Projective Morphisms (Part 1)

Projective morphisms, part 1 We now describe projective morphisms, starting over an affine base. 1 Proj of a graded ring The construction of Proj of a graded ring was assigned as an exercise; let me now recall the result of that exercise. Let S = ⊕∞ n=0 Sn be a graded ring, i.e., a ring such that each Sn is closed under addition, and Sm Sn ⊆ Sm+n . An element of Sn is said to be homogeneous of degree n; the elements of S0 form a subring of S, and each Sn is an S0 -module. (One could also define a graded ring to allow negative degrees; on the few occasions where I’ll need that construction, I’ll call it a graded ring with negative degrees.) Let S + denote the ideal ⊕∞ n=1 Sn . Let Proj S be the set of all homogeneous prime ideals of S not containing S+ . For each positive integer n and each f ∈ Sn , we may view the localization Sf as a graded ring with negative degrees, by placing g/f k in degree m − kn whenever g ∈ Sm . We may then identify the set D(f ) = {p ∈ Proj S : f ∈ / p} with Spec Sf,0 , where Sf,0 is the degree zero subring of Sf . These glue to equip Proj S with the structure of a scheme (note that D(f ) ∩ D(g) = D(f g)). In the case S = A[x0 , . . . , xn ] where each of x0 , . . . , xn is homogeneous of degree 1, this simply produces the projective space PnA . Any morphism S → T of graded rings induces a morphism Proj T → Proj S of schemes. For example, we say an ideal I of S is homogeneous if as abelian groups we have I = ⊕∞ n=0 (I ∩ Sn ). In other words, if we split each element of I into homogeneous components, the components themselves belong to I. Then S/I may also be viewed as a graded ring, the projection S → S/I induces a morphism Proj S/I → Proj S, and this morphism is a closed immersion (as we see immediately by checking on a D(f )). Beware that the scheme Proj S does not by itself determine the graded ring S. For instance, omitting S1 gives another graded ring with the same Proj. We’ll come back to this point later. More generally, if M = ⊕∞ n=−∞ is a graded S-module, i.e., Sm Mn ⊆ Mm+n for all m, n, we can convert M into a quasicoherent sheaf M̃ on Proj S by doing so on each D(f ) (using the degree-zero subset of Mf ) and then glueing. For a converse, see below. 1 2 The sheaf O(1) For S a graded ring, n a nonnegative integer, and M a graded S-module, let M(n) denote the shifted module M(n)i = Mn+i . Let OX (n) be the quasicoherent sheaf on X = Proj S defined by the graded module S(n). In particular, OX (0) = OX . More generally, for any quasicoherent sheaf F of OX -modules, put F (n) = F ⊗OX OX (n). Lemma. Suppose that S is generated by S1 as an S0 -algebra. Then the sheaves OX (n) on Proj S are locally free of rank 1, and OX (m) ⊗OX OX (n) is canonically isomorphic to OX (m + n). Proof. See Hartshorne, Proposition II.5.12. Note: a quasicoherent sheaf F on a locally ringed space X which is locally free of rank 1 is also called an invertible sheaf. That is because there is a unique sheaf F ∨ such that F ⊗OX F ∨ ∼ = OX , the dual of X (exercise). In this case, the dual of OX (n) is in fact OX (−n). This gives us an explanation for what x0 , . . . , xn are on the projective space Proj A[x0 , . . . , xn ]: they are global sections not of the sheaf OX , but rather of the sheaf OX (1). Theorem 1. Suppose that S is finitely generated by S1 as an S0 -algebra. Then each quasicoherent sheaf on Proj S can be written as M̃ for a canonical choice of M. The finitely generated hypothesis is needed to ensure that Proj S is quasicompact; we will impose it pretty consistently hereafter. Proof. Let F be a quasicoherent sheaf on M. Then the module we want is Γ∗ (F ) = ⊕n∈Z Γ(X, F (n)), where F (n) = F ⊗OX OX (n). For the rest of the proof, see Hartshorne, Proposition II.5.15. Beware that this does not imply that S = ⊕∞ n=0 Γ(X, OX (n)) in general. For a stupid example, take S = A[x], in which case the sheaves OX (n) are all free and so Γ(X, OX (n)) = 6 0 even when n < 0. For less stupid examples, see Hartshorne exercise II.5.14. However, the following is true. Lemma. Let n ≥ 1 be an integer. For S = A[x0 , . . . , xn ] with the usual grading (by total degree), we have S = ⊕∞ n=0 Γ(X, OX (n)). Proof. Exercise, or see Hartshorne Proposition II.5.13. 2 3 Closed subschemes of projective spaces Proposition. For n ≥ 1, any closed immersion into PnA is defined by some homogeneous ideal of A[x0 , . . . , xn ]. Proof. In fact, there is a canonical way to pick out the ideal. Let I be the ideal sheaf defining the closed immersion; then Γ∗ (I) is an ideal of Γ∗ (OX ), but we already identified the latter with S = A[x0 , . . . , xn ]. (This identification uses the fact that S is finitely generated by S1 as an S0 -algebra, in order to invoke the previous theorem. In fact, it is part of the proof of that theorem; see Hartshorne Proposition II.5.13.) In general, there may be multiple homogeneous ideals defining the same closed subscheme of PnA . If we start with an ideal I, pass to the closed subscheme, then use the previous proposition to get back, we get the saturation of I, namely, the set of all elements f ∈ A[x0 , . . . , xn ] such that xj0 f, . . . , xjn f ∈ I for some nonnegative integer j. We thus obtain a one-to-one correspondence between closed subschemes of PnA and saturated (equal to their saturation) homogeneous ideals. Corollary. For n ≥ 1, let I be a homogeneous ideal of S = A[x0 , . . . , xn ]. The following conditions are equivalent. (a) The subscheme of PnA defined by I is empty. (b) The saturation of I equals S + . (c) For some n0 , we have Sn ⊆ I for all n ≥ n0 . Proof. We just proved the equivalence of (a) and (b). It is clear that (c) implies (b). Let us check that (b) implies (c). Given (b), each f ∈ {x0 , . . . , xn } has the property that xj0 f, . . . , xjn f ∈ I for some j. In particular, we have xj0 , . . . , xjn ∈ I for some j. This in turn implies S(n+1)(j−1)+1 ⊆ I since each monomial of degree (n + 1)(j − 1) + 1 is divisible by one of xj0 , . . . , xjn (pigeonhole principle!). 4 Projective implies proper We are now ready to complete the proof that f : PnZ → Spec Z is proper. Recall that the missing step was to show that f is universally closed, i.e., for any scheme X, the map PnX → X is closed. It is enough to do this in case X = Spec A is affine. Moreover, we may assume n ≥ 1, as the case n = 0 is stupid (because f is an isomorphism). Let Z be a closed subset of PnX , suppose z ∈ X is not in the image of Z, and put k = κ(z). We must exhibit an open neighborhood U of x in X such that Z ∩ PnU = ∅. Let I = ⊕∞ n=0 In be the saturated homogeneous ideal in S = A[x0 , . . . , xn ] defining Z. Then I ⊗A k defines the empty subscheme of Proj k[x0 , . . . , xn ], but may not be saturated. Nonetheless, for some m, we have that In ⊗A k = Sn ⊗A k, and so (Sn /In ) ⊗A k = 0. 3 Since Sn /In is a finitely generated A-module, by Nakayama’s lemma, (Sn /In ) ⊗A Ap = 0 for p the prime ideal of A defining z. Again since Sn /In is finitely generated, we have (Sn /In ) ⊗A Ag = 0 for some g ∈ A \ p. Then z ∈ D(g) and D(g) is disjoint from the image of Z, proving the claim. 5 What is a projective morphism? Several authors (Hartshorne, Eisenbud-Harris) define a morphism f : Y → X to be projective n if it is the composition of a closed immersion Y → PnX with the projection PX for some nonnegative integer n. This definition is evidently stable under base change, but it is not local on the base! Better to say that such a morphism is globally projective, and to say that f is locally projective if each x ∈ X admits an open neighborhood U such that f : Y ×X U → U is globally projective. This is not such a serious distinction in practice, as globally projective equals locally projective if X is “not too large”. For instance, this occurs if X is itself globally quasiprojective over an affine scheme. (A morphism is globally quasiprojective if it factors as an open immersion followed by a globally projective morphism. Again, this is stable under base change but not local on the base; the version where we force locality on the base is a quasiprojective morphism.) The definition of projective given in EGA is in fact somewhere between locally and glob­ ally projective. More on that later. (Warning: Eisenbud-Harris claim that locally projective and projective are the same. They aren’t, but counterexamples are rather pathological.) 4