2024-09-19

Affine communication lemma

Let be a scheme and be affine opens. Then such that where and for all .

#TODO Picture.

Proof

Let . Therefore there exists such that . But now, is an open. So there exists such that . These distinguished opens may not be equal, but we can fix that.

The inclusion maps come from ring homomorphisms (because they’re all affine schemes)

and must be the localisation map as it arises from the inclusion . In , we do not know if we can invert . So we set and write , . Observe now that we have an isomorphism

Why? Because we have the diagram

where all the dashed maps exist and are unique by the universal property of localisation:

  1. exists because the image of is invertible in .
  2. exists because is the localisation map, and therefore the image of in is , which is invertible.
  3. is the same map as .

Hence .

Aside

Local properties of morphisms of schemes

“Relative” versions.

Definition (Affine-local)

A property of ring homomorphisms is called affine-local if

  1. is implies is for all .
  2. is implies is for all .
  3. Given and such that is for all , then is .
  4. Given and such that is for all , then is .

An affine-local property is stable under base change if is implies is for all .

If is stable under base change, then 1. is redundant (because localisation is a base change).

Examples

The following are affine-local and stable under base change:

  1. Finite type ( is a finitely-generated -algebra).
  2. Finitely presented ( is a finitely-presented -algebra).
  3. Flat ( is a flat -algebra).
  4. Finite fibres ( for all , is a finite-dimensional -vector space).
  5. Geometrically reduced fibres (for all finite extensions for all , is a reduced ring).

Proposition

Let be a morphism of schemes. Let be an affine-local property of ring homomorphisms. Then the following are equivalent:

  1. and such that is for all .
  2. For all affine open and affine open, is .

In this case, we say that is locally . Moreover, if is stable under base change and is locally then is locally for all .

Example

Let be a locally Noetherian scheme. Let be locally of finite type. Then is a locally Noetherian scheme.

Proof

Let be an affine open. Then is a Noetherian ring. Let be an affine open. Then is an -algebra of finite type (finitely-generated -algebra). By Hilbert’s Basis Theorem, is a Noetherian ring. Hence, is locally Noetherian.

Definition (Finite-type, finitely-presented)

Let be a morphism of schemes. We say that is finite-type if it is locally finite-type and quasi-compact. We say that is finitely-presented if it is locally of finite-presentation and quasi-compact and quasi-separated.

Global properties

Proposition (Affine morphism)

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

  1. and .
  2. For all affine open, is an affine scheme.

We call such an affine.

Proof

2. 1. is clear. We focus on 1. 2.. We note for now that that 1. implies that is quasi-compact and quasi-separated (by Corollary 1 and Corollary 3). Therefore is a quasi-coherent -algebra.

Let be an affine open. By the Affine communication lemma, where . Therefore, by assumption that , pulling back these distinguished opens results in distinguished opens:

That is to say, . Also observe that since cover . By taking the open , we thus conclude that , allowing us to reduce to the case by replacing with .

To prove that is affine, it now suffices to prove that the natural map (arising from the adjunction ) is an isomorphism over . We get a map

which induces a map in the following diagram

Observe first that by definition we have

Now, from before, we noted that is a quasi-coherent sheaf on , so for some -module , and we have

So finally recalling that , we observe that

Therefore, (since it is locally an isomorphism).

Corollary (Integral/finite/closed immersions)

  1. Same as in the above Proposition (Affine morphism), but (respectively ) is an integral/finite (respectively )-algebra. We call these integral/finite morphisms.

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

  1. and , and the induced is an integral -algebra.
  2. For all affine open, , and the induced is an integral -algebra.
    We call such an integral.

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

  1. and , and the induced is an finite -algebra.
  2. For all affine open, , and the induced is an finite -algebra.
    We call such an finite.

is a finite -algebra if is finitely-generated as an -module.
is an integral -algebra if is integral over .

  1. Same as in the above Proposition (Affine morphism), but (respectively ). We call these closed immersions.

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

  1. and , and the induced is surjective.
  2. For all affine open, , and the induced is surjective.
    We call such an a closed immersion.