2024-09-06

Definition (Adjoint functors)

Two functors and are adjoint if there exists a natural bijection

for all objects and .

Examples

  1. . The left adjoint is and the right adjoint is .
  2. Given a continuous map between topological spaces, we have
  3. Forgetful functor: .
  4. .
  5. For a topological space, the sheafification functor
    is the left adjoint to the forgetful functor
    That is,
    This is also expressed as the following. If is a presheaf and is a sheaf then for any map of presheaves , there exists a unique map of sheaves through which factorises:
presheafFGsheafFshuniversal9!

The sheafification functor is defined to be the following: If is a presheaf, define for each

Some remarks:

  • The collection of systems of coverings is filtered: we can order coverings by inclusion, and coverings have refinements.
  • So that theorem from before applies.

Now, is a presheaf and with a map .

  • is a separated presheaf. #TODO Exercise.
  • If is separated, then is a sheaf.

#TODO Exercise: .
6. Let be a ringed space. Let and be -modules. We can define

#TODO Exercise: Explain why is not a sheaf in general. Hence, we have a sheafified version of Tensor-Hom adjunction

  1. Let be continuous. For a sheaf , We define
    We now have for sheaves and ,

    Composition of the adjoints is the adjoint of the composition: on the left we have , on the right we have .

  2. Let be a topological space and . Let be a presheaf.
    Claim: .
    Proof. Consider the inclusion . We have by Yoneda:
    So . □
  3. Let be a map of ringed spaces. Then we can define a pullback
    We can tensor over , recalling that that map of ringed spaces has the data by adjunction. Therefore
  4. Let be an open subset. Let be a presheaf on . Then we have a unique map
(Fsh)UFU(FU)sh»9!

which is in fact an isomorphism (since we have bijections on stalks). This says that sheafification is “local”. Alternatively, we have

Sh(U)Sh(X)PreSh(U)PreSh(X)j¤forgetj¡1forgetshj¤shj¡1p

Since the purple square commutes, taking their adjoints, the blue square also commutes.

Proposition

Let be an adjoint pair. Then preserves all small colimits and preserves all small limits.

Proof

#TODO Exercise.

Corollary

Let be a topological space. Then has all small limits and colimits. Moreover:

  1. Limits are presheaf limits.
  2. Colimits are sheafifications of presheaf colimits.
  3. Sheafification preserves finite limits.

Fibre products of schemes

Theorem

Finite limits of schemes exist: if and are morphisms of schemes, then there is a commutative diagram of schemes

Proof

First step: is an inclusion of an open subset

Claim: in the following diagram

is Cartesian in locally ringed spaces.
Proof. Because is an inclusion, we just need to show that factors through . But this is just because maps to , and and agree down to . So we obtain that is the fibre product in . Now restricting ourselves to schemes, open subsets of schemes are schemes, so the whole diagram lands inside . So is also the limit in .

Next step: Definition (Open immersion)

An open immersion of locally ringed spaces is a map that factors as , where is an open subset.

Therefore we are done for an open immersion.

Second step: , , affine

The functor

is a right adjoint to . Therefore a colimit in of is sent to a limit of in by Proposition, i.e.,

is a fibre diagram in .

Examples

The following are fibre diagrams:

where is a prime ideal. In fact, we also know that is an embedding, and so represents the fibre of the map over the point .