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The Proj Construction

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The Proj Construction The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of Projective Space 12 1 Basic Properties Definition 1. Let A be a ring. A graded A-algebra is an A-algebra R which is also a graded ring in such a way that if r ∈ Rd then ar ∈ Rd for all a ∈ A. That is, Rd is an A-submodule of R for all d ≥ 0. Equivalently a graded A-algebra is a morphism of graded rings A −→ R where we grade A by setting A0 = A, An = 0 for n > 0. A morphism of graded A-algebras is a morphism of A-algebras which preserves grade. Equivalently, this is a morphism of graded rings which is also a morphism of A-modules, or a morphism of graded rings R −→ S making the following diagram commute R / S _@@ ? @@ @@ @ A Let S be a graded A-algebra, and let X = P rojS. Then there is a canonical morphism of rings θ : A −→ Γ(X, P rojS) defined by θ(a)(p) = a/1 ∈ S(p) Where as usual, a denotes the element a · 1 of S. By Ex 2.4 this induces a morphism of schemes f : P rojS −→ SpecA. If γ : A −→ S gives the A-algebra structure on S, then an element a ∈ A is mapped to the maximal ideal of OP rojS,p via A −→ Γ(X, P rojS) −→ OP rojS,p if and −1 only if a ∈ γ p. We denote this prime ideal by A ∩ p. Let κp : Af(p) −→ S(p) be defined by κp(a/s) = a/s. Then f : P rojS −→ SpecA f(p) = p ∩ A # fU (t)(p) = κp(t(f(p))) 1 So the projective space over any graded A-algebra is a scheme over A. In particular there is a n canonical morphism of schemes PA −→ SpecA. Notice that if e ∈ S+ is homogenous then the ∼ composite of the open immersion Spec(S(e)) = D+(e) −→ P rojS with f is the morphism of schemes corresponding to the canonical map A −→ S(e) defined by a 7→ a/1. Proposition 1. If S is a graded A-algebra then the morphism f : P rojS −→ SpecA is separated. In particular P rojS is separated for any graded ring S. Proof. The open sets D+(a) for homogenous a ∈ S+ are an open cover of P rojS. Therefore the open sets D+(a) ×A D+(b) are an open cover of P rojS ×A P rojS, and the inverse image of this open set under the diagonal P rojS −→ P rojS ×A P rojS is clearly D+(a) ∩ D+(b) = D+(ab). Since the property of being a closed immersion is local on the base, we reduce to showing that the morphism D+(ab) −→ D+(a) ×A D+(b) is a closed immersion. Since we know that this gives the canonical morphisms D+(ab) −→ D+(a),D+(ab) −→ D+(b) when composed with the projections, the corresponding morphism of A-algebras is the morphism α : S(a) ⊗A S(b) −→ S(ab) induced by the canonical morphisms S(a) −→ S(ab),S(b) −→ S(ab). Since α is clearly surjective, the proof is complete. Proposition 2. If S is a noetherian graded ring then P rojS is a noetherian scheme. Proof. For any homogenous f ∈ S+ the ring S(f) is noetherian (GRM,Corollary 15). Since S is noetherian the ideal S+ admits a finite set of homogenous generators f1, . , fp and the schemes ∼ D+(fi) = SpecS(fi) cover P rojS. Therefore P rojS is noetherian. Proposition 3. If S is a finitely generated graded A-algebra, then the scheme P rojS is of finite type over SpecA. Proof. The hypothesis imply that S0 is a finitely generated A-algebra, and that S is a finitely generated S0-algebra. Therefore S+ is a finitely generated ideal (GRM,Corollary 8). So using the argument of Proposition 2 we reduce to showing that for homogenous f ∈ S+, S(f) is a finitely generated A-algebra. By (GRM,Proposition 14) it suffices to show that S(d) is a finitely generated A-algebra, which follows from (GRM,Lemma 11). Proposition 4. If S is a graded domain with S+ 6= 0 then P rojS is an integral scheme. Proof. By (3.1) it is enough to show that P rojS is reduced and irreducible. By assumption the zero ideal 0 is a homogenous prime ideal, so P rojS has a generic point and is therefore irreducible. The scheme P rojS is covered by open subsets isomorphic to schemes Spec(S(f)) for nonzero homogenous f ∈ S+, and it is clear that S(f) is an integral domain. This shows that P rojS is reduced and therefore integral. n Corollary 5. Let A be a ring and consider the scheme PA for n ≥ 1. Then n (i) The morphism PA −→ SpecA is separated and of finite type. n (ii) If A is noetherian then so is PA. n (iii) If A is an integral domain, then PA is an integral scheme. 2 Functorial Properties Let ϕ : S −→ T be a morphism of graded rings, G(ϕ) the open subset {p ∈ P rojT | p + ϕ(S+)} and induce the following morphism of schemes using Ex 2.14: Φ: G(ϕ) −→ P rojS Φ(p) = ϕ−1p # −1 ΦV (s)(p) = ϕ(p)(s(ϕ p)) 2 Here ϕ(p) : S(ϕ−1p) −→ T(p) is defined by a/s 7→ ϕ(a)/ϕ(s). If ϕ is an isomorphism of graded rings with inverse ψ then clearly G(ϕ) = P rojT , G(ψ) = P rojS and one checks that if Ψ : P rojS −→ P rojT is induced by ψ then ΦΨ = 1 and ΨΦ = 1 so that P rojS =∼ P rojT . More generally, let ϕ : S −→ T and ψ : T −→ Z be morphisms of graded rings. Then G(ψϕ) ⊆ G(ψ). Let Φ: G(ϕ) −→ P rojS and Ψ : G(ψ) −→ P rojT be induced as above, and let Ω be induced by the composite ψϕ. Then ΨG(ψϕ) ⊆ G(ϕ) and it is readily checked that the following diagram of schemes commutes: Ω P rojS o G(ψϕ) cHH v HH vv HH vv Φ HH vvΨ|G(ψϕ) H {vv G(ϕ) If ϕ : S −→ T is a morphism of graded A-algebras for some ring A, then Φ : G(ϕ) −→ P rojS is a morphism of schemes over A, where G(ϕ), P rojS are A-schemes in the canonical way. Also if ϕ : S −→ S is the identity, clearly G(ϕ) = P rojS and Φ is the identity. ∼ For any graded ring S and homogenous f ∈ S+ there is an isomorphism of schemes Spec(S(f)) = D+(f). We claim that this isomorphism is natural with respect to morphisms of graded rings, in the following sense. Lemma 6. Let ϕ : S −→ T be a morphism of graded rings and Φ: G(ϕ) −→ P rojS the induced −1 morphism of schemes. If f ∈ S+ is homogenous then Φ D+(f) = D+(ϕ(f)) and the following diagram commutes: Spec(T(ϕ(f))) / Spec(S(f)) (1) KS KS D+(ϕ(f)) / D+(f) n n where the top morphism corresponds to ϕ(f) : S(f) −→ T(ϕ(f)) defined by s/f 7→ ϕ(s)/ϕ(f) , and the bottom morphism is the restriction of Φ. Proof. By (H, Ex.2.4) it suffices to check that the diagram commutes on global sections of Spec(S(f)), which follows immediately from the definition of Φ and the explicit form of the vertical isomorphisms given in our Section 2.2 notes. Lemma 7. Let S be a graded ring and f, g ∈ S+ homogenous elements. Then the following diagram commutes Spec(S(f)) +3 D+(f) O O Spec(S(fg)) +3 D+(fg) where the vertical morphism on the left corresponds to the ring morphism S(f) −→ S(fg) defined by s/f n 7→ sgn/(fg)n. Proof. By (H, Ex.2.4) it suffices to check that the diagram commutes on global sections of Spec(S(f)). But this is easy to check, using the explicit form of the isomorphisms given in our Section 2.2 notes. P d Let H be a graded ring and suppose f ∈ H0. Then ϕf (h) = d≥0 f hd defines a morphism of graded rings ϕf : H −→ H. If f is a unit with inverse g, then ϕf is an isomorphism with −1 inverse ϕg. If f is a unit, then it is easy to see that for a homogenous prime ideal p, ϕf p = p and the induced morphism (ϕf )(p) : H(p) −→ H(p) is the identity. Therefore ϕf induces the identity morphism P rojH −→ P rojH. This simple observation has the following important consequence 3 Lemma 8. Let ϕ : S −→ T be a morphism of graded rings and suppose f ∈ T0 is a unit. Let ϕf : S −→ T be the following morphism of graded rings X d ϕf (s) = f ϕ(sd) d≥0 Then G(ϕ) = G(ϕf ) and the induced morphisms G(ϕ) −→ P rojS are the same. Proposition 9. Let S be a graded ring and e > 0. The morphism of graded rings ϕ : S[e] −→ S induces an isomorphism of schemes Φ: P rojS −→ P rojS[e] natural in S. Proof. We defined the graded ring S[e] in (GRM,Definition 6).
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