In this post I’d like to establish some basic properties of schemes and prove a really important theorem – the category of affine schemes is the categorical dual of the category of commutative rings!

**Basic Technical Results**

So, let’s begin:

**Theorem**: Let be a commutative ring and the affine scheme obtained from (here I use to mean the ringed space with the sheaf of rings hiding somewhere in the context!). Let be any point (i.e. any prime ideal of ). Then the stalk is isomorphic to the localisation by a unique isomorphism.

**Proof**: Recall that the stalk at is the colimit where runs over all open sets of . Since we have a base for our topology consisting of the basic open sets , we may as well take the colimit over this instead:

Now for any , there is a natural ring homomorphism sending each element of the form to itself in ; this is well-defined because every element of is a unit in . Clearly this is also compatible with the restriction maps, so by the universal property of the colimit there is a unique ring homomorphism .

Let’s call this morphism . Now every element of is of the form for some , and , and therefore is a surjection (because we can find this element in the localisation and then include it into the stalk). Now suppose that is sent to ; by definition of being zero in a localisation this means there exists such that . But then , and thus its image in the stalk is zero. Hence , so is injective. Thus

This establishes that the affine scheme is not only a ringed topological space but a *locally ringed space*, which means that the stalks at each of its points are local rings.

Here’s a useful consequence of the previous result:

**Lemma**: Let be an integral domain with field of fractions , and let be the point of corresponding to the prime ideal . Then . Furthermore, every nonempty open subset contains and the universal inclusions into the colimit are injective. Hence if then the restriction is injective.

**Proof**: We have . For a principal open subset we have , so there is a natural injection . Now suppose that is a general open subset of and suppose a section is mapped to zero in . Then using the diagram

we know the restriction of to each is zero. Since is a sheaf on , the restriction is a sheaf on and hence the identity sheaf axiom implies in . Thus is injective. The commutativity of the restrictions and inclusions into the stalk implies that each restriction map is injective.

Now I want to tell you two more magical properties of my functor (we can forget about the ringed space structure for a moment).

**Lemma**: Let be a ring homomorphism and the corresponding continuous map (which I wrote about in this post). Then:

- If is surjective then induces a homeomorphism .
- If is a localisation morphism then induces a homeomorphism from onto the subspace

.

**Proof**:

(1): Since as is surjective, the prime ideals of are in bijective correspondence with the prime ideals of , which are themselves in bijective correspondence with the prime ideals of containing . This establishes a bijection of sets ; since is continuous, this is a continuous bijection. Furthermore for an ideal we have

This tells us that the image of a closed set under is closed, so is closed. Therefore is a homeomorphism.

(2): The prime ideals of are in bijection with the prime ideals of containing no elements of because all these elements become units in the localisation . Thus establishes a continuous bijection from to . Now let be an ideal; then

Again this proves that is closed and so it is a homeomorphism.

We can use this to prove a useful lemma I really ought to have proved last time:

**Lemma**: Let be an affine scheme and let . Then the open subset with the induced ringed space structure from is an affine scheme isomorphic to .

**Proof**: By the previous lemma there is an open immersion whose image is . Suppose , and let be the image of in . Then we have

Now the ‘s form a base of open subsets for ; because we have well-defined restriction maps between these basic open sets, we see we obtain an isomorphism of sheaves

It’s easy to check that the resulting isomorphism of ringed spaces is local, giving an isomorphism of schemes.

The next lemma generalises this:

**Lemma**: Let be a scheme (not necessarily affine). Then for any open subset the ringed space is also a scheme.

**Proof**: Since is a scheme there exist open sets , each isomorphic to an affine scheme, such that . Since the principal open sets form a basis for the topology on , each intersection is a union of principal open sets. We’ve just seen each of these is isomorphic to an affine scheme, so it follows that is a scheme.

If is a scheme and is an open subset as above we call an **open subscheme** of ; if it turns out that is an affine scheme then (duh!) we call an **affine open subscheme**.

**Generalised basic open sets**

Now let’s generalise the notion of principal open subset; in the affine case , we took a ring element and looked at the open set of prime ideals not containing it. Now let’s apply our “reverse geometry” that I’ve alluded to before: we want to consider as a function on , and the open set as the set of points on which its value is nonzero. To formalise this, for any prime ideal we set

So the value of the “function” at the point is the value of in the quotient ring . Clearly this isn’t a function in the usual sense because *the rings it takes values in vary from point to point*! But what makes sense – just about the only thing that does – is that we can talk about whether is zero at a point or not. And so on , we see that the “function” is never zero, so we consider this the subset on which the function is defined. Indeed, for any prime ideal we saw above that , so if then and hence the image of in is a unit. It is this fact we’ll use to generalise principal open sets.

We’ve now made a suitable definition of “scheme” built up from affine schemes. In the affine case we picked global sections to define our principal open subsets , but there’s no reason in the general nonaffine case that we can’t do the same for global sections . With this in mind, let’s make the following definition:

**Definition**: Let be a scheme and let be a global section of the structure sheaf. We define a **generalised principal open set** to be

where is the **group of units** of the stalk .

We see that for an affine scheme the ring of global sections is just . Choosing and defining as above just gives because is a unit in all the stalks of elements in , and if is a unit in a stalk of the form then . So this definition really does generalise principal open sets.

**Proposition**: Let be a scheme and . Then is an open subset of .

**Proof**: We’ll show that any point in is contained within an open neighbourhood contained in . Take ; then , so there exists an open neighbourhood and a section with . But equality in the stalk means there is an open subset with such that . Hence is a unit in and hence is a unit in the stalk for any point . Thus . Therefore every point of is an interior point, so is open.

We’ll now construct the inverse for (the image of) in . By the construction above, for each we have an open set and a section with . Suppose that the two open sets and have nonempty intersection. Write and for the corresponding restriction maps. Then in we have equalities

Hence we have

Since is a unit, we can multiply through by its inverse to obtain All this was just a long-winded procedure to prove that the restrictions of the sections agree on the overlaps between open sets and . Therefore by the sheaf axiom all these sections glue together to give a unique section such that . We see that , so we’ve constructed the inverse of in .

Finally, note that since is open there is a restriction map and we’ve seen this sends to a unit in . Therefore it sends the multiplicative set

to units in . By the universal property of localisation there is a unique ring homomorphism

such that , where is the canonical localisation homomorphism.

We’ll see later that under certain “covering” conditions this homomorphism is an isomorphism. For now, let’s press on with something more fundamental.

**Morphisms of Schemes**

A **morphism** of schemes is just a morphism of locally ringed spaces; isomorphisms are defined similarly. A closed/open immersion of schemes is a closed/open immersion of ringed topological spaces. We’ll now show how we can construct scheme morphisms from our continuous maps defined previously – i.e. how to make them work nicely with the ringed space structures to give morphisms of locally ringed spaces.

**Theorem A**: Let be a ring homomorphism and write for the corresponding continuous map of spectra (this isn’t currently a morphism of locally ringed spaces). Then there exists a morphism of schemes

such that .

**Proof**: Let and . It is easy to check that for any we have

Therefore this shows that . Define the ring homomorphism

Then sends every element of the multiplicative set to a unit in , and hence there exists a unique ring homomorphism

such that , where is the canonical localisation map. This is compatible with restrictions of basic open sets so it gives a morphism of sheaves on the base of open sets which extends to a unique morphism of sheaves on :

Moreover let be a prime ideal of . Then the morphism of stalks induced by

is a local homomorphism equal to . Hence is a morphism of locally ringed topological spaces, and hence of schemes. Finally, we clearly have . This concludes the proof

So any ring homomorphism induces a morphism of the affine schemes in the opposite direction, and the original ring homomorphism can be recovered as the component of on global sections.

Therefore we can redefine our functor to be

where is the category of schemes and is the category of locally ringed topological spaces. I’ll now prove the big theorem!

**Lemma**: Let be two schemes and write for the set of scheme morphisms from to . We write for the set of ring homomorphisms from to . Then for all schemes there is a natural transformation

where and are functors .

**Proof**: The functor is self-explanatory, but the functor needs some clarification. It sends every scheme to the set of ring homomorphisms to on the rings of global sections, and every morphism of schemes

to the function given by

The component of at is

But then for a morphism of schemes

Hence is indeed natural.

**BIG THEOREM**: For any ring the natural transformation

is a natural isomorphism (here I’ve implicitly used that ).

**Proof**: Proving that is a natural isomorphism amounts to proving that for any scheme the component is a bijection of sets (seeing as we’ve already proved it’s natural). To do this, we’ll construct an explicit inverse function . To ease notation let and let be a ring homomorphism. For any point we let denote the unique maximal ideal in the local ring . Then the preimage of under the composition

is a prime ideal of , where is the restriction to the stalk. Hence we obtain a map of the underlying point sets

where is just the point set of , considered without topology or ringed space structure. But in fact this map is also continuous with respect to the Zariski topology. To see this, take and consider the principal basic open set . We want to show that is open. We have

where is one of our generalised basic open sets defined above, and is indeed open. Hence is a continuous map on the underlying topological spaces. We now need to define a morphism of sheaves.

For each basic open set define

as the composition

where the first morphism is the obvious map obtained from by localising and the second map is the canonical morphism which we proved the existence of above with regards to generalised basic open sets. We have a commutative diagram

from which it is clear that this defines a morphism of sheaves on the base (as we can do our “localise again” trick to deal with restrictions ) and thus by glueing these all together a uniquely defined morphism of sheaves

It is also clear from the above diagram that the induced homomorphism on global sections is equal to . Now suppose that . The diagram below commutes because it expresses that is a sheaf morphism:

We want to show that which will prove that is a *local homomorphism*. To do this, let and BWOC suppose that is a unit in . Let be the complement of in so that is multiplicative. Then is multiplicative and the image of in must contain only units, because consists only of units (here I use to mean the image of in ). But contains an element which localises to , which we assumed was not a unit in . This is a contradiction, and hence . Thus is a local homomorphism.

Hence is a morphism of locally ringed spaces and hence of schemes. We therefore define the inverse to to be the function

where is constructed as above.

Now we just need to prove that and are mutually inverse. **Theorem A **above gives that

Starting with a morphism of schemes we can apply to get a ring homomorphism on global sections. Applying then sends to the scheme morphism with and

[This notation will freak you out if you don’t stop and think what’s what – remember, so is a prime ideal of .] Clearly if then its image in the localisation is contained within and so applying and recalling that it is a local homomorphism we obtain an element of ; this proves that . Now take ; then so by locality. But then , so . This proves that

Thus this gives us a natural isomorphism. This concludes the proof.

**Translation**: Restricting to an affine scheme, we see that the morphisms between two ring spectra are in natural bijection with the ring homomorphisms between the rings, with the direction of the arrows reversed. Therefore the category of affine schemes is the categorical dual of the category of commutative rings !!!

I think I’ll stop here. Next time I’ll derive some more technical properties of schemes and then try to draw some pretty pictures. After that, I’d like to look at an alternative – and more abstract – way of defining schemes via their *functors of points.*