## General Schemes

Definition A ringed space is a pair (X, OX) consisting of a topological space X and a sheaf of rings OX defined on X. A morphism of ringed spaces from (Y, OY) to (X, OX) is a pair (f, f#) where f : Y X is a continuous function and f# : OX f*OY is a morphism of sheaves on X.

Remark We will often be careless about notation and talk about a ringed space X, tacitly assuming the existence of a structure sheaf OX. Similarly, we'll usually just write f : Y X for a morphism of ringed spaces, assuming that f# is the name of the corresponding morphism of sheaves.

Remark We have already seen that the spectrum of a commutative ring is a ringed space; that's the motivation for this definition. In addition, every homomorphism of commutative rings induces a morphism of ringed spaces in the other direction. It is important to note, however, that these morphisms have an extra property. To describe it, let φ : R S be a homomorphism of rings, inducing a morphism of ringed spaces f : Y=Spec(S) X=Spec(R). Let y Y be a point corresponding to a prime ideal Q S. Write x=f(y) X for the image, which corresponds to the prime ideal P = φ-1(Q) R. Look at an open neighborhood x U X. Its inverse image V = f-1(U) is an open neighborhood of y in Y. Composing the sheaf maps with the map that computes germs at y gives a system of homomorphisms

germy°f#U : OX(U) f*OY(U) = OY(V) OY,y.

Because these homomorphisms are compatible with the restriction homomorphisms, the univeral property of the direct limit produces a homomorphism

OX,f(y) OY,y.

The construction of these homomorphisms makes sense for arbitrary morphisms for ringed spaces. In the case of a morphism between ring spectra arising from an underlying homomorphism of rings, however, we can identify the stalks and the map more completely; it's just the ring homomorphism

φQ : RP SQ,

where P=φ-1(Q). The special property that this homomorphism of local rings has is that the inverse image of the maximal ideal QSQ is the entire maximal ideal PRP, and not some smaller prime ideal. A homomorphism of local rings with this property is called a local homomorphism. Not every ring homomorphism between local rings is a local homomorphism; consider, for example, the inclusion map Z(p) Q from the localization of the integers at the prime ideal (p) to the rational numbers.

Definition A locally ringed space is a ringed space (X, OX) with the property that all of the stalks OX,x are local rings. A morphism of locally ringed spaces is a morphism of ringed spaces

(f, f#) : (Y, OY) (X, OX)

such that all the induced maps OX,f(y) OY,y are local homomorphisms. An isomorphism of locally ringed spaces is a morphism with a two-sided inverse.

Remark Let X be a locally ringed space and let U X be an open subset. The functor OX on the category TOP(X)op restricts in a natural way to define a ring-valued functor OU = OX|U on TOP(U)op. The stalks of the restriction at points of U must be the same for these two functors, since they only depend on sufficiently small open neighborhoods of the point. Thus, (U, OU) is also a locally ringed space.

Definition A scheme is a locally ringed space X that has an open cover {Ui} such that each Ui is isomorphic as a locally ringed space to Spec(Ri) for some commutative ring Ri. A morphism of schemes is the same as a morphism of locally ringed spaces.

Definition A scheme X is called an affine scheme if it is isomorphic (as a scheme, and hence as a locally ringed space) to Spec(R) for some commutative ring R.

Example Here is the simplest (and smallest) example of a non-affine scheme. Start with a discrete valuation ring R, and let K denote its field of fractions. The affine scheme X=Spec(R) contains two points: The maximal ideal M R corresponds to a closed point x X; the zero ideal in R corresponds to the generic point η X, which is both open and dense. So, X contains exactly three open sets, and the structure sheaf is defined by
 ∅ |→ 0 {η} |→ K X |→ R,

with the obvious restriction maps. From this information, we can easily read off the stalks:
 OX,x = R, OX,η = K.

Now start with two copies of X, and glue them together by identifying the two generic points. The result is a topological space Y containing three points, five open sets, and a structure sheaf
 ∅ |→ 0 {η} |→ K {η, x1} |→ R {η, x2} |→ R Y = {η, x1, x2} |→ R.

(The restriction maps are the obvious ones.) It's clear that this defines a scheme, since the two open subsets with two points are both scheme-theoretically isomorphic to Spec(R). However, it can't be an affine scheme. After all, if Y were isomorphic to the affine scheme Spec(S), then we would have R = OY(Y) = Γ(Y, OY) = S, which is impossible.

Let's now look at a more interesting class of examples. Start with a graded ring S. This means, among other things, that we can decompose S = d0 Sd as a direct sum of abelian groups Sd. Elements of Sd are called homogeneous elements of degree d in S. Multiplication in S also satisfies: for each d, e we have Sd . Se Sd+e. We will write S+ = d>0 Sd for the ideal generated by all homogeneous elements of positive degree.

As a set, define Proj(S) to be the set of homogeneous prime ideals P S that do not contain the ideal S+. Given any homogeneous ideal J S, we define the set Z(J) = { P Proj(S) : P J }. As before, the Z(J) are the closed sets of a topology (called the Zariski topology) on Proj(S).

If P is a homogeneous prime ideal in S, write T for the multiplicative set of all homogeneous elements in S \ P. Because we are only inverting homogeneous elements, the localized ring T-1S has a natural grading. Write S(P) for the set of elements of degree zero in T-1S. Now we can define a structure sheaf on X=Proj(S) by taking the sections on an open subset U to be the set of functions s: U PU S(P) such that for each P U, one has s(P) S(P) and there exists an open neighborhood V of P in U and homogeneous elements a, f in S of the same degree such that for all Q V, one has s(Q) = a/f S(Q).

Phew. Of course, that's exactly the sort of definition we've seen twice before. It's clear that it gives a sheaf of rings, making Proj(S) into a ringed space.

Theorem With these definitions, X=Proj(S) is a scheme.

Proof: One needs to verify these facts:
1. The stalk OX, P is naturally isomorphic to the degree zero localization S(P).
2. For any homogeneous element f S+, let D+(f) = Proj(S) \ Z(f). Each D+(f) is open in Proj(S), and the collection of all D+(f) covers Proj(S).
3. For any homogeneous element f S+, let S(f) be the subring of elements of degree zero in the localized ring Sf. There is an isomorphism of locally ringed spaces between D+(f) and Spec(S(f)).

(i) Any element of the stalk is represented in some neighborhood as a degree zero fraction, and thus gives rise to an element of the localized ring. This construction clearly defines a surjection. The proof of injectivity is then similar to the argument we gave in the affine case. Note that, as a consequence, we can conclude that Proj(S) is a locally ringed space.

(ii) The D+(f) are clearly open. If P is a homogeneous prime ideal that does not contain all of S+, then we can choose an element f S+ \ P. Then P D+(f).

(iii) Given any homogeneous ideal J, define φ(J) = (JSf) S(f). When P is prime and fP, then φ(P) is also prime, thus defining a set-theoretic map D+(f) Spec(S(f)). The properties of localization show that this is a bijection; by looking at properties of containment, one also sees that it is a homeomorphism. Finally, the stalks of the structure sheaves on these spaces are given by the isomorphic local rings S(P) and (S(f))φ(P).

Example Let R be an arbitrary commutative ring, and let S = R[X0, X1, , Xn] be the polynomial ring over R with its usual grading. Define projective n-space over R to be PnR = Proj(S). In particular, if k is an algebraically closed field, one can check that the closed points of this scheme are exactly the same as the points of the variety we previously constructed. In the general case, we have a finite open cover given by the affine open subsets D+(Xi) R[X0, , Xn](Xi). Fixing the index i, we can write xj = Xj/Xi; it is easy to see that the degree zero localization is isomorphic to R[x0, , xi-1, xi+1, , xn]. Thus, D+(Xi) is isomorphic to affine n-space, AnR.

Proposition Let X be an arbitrary scheme and let R be a commutative ring. Then there is a natural bijection:

α : Mor(X, Spec(R)) Hom(R, Γ(X, OX)).

Proof: Given a morphism f : X Y=Spec(R), the sheaf-theoretic part of the morphism induces a map on global section by

α(f) : R = OY(Y) f*OX(Y) = OX(f-1(Y)) = OX(X) = Γ(X, OX).

Thus, to any scheme morphism we can associate a ring homomorphism. To go the other direction, start with a ring homomorphism φ : R Γ(X, OX). Let U = Spec(A) X be any affine open subset. Composing with the restriction map ρXU, we get a ring homomorphism R A. As before, this allows us to construct a morphism of schemes Spec(A) Spec(R). It is clear that if V=Spec(B) is another affine open subset contatined in U, then the chain of ring homomorphisms R A B induces compatible scheme morphisms Spec(B) Spec(A) Spec(R). Thus, we have a morphism defined on each open affine of X, and these morphisms are compatible on affine subsets inside the intersection of two affines. We can glue these together to produce a well-defined morphism on all of X. It is now straightforward to check that the two constructions are inverses, yielding the desired bijection.

Remark I've cheated here, because I've hidden one of the key points: this is where the local homomorphism property of maps between locally ringed spaces needs to be used.

Corollary Every morphism of affine schemes is induced from a ring homomorphism.

Corollary The functor R |→ Spec(R) is a fully faithful contravariant functor from the category RG of commutative rings into the category SCH of schemes; the image is naturally equivalent to the category of affine schemes.

Proof: We already know that we can recover the ring R (up to isomorphism) from the scheme Spec(R) by looking at the global sections of its structure sheaf. The previous proposition says that every morphism between affine schemes arises in a unique way from a ring homomorphism, yielding the result.

Corollary Spec(Z) is a final object in the category of schemes.

Proof: Recall that an object in a category is called final if there is a unique morphism to that object from every other object in the category. Well,

Mor(X, Spec(Z)) = Hom(Z, Γ(X, OX)),

and there is a unique ring homomorphism from Z to any other commutative ring with unity.