2024-09-10

Schemes

We want to construct the fibre product for schemes:

We have already done the following two cases:

  • , , , all affine: .
  • When is an inclusion of an open subset, an immersion: .

Aside

Let be a category. Then the Yoneda embedding

preserves all limits (in particular, fibre products). So if we can construct the limit in (which know how to do), and then show that the limit is representable, that will be the limit in .

Hence, it suffices to prove that the presheaf on schemes

is representable.

Definition (Relatively representable)

A morphism of presheaves is relatively representable if for all schemes and morphisms , the fibre product is representable by some :

#TODO2

Example

#TODO2
If , , then Yoneda implies that is of the form for some , and comes from some .

Then

from above. In particular, we don’t yet know that is relatively representable! Except* when is an open immersion.

Definition (Representable by open immersion)

We say that (in the general case) is representable by open immersion if the induced map (from ) is an open immersion of schemes for all .

Of course, if is an open immersion, then is representable by open immersion.

We will just call these open immersions instead of representable by open immersion.

Definition (Zariski open covering)

#TODO2
Let be a presheaf of sets on . A Zariski open covering of is a family of open immersions such that for all schemes and morphisms , the induced family of open immersions is a Zariski open covering of schemes.

To be more precise, the family consists of morphisms of presheaves which are representable by open immersions. Thus, each corresponds to an open immersion , where is a scheme. Therefore it makes sense to talk about being an open cover in the Zariski topology.

Exercise

A Zariski sheaf on that admits a representable open covering is representable. A presheaf is Zariski if is a sheaf for all schemes .

Back to the original problem

Let be an open immersion. Then

is an open immersion. Why? Observe the following diagram:

Because is an open immersion, then its base change is also an open immersion: we just use the square that we get from as shown in the diagram above.

Similarly, if is a Zariski open covering, then is a Zariski open covering.

Next

If is an open immersion, then we have an induced isomorphism

Proof

We know that is a monomorphism of schemes, and therefore is an inclusion. So we have a bijection of sets

Claim

Recall the setup:

We claim the following:

  1. If is an open immersion, then
    is an open immersion.
  2. If is an open covering, the same is true of

Proof

It suffices to write as a composition of open immersions. Since is an open immersion, we first have

But now, is an open immersion, and therefore

is an open immersion. By symmetry in and , the open immersion gives us the open immersion

Put all together, we have the composition

For a covering, the argument applies to each variable just the same.

More

Still with an open covering, we now have that if and are open coverings, then

is an open covering.

Last step

Take

Therefore

is an open covering by the above. Hence, it suffices to show that are representable. But

and so we’re done.

Additive and abelian categories

Definition (Initial and final objects)

Let be a category.

  • is an initial object if for all , there exists a unique map .
  • is a final object if for all , there exists a unique map .
  • is a zero object if it is both initial and final.

Examples

  1. is initial in .
  2. is final in (even in ) because .
  3. The module is the object of for all rings .
  4. Let be a topological space. The presheaf is the initial presheaf but is the initial sheaf. If is a ringed space, is the object in .
  5. If initial/final/zero objects exist, then they are unique up to unique isomorphism.

Definition (Additive category)

An additive category is a category that

  1. Has a object.
  2. Finite products and coproducts exist and the natural map is an isomorphism.
  3. 2. gives a monoid structure on the hom-sets, and we require this to be an abelian group.

Being an additive category is a property, not extra structure. (Although technically saying that the hom-sets form an abelian group is putting extra structure on the hom-sets).