We're on the verge of being able to define an affine scheme. It's more than the set Spec(R); it's even more than the Zariski topology on this set; it's all that plus a structure sheaf.

Definition Let X=Spec(R) be the spectrum of a commutative ring. We define the structure sheaf on X to be the sheaf O_X whose ring of sections O_X(U) on an open U ⊂ X consists of those functions

s : U → ∐_P∈U R_P

that satisfy the following properties:

for all P ∈ U, one has s(P) ∈ R_P; and
for all P ∈ U, there exists an open neighborhood P ∈ V ⊂ U and elements f, g ∈ R such that for all Q ∈ V, one has f ∉Q and s(Q) = g/f ∈ R_Q.

Remark This definition tells us the sections of the structure sheaf over open sets. It leaves us to infer how to define the restriction maps to get a presheaf structure. (Since the sections are actual functions of actual sets, you can just restrict them in the usual way.) It also leaves us to infer that the resulting presheaf actually satisfies the sheaf axioms. This is reasonably straightforward to check: In terms of its logical structure (in particular, the number and placement of quantifiers), the definition of the structure sheaf precisely parallels the definition we used when constructing the sheaf associated to a presheaf. By analogy, one also suspects that the local rings R_P should become isomorphic to the stalks of the structure sheaf. The only way to verify this, however, is to get a better understanding of the ring of sections over some interesting open sets.

Proof: An element of R_f can always be represented as a fraction g/f^N with g ∈ R and N a nonnegative integer. If Q ∈ D(f) = D(f^N), then f^N ∉ Q, and so the fraction g/f^N determines a well-defined element in each of the local rings R_Q. We define α_f(g/f^N) to be the function that assigns to a prime ideal Q the element g/f^N ∈ R_Q.

Proof: Suppose s=g/f^N ∈ R_f and α_f(s) = 0. That means, for every prime ideal Q ∈ D(f), the fraction g/f^N represents zero when considered as an element of the local ring R_Q. So, there exist elements h_Q (depending on Q, of course), such that h_Q g = 0 ∈ R. From this relation, we can conclude that the fraction g/f^N actually represents 0 as soon as both f and h_Q have been inverted; i.e., in R_{fh_Q}, and hence as functions on D(f) ∩ D(h_Q) = D(fh_Q). Without loss of generality, we can replace h_Q by fh_Q. Now we have such a relation for every point Q ∈ D(f). So, the collection of standard open sets of the form D(h_Q) is an open cover of D(f). By the proof of quasicompactness, this means that we have a (finite partition of unity) relation of the form

f^M = ∑ e_Q h_Q.

Multiplying this relation through by g, we get

f^M g = ∑ e_Q h_Q g = ∑ 0 = 0.

But that equation tells us that g/f^N already represents the zero element when viewed in the ring R_f, which was exactly what we needed to show.

Proof: Since we've already proved that they are injective, we just need to verify surjectivity. So, take a section s ∈ O_X(D(f)). By definition, near each point of D(f) there is an open neighborhood on which s can be represented as a quotient of elements of R. Using the facts that the standard open sets form a basis for the topology and that D(f) is quasicompact, we reduce to considering the following situation. We are given a finite collection of elements f_i, g_i ∈ R and nonnegative integers m_i such that:

The sets D(f_i) cover D(f);
The fractions g_i/f_i^m_i ∈ R_{f_i} define sections in O_X(D(f_i)) that agree when restricted to the intersections D(f_i) ∩ D(f_j) for different i and j.

The first simplification is to observe that we can replace the various exponents m_i appearing in the denominators with a single nonnegative integer M. (Just multiply numerator and denominator by the right power of f_i, which we can do by finiteness.) Using the fact that D(f) = D(f^M), we can then make things even simpler by assuming that M=1. Now the restrictions in question live in O_X(D(f_i f_j)). The elements being restricted all live in the image of α_{f_i f_j}. Since that map is injective, the second condition reduces to the statement that

[g_i / f_i] = [g_j / f_j] ∈ R_{f_i f_j}.

Translating this into a condition in R, we learn that there exist nonegative integers N (depending on i and j, but we can ignore that because of finiteness) such that

(f_i f_j)^N (g_i f_j - g_j f_i) = 0 ∈ R.

Rewriting this expression a tad, we get

[g_i f_i^N] [f_j^N+1] - [g_j f_j^N] [f_i^N+1] = 0.

Replacing our original pairs of data (f_i, g_i) by the new pairs (f_i^N+1, g_i f_i^N), we can then assume that N=0 as well. After these simplifications, we can turn our attention to the first condition. Because we have a cover, the proof of quasicompactness gives us a partition of unity relation of the form

f^T = ∑ e_i f_i.

Define

g = ∑ e_i g_i.

Multiply by f_j and compute:

g f_j = ∑ e_i g_i f_j = ∑ e_i f_i g_j = f^T g_j ∈ R.

In other words, there is an equality of fractions g/f^T = g_j/f_j in all of the rings R_{f_j} showing that α_f(g/f^T) must hit our original section s.

Before stating some of the important corollaries of this result, I need to introduce yet another piece of notation.

Remark It seems sort of silly to take a simple notation like F(U) and expand it to the more complicated looking Γ(U, F). Why do we need two names for the same thing? Well, the first notation is particularly useful when we are thinking about a fixed sheaf F, and want to emphasize the fact that it defines a functor of the variable U. The second notation is more useful when we are considering more than one sheaf at a time, and want to emphasize how the group of sections for a fixed U (most often taking U=X itself) changes as the sheaf varies.

Proof: This is a simple restatement of the proposition.

Proof: This is a special case of the previous corollary, since X=D(1).

Proof: Because the standard open sets form a basis for the topology, we can compute the direct limit that defines the stalk by looking at the sections of O_X over those D(f) that contain P. However, P ∈ D(f) if and only if f ∉ P. So, the stalk is the limit of O_X(D(f)) = R_f = R[1/f], where we end up inverting precisely the elements of R \ P.

Now let's look at a pair of rings and a homomorphism φ : R → S between them. Since we're supposed to think of elements of the rings as functions on the corresponding spectra, we can at least hope that φ is related in some sensible way to a map on spectra that goes in the other direction (f : Spec(S) → Spec(R)). After a moment's reflection, we can see what this map should be on the level of sets: Given a prime ideal P in S, its inverse image φ^-1(P) is a prime ideal in R.

Proof: Under the map f, the inverse image f^-1(Z(J)) of the closed set defined by an ideal J ⊂R consists of the set of prime ideals P ⊂ S such that f(P) ∈ Z(J). Equivalently, φ^-1(P) ⊃ J or P ⊃ φ(J). In other words, the inverse image of Z(J) is just the closed set Z(φ(J)).

So far, we've defined a map on spectra, coming from a homomorphism, that reflects the topological part of the structure. What can we expect on the sheaf-theoretic part? Well, let's write X=Spec(R) and Y=Spec(S), and keep the rest of our notation the same, so that φ : R → S is a ring homomorphism that defines a continuous function f : Y → X. Take an open subset U ⊂ X and a section s ∈ O_X(U). Write, for the moment, V=f^-1(U). By composing the set-theoretic functions that underly the whole structure, we get a function

There appear to be some technical difficulties with using this construction to understand how the structure sheaves on the two spectra are related. First, we haven't explained what to do with points P ∈ U that are not in the image of f. Second, we've only described what's going on with sections on open sets of Y of the form V = f^-1(U), and not on general open sets.

Now we can correctly interpret the construction we started describing earlier. Given a ring homomorphism φ : R → S, it not only determines a continuous function on spectra f : Y = Spec(S) → X = Spec(R), but it also defines a morphism of sheaves

Affine Schemes