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 homomorphismOX,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 homomorphismφQ : 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.(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.∅ | |→ | 0 |
{η} | |→ | K |
X | |→ | R, |
OX,x = R, | OX,η = K. |
∅ | |→ | 0 |
{η} | |→ | K |
{η, x1} | |→ | R |
{η, x2} | |→ | R |
Y = {η, x1, x2} | |→ | R. |
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 = ⊕d≥0 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 → ∐P∈U 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.
(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 f∉P, 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).
α : Mor(X, Spec(R)) → Hom(R, Γ(X, OX)).
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.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.Comments on this web site should be addressed to the author:
Kevin R. Coombes