be a morphism of schemes. Then the following are equivalent:

  1. is quasi-separated.
  2. is quasi-compact.
  3. If then is quasi-separated.
  4. and and is a finite union of affines.

Theorem

Let be a quasi-compact, quasi-separated morphisms of schemes. Then the restriction of

to factors through . In particular,

is an adjoint pair.

We can always just construct an adjoint pair using categorical nonsense, but then we don’t know much about the resulting objects. But in fact, above has an explicit formula: .

Proof

Let be a quasi-coherent -module.

Step 1: Both and affine

Let . We know that for some -module (by a lemma from earlier). So

where is regarded as an -module via . That is, which is quasi-coherent on .

Step 2: is affine

Let .

  • Quasi-compactness of implies that .
  • Quasi-separatedness of implies that .
  • Let and .

Now we have an exact sequence of -modules (by the sheaf condition)

We can apply to the above sequence: is left exact (since it’s the right adjoint to ), so we obtain an exact sequence

Note that and . However, we note that

and therefore and are maps of affines. So and are quasi-coherent by Step 1, and therefore is quasi-coherent (as the kernel of a map of quasi-coherent modules).

Step 3

Let . We look at the inclusion of an affine :

Here, . Now, . So is quasi-coherent by Step 2 (quasi-coherent is a local property).

Properties of schemes

We want to move properties of rings to properties of schemes. “Absolute” properties.

Definition (Affine-local)

A property of rings is called affine-local if:

  1. (Restricting to distinguished opens) If is then is for all .
  2. (Cover by distinguished opens) If is a ring with , and satisfy for all , then satisfies .

Proposition

Let be a scheme and be an affine-local property of rings. Then the following are equivalent:

  1. (opens), where is .
  2. For all open, is .

If one of these is satisfied, then we say that is a scheme*.

Proof

2. 1.

OK.

1 2.

#TODO diagram. We have a bunch of s, and a . We want to get the information about the s onto . From the definition of affine-local, the only thing we know how to do is localisation.

We want such that . Because then is , and so is . So now has a cover by , and then is . This will follow from the affine communication lemma.

Examples

  1. Reduced, i.e., no nilpotents.
  2. Local rings are normal domains. A domain is normal if it is integrally closed in its field of fractions.
  3. Noetherian: a scheme satisfying this called a “locally Noetherian scheme”, not a “Noetherian scheme”. A Noetherian scheme is a locally Noetherian scheme that is quasi-compact.