Let be a topological space and a presheaf. For each , we define the stalk of at to be

where we say that if and only if there exists open such that and .

Let be a ring and an -module. For , , what is ?

There exists a natural isomorphism .

Let us first define a map

If , then acts invertibly on . To see this, consider the pair for some and . We have the equivalence , and and . So we find that

Therefore there exists a unique map

For the inverse map, we send the element with and to its image via the uniquely induced map

After some verification, this map works as the inverse isomorphism.

Let continuous and . If is a presheaf on , then there is an induced map of stalks

In particular, if is a map of ringed spaces, then there is an induced map given by the composition

Let be a ring homomorphism. Consider the induced map (of ringed spaces) on the spectra

Let . Then is the induced map (using the isomorphism in Example $( wt M)_ mfp$Example: )

since elements in are sent to . We note additionally that both and are local rings, with . This is because , and :

Such a homomorphism is called a local homomorphism.

A locally ringed space is a ringed space such that for all , the ring is local with maximal ideal . We define

to be the residue field at .

Since the stalks of a locally ringed space are local rings, they have maximal ideals. In particular, the local rings are always non-zero when the space is non-empty. The structure sheaf “lives everywhere”.

Let be a locally ringed space. Let . Define

Note that if , then is open.

is open.

The do not in general form a basis of the topology on . The locally ringed spaces for which this holds are called quasi-affine.

Let and be the image of in . Then by definition. So since is a local ring, there must exist such that in . We can write for some open neighbourhood of and . The equation means that holds on some open , , and therefore in . So for all .

A map of locally ringed spaces is a map of ringed spaces such that the induced map on stalksinduced map on stalks

is local, i.e., we have a containment . This means there is an induced map on the residue field.

Evaluating a function is like look at its residue in the local ring. A map of locally ringed spaces is like moving functions from one space to another.

For a ring homomorphism , the induced map

is a map of locally ringed spaces.

Let be a ring and a locally ringed space. Then the induced map

is a bijection. If we consider , then

In this sense, is an adjoint.

We produce an inverse to the above map. Given a ring homomorphism , there is an induced map

Let be the prime ideal formed by taking the preimage of . Define the following map (of sets)

It suffices to show that is open for all . We have

is open by Generalisation of distinguished opensGeneralisation of distinguished opens. □

We now need to produce the map on the sheaves of rings

That is, for all , we need to produce a map

This is equivalent to specifying a map such that maps to a unit in :

We take the composition

Let and be rings. Then there is a bijection

In other words, is fully faithful.

A scheme is a locally ringed space that admits an open cover such that there exist rings with as locally ringed spaces for all .

The above CorollaryCorollary tells us that fully faithfulness of means that the gluing of the charts must happen algebraically: they must arise from a map of the underlying rings.