## Affine Schemes

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 OX whose ring of sections OX(U) on an open U X consists of those functions

s : U PU RP

that satisfy the following properties:
1. for all P U, one has s(P) RP; and
2. 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 RQ.

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 RP 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.

Lemma Let X=Spec(R) and let f R. There is a natural ring homomorphism αf : Rf OX(D(f)).

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

Lemma The maps αf are injective.

Proof: Suppose s=g/fN Rf and αf(s) = 0. That means, for every prime ideal Q D(f), the fraction g/fN represents zero when considered as an element of the local ring RQ. So, there exist elements hQ (depending on Q, of course), such that hQ g = 0 R. From this relation, we can conclude that the fraction g/fN actually represents 0 as soon as both f and hQ have been inverted; i.e., in RfhQ, and hence as functions on D(f) D(hQ) = D(fhQ). Without loss of generality, we can replace hQ by fhQ. Now we have such a relation for every point Q D(f). So, the collection of standard open sets of the form D(hQ) 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

fM = eQ hQ.

Multiplying this relation through by g, we get

fM g = eQ hQ g = 0 = 0.

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

Proposition The maps αf are isomorphisms.

Proof: Since we've already proved that they are injective, we just need to verify surjectivity. So, take a section s OX(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 fi, gi R and nonnegative integers mi such that:
1. The sets D(fi) cover D(f);
2. The fractions gi/fimi Rfi define sections in OX(D(fi)) that agree when restricted to the intersections D(fi) D(fj) for different i and j.
The first simplification is to observe that we can replace the various exponents mi appearing in the denominators with a single nonnegative integer M. (Just multiply numerator and denominator by the right power of fi, which we can do by finiteness.) Using the fact that D(f) = D(fM), we can then make things even simpler by assuming that M=1. Now the restrictions in question live in OX(D(fi fj)). The elements being restricted all live in the image of αfi fj. Since that map is injective, the second condition reduces to the statement that

[gi / fi] = [gj / fj] Rfi fj.

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

(fi fj)N (gi fj - gj fi) = 0 R.

Rewriting this expression a tad, we get

[gi fiN] [fjN+1] - [gj fjN] [fiN+1] = 0.

Replacing our original pairs of data (fi, gi) by the new pairs (fiN+1, gi fiN), 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

fT = ei fi.

Define

g = ei gi.

Multiply by fj and compute:

g fj = ei gi fj = ei fi gj = fT gj R.

In other words, there is an equality of fractions g/fT = gj/fj in all of the rings Rfj showing that αf(g/fT) 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.

Definition Let F be a sheaf of abelian groups on a topological space X. For each open U X, define Γ(U, F) = F(U).

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.

Corollary Γ(D(f), OX) = Rf.

Proof: This is a simple restatement of the proposition.

Corollary Γ(X, OX) = R.

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

Corollary Let x X = Spec(R) correspond to the prime ideal P R. Then the stalk OX,x is isomorphic to the local ring RP.

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 OX over those D(f) that contain P. However, P D(f) if and only if f P. So, the stalk is the limit of OX(D(f)) = Rf = R[1/f], where we end up inverting precisely the elements of R \ P.

Remark Let R be an integral domain, with generic point η. The stalk of the structure sheaf at the generic point is just the field of fractions of R.

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.

Lemma Given a homomorphism of rings, φ : R S, it induces a function f: Spec(S) Spec(R) on spectra, given by f : P |→ φ-1(P). Moreover, f is a continuous map of topological spaces.

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 OX(U). Write, for the moment, V=f-1(U). By composing the set-theoretic functions that underly the whole structure, we get a function

s°f : V PU RP.

Let's now think about a point Q V. This point represents a prime ideal of S such that P := f(Q) = φ-1(Q). The ring homomorphism φ : R S can be composed with the localization map S SQ to give a natural homomorphism R SQ which just happens to take every element of R \ P to an invertible element of SQ. By the universal property of localizations, we get a naturally induced homomorphism RP SQ. Composing yet again, we get

"φ" °s°f : V QV SQ.

It is (relatively) easy to check that this composite represents a section of OY(f-1(U)).

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.

Definition Let f : Y X be a continuous function between two topological spaces, and let F be a sheaf of abelian groups defined on Y. Define the direct image of F along f to be the sheaf f*F on X whose sections on an open set U X are

f*F(U) = F(f-1(U))

Remark As is so often the case, I've hidden a proposition in the statement of this definition. It's easy to see that this formula defines a presheaf on X; you still need to work a little bit to verify that this presheaf satisfies the sheaf axioms.

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

f# : OX f*OY

by composing any section of the structure sheaf of X with f on one side and with the localizations of φ on the other.

Lemma With this notation, f#X : OX(X) OY(Y=f-1(X)) is nothing other than the original ring homomorphism φ.

Proof: By our earlier computation, we can at least see that f#X is a homomorphism from R to S. Now an element rR is the section that assigns to each prime PR the element rRP. Given any prime QS, the composition defining f#(r) assigns to Q the element φ(r)SQ. But this is exactly the same as the section corresponding to φ(r)S.