If I and J are graded ideals, then since Ĩ, J are ideal sheaves, there is a canonical homomorphism J OX. It is equal to the composite

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1 Synopsis of material from EGA Chapter II, Sheaf associated to a graded module. (2.5.1). If M is a graded S module, then M (f) is an S (f) module, giving a quasi-coherent sheaf M (f) on Spec(S (f) ) = D + (f) Proj(S) (I, 1.3.4). Proposition (2.5.2). Given a graded S module M, there is a unique quasi-coherent sheaf M of O X modules on X = Proj(S) such that Γ(D + (f), M) = M (f) for every homogeneous f S +, with restriction from D + (f) to D + (fg) given by the canonical homomorphism M (f) M (fg). Definition (2.5.3). M in (2.5.2) is the sheaf associated to the graded S module M. Proposition (2.5.4). M M is an exact functor which commutes with inductive limits and arbitrary direct sums. Proposition (2.5.5). For all p Proj(S), we have M p = M (p). Proposition (2.5.6). Suppose that for every z M and every homogeneous f S +, some power of f annihilates z. Then M = 0. If S 1 generates S as an S 0 -algebra, the converse holds. Proposition (2.5.7). Let f S d, d > 0. For every integer n, the sheaf S(nd) D + (f) is isomorphic to O X D + (f). Corollary (2.5.8). The restriction of S(nd) to the open set U = f S d D + (f) is invertible [i.e., locally free of rank 1 (0, 5.4.1)]. Corollary (2.5.9). If S 1 generates S +, then S(n) is an invertible sheaf on X = Proj(S) for every n. (2.5.10). From now on we use the notation ( ) O X (n) = S(n) and also, for any open U X and sheaf of O X U modules F, ( ) F(n) = F OX U (O(n) U). If S 1 generates S + then the functor F F(n) is exact. (2.5.11). Given graded modules M, N, there are canonical functorial homomorphisms ( ) λ (f) : M (f) S(f) N (f) (M S N) (f), and hence ( ) λ: M OX Ñ (M S N). If I and J are graded ideals, then since Ĩ, J are ideal sheaves, there is a canonical homomorphism Ĩ O X J OX. It is equal to the composite ( ) Ĩ OX J λ (I S J ) O X. 1

2 2 Finally, given three graded modules, there is a canonical homomorphism ( ) M OX Ñ OX P (M S N S P ) given by λ (λ 1) = λ (1 λ). (2.5.12). Similarly, there is a canonical functorial homomorphism of S (f) modules ( ) µ (f) : Hom S (M, N) (f) Hom S(f) (M (f), N (f) ), and hence, using (I, 1.3.8), a canonical homomorphism of O X module sheaves ( ) µ: Hom S (M, N) Hom OX ( M, Ñ). Proposition (2.5.13). Suppose S 1 generates S +. Then λ in ( ) is an isomorphism; and if M is finitely presented (2.1.1), then so is µ in ( ). If I is a graded ideal, then Ĩ M = (IM). Corollary (2.5.14). If S 1 generates S +, then there are canonical isomorphisms ( ) ( ) O X (m) O X (n) = O X (m + n) O X (n) = (O X (1)) n for all integers m, n. Corollary (2.5.15). If S 1 generates S +, then there is a canonical isomorphism M(n) = M(n), for every graded module M. (2.5.16). Under the identifications X = Proj(S) = X = Proj(S ) = X (d) = Proj(S (d) ) of (2.4.7), we have O X (n) = O X (n) and O X (d)(n) = O X (nd). Proposition (2.5.17). The canonical homomorphisms O X (nd) O O X (md) O X ((m+ n)d) restrict to isomorphisms on U = f S d D + (f) Graded S module associated to a sheaf on Proj(S). In this section we assume that S 1 generates the ideal S +, and put X = Proj(S). (2.6.1). By (2.5.9), the sheaf O X (1) is invertible. For any O X module sheaf F we define as in (0, 5.4.6) ( ) Γ (F) = Γ (O X (1), F) = n Z Γ(X, F(n)), the second equality following from ( ). Then (0, 5.4.6) Γ (O X ) is a graded ring and Γ (F) is a graded Γ (O X ) module sheaf. Since O X (n) is locally free, F Γ (F) is a left exact functor. In particular, if I is an ideal sheaf, then Γ (I) is a graded ideal in Γ (O X ).

3 (2.6.2). The map x x/1: M 0 M (f) induces maps M 0 Γ(D + (f), M) for all homogeneous f S +, compatible with restricions, and hence a map ( ) α 0 : M 0 Γ(X, M). Applying this to M(n) and using (2.5.15), we get ( ) α n : M n = M(n) 0 Γ(X, M(n)), and hence a homomorphism of graded abelian groups ( ) α: M Γ ( M). The map α: S Γ (O X ) is a graded ring homomorphism, and ( ) is an S module homomorphism. Proposition (2.6.3). For every f S d (d > 0), the open set D + (f) is the non-vanishing locus of the section α(f) of O X (d) (0, 5.5.2). (2.6.4). Set M = Γ (F), which we may consider as an S module via S Γ (O X ). By (2.6.3), the section α d (f) of O X (d) is invertible on D + (f). Hence there is an S (f) module homomorphism ( ) β (f) : M (f) Γ(D + (f), F) given by z/f n (z D + (f))/(α d (f) D + (f)) n. This is compatible with restriction to D + (fg), giving a canonical homomorphism of sheaves of O X modules ( ) β : Γ (F) F. Proposition (2.6.5). For any graded S module M and sheaf of O X modules F, each of the following maps is the identity: M α Γ ( M) β M, 2.7. Finiteness conditions. Γ (F) α Γ (Γ (F) ) Γ (β) Γ (F) Proposition (2.7.1). (i) If S is a Noetherian graded ring, then X = Proj(S) is a Noetherian scheme. (ii) If S is a finitely-generated graded A-algebra, then X is a scheme of finite type over Y = Spec(A). (2.7.2). Consider two conditions on a graded S module M: (TF) There exists n such that k n M k is a finitely generated S module; (TN) There exists n such that M k = 0 for k n. A graded S module homomorphism u will be called (TN)-injective (resp. (TN)-surjective, (TN)-bijective) if its kernel (resp. cokernel, both) satisfies (TN). By (2.5.4), this implies that ũ is injective (resp. surjective, bijective). 3

4 4 Proposition (2.7.3). Assume that S + is a finitely generated ideal. (i) If M satisfies (TF), then M is an O X module of finite type. (ii) If M satisfies (TF), then M = 0 if and only if M satisfies (TN). Corollary (2.7.4). If S + is finitely generated, then Proj(S) = iff there is an n such that S k = 0 for all k n. Theorem (2.7.5). Let X = Proj(S), where S + is generated by finitely many elements, homogeneous of degree 1. Then for every quasi-coherent sheaf of O X modules F, the canonical homomorphism β : Γ (F) F (2.6.4) is an isomorphism. Remark (2.7.6). If S is Noetherian and S 1 generates S +, then the hypotheses of (2.7.5) hold. Corollary (2.7.7). Under the hypotheses of (2.7.5), every quasi-coherent O X module F is isomorphic to M for some graded S module M. Corollary (2.7.8). Under the hypotheses of (2.7.5), every quasi-coherent O X module F of finite type is isomorphic to Ñ for some finitely generated graded S module N. Corollary (2.7.9). Under the hypotheses of (2.7.5), let F be a quasi-coherent O X module of finite type. Then there exists n 0 such that for all n n 0, F(n) is isomorphic to a quotient of OX k (where k depends on n), i.e., F(n) is generated by finitely many global sections (0, 5.1.1). Corollary (2.7.10). Under the hypotheses of (2.7.5), let F be a quasi-coherent O X module of finite type. Then there exists n 0 such that for all n n 0, F is isomorphic to a quotient of O X ( n) k (where k depends on n). Proposition (2.7.11). Assume the hypotheses of (2.7.5) hold, and let M be a graded S module. (i) The canonical homomorphism α: M Γ ( M) is an isomorphism. (ii) Let G M be a quasi-coherent O X submodule sheaf, and let N M be the preimage of Γ (G) Γ ( M) via α. Then Ñ = G Functorial behavior. (2.8.1). Let φ: S S be a graded ring homomorphism. Let G(φ) denote the complement of V + (φ(s +)) in X = Proj(S), that is, the union of the open sets D + (φ(f )) for homogeneous f S +. Then a φ: Spec(S) Spec(S ) induces a continuous map a φ: G(φ) Proj(S ) such that ( ) a φ 1 (D + (f )) = D + (φ(f )). Let f = φ(f ). Then φ induces φ f : S f S φ(f ) and φ (f) : S (f ) S (φ(f )), hence a morphism a φ (f) : D + (f) D + (f ), which on the underlying space is the restriction of a φ to the open sets in ( ). These are compatible with restriction to D + (fg).

5 Proposition (2.8.2). There is a unique morphism ( a φ, φ): G(φ) Proj(S ) (called the morphism associated to φ and denoted Proj(φ)) whose restriction to each D + (φ(f )) coincides with a φ (f). Corollary (2.8.3). (i) Proj(φ) is an affine morphism. (ii) If ker(φ) is nilpotent (in particular, if φ is injective), then Proj(φ) is dominant. In general a morphism Proj(S) Proj(S ) need not be affine, hence not of the form Proj(φ). An example is Proj(S) Spec(A) = Proj(A[t]) when S is an A-algebra. (2.8.4). Given a third ring S and φ : S S, let φ = φ φ. Then G(φ ) G(φ), and if Φ, Φ, Φ are the associated morphisms, then Φ = Φ (Φ G(φ )). (2.8.5). Suppose S (resp. S ) is a graded A-algebra (resp. A -algebra), and ψ : A A commutes with φ: S S. Then G(φ) and Proj(S ) are schemes over Spec(A) and Spec(A ) respectively, and the the corresponding diagram commutes. (2.8.6). Let M be a graded S module, which we may consider as a graded S module M [φ]. Proposition (2.8.7). There is a canonical functorial isomorphism (M [φ] ) = Φ ( M G(φ)), where Φ = Proj(φ). Proposition (2.8.8). Let M be a graded S module. There is a canonical functorial homomorphism ν : Φ ( M ) (M S S) G(φ). If S 1 generates S +, then ν is an isomorphism. (2.8.9). Let ψ : A A be a ring homomorphism, Ψ: Y = Spec(A) Spec(A ) = Y its associated morphism. Let S be a positively graded A -algebra; then S = S A A is a positively graded A-algebra. We have the ring homomorphism φ: S S, φ(s ) = s 1, and φ(s +) generates S + as an A module, hence G(φ) = Proj(S) = X. Set X = Proj(S ). Further, let M be a graded S module, and set M = M A A = M S S. Proposition (2.8.10). With the notation of (2.8.9), we have X = X Y Y, and the canonical homomorphism ν : Φ ( M ) M (2.8.8) is an isomorphism. Corollary (2.8.11). For all n Z, M(n) is identified with Φ ( M (n)) = M (n) Y O Y. In particular, O X (n) = Φ O X (n) = O X (n) Y O Y. (2.8.12). For f S d (d > 0) and f = φ(f ), the canonical map M (f ) M (f) is identified with M (d) /(f 1)M (d) M (d) /(f 1)M (d) by (2.2.5). (2.8.13). In the setting of (2.8.9), let F be an O X module, and set F = Φ (F ). Then F(n) = Φ (F (n)) by (2.8.11) and (0, 4.3.3). From (0, 4.4.3) we have Γ(ρ): Γ(X, F (n)) Γ(X, F(n)) for all n Z, giving a homomorphism of graded modules Γ (F ) Γ (F). 5

6 6 If S 1 generates S + and F = M, then F = M, where M = M A A, and we have commutative diagrams ( ) M α M M Γ ( M ) α M Γ ( M), ( ) Γ (F ) β F F Γ (F) β F F, in which the vertical arrows are Φ-morphisms. (2.8.14). Given a second graded S module N, we have a canonical homomorphism ( ) Φ ((M S N ) ) (M S N), and a commutative diagram ( ) Φ ( M OX Ñ ) Φ (λ) M OX Ñ λ Φ ((M S N ) ) (M S N), where the top row is the canonical isomorphism (0, 4.3.3). If S 1 generates S +, then S 1 generates S +, the vertical arrows are isomorphisms by (2.5.13), and hence ( ) is an isomorphism. Similarly, there is a commutative diagram Φ (Hom S (M, N ) ) Φ (µ) µ Hom S (M, N) Φ (Hom OX ( M, Ñ )) Hom OX ( M, Ñ), with bottom row given by (0, 4.4.6) and vertical arrows by (2.5.12). (2.8.15). One can replace S 0 and S 0 by Z, or replace S and S by S (d) and S (d) as in (2.4.7), without changing Φ Closed subschemes of Proj(S). (2.9.1). If φ: S S is (TN)-injective (resp. (TN)-surjective, (TN)-bijective) (2.7.2), then (2.8.15) shows that where Φ is concerned we can reduce to the case that φ is actually injective (resp. surjective, bijective).

7 Proposition (2.9.2). Let X = Proj(S). (i) If φ: S S is (TN)-surjective, then the associated morphism Φ is defined on all of Proj(S ) and is a closed immersion into X. If I = ker(φ), the image of Φ is the closed subscheme defined by the ideal sheaf Ĩ. (ii) Suppose further that S + is generated by finitely many elements, homogeneous of degree 1. Let X X be a closed subscheme, defined by a quasi-coherent sheaf of ideals J, and let I S be the preimage of Γ (J ) under α: S Γ (O X ) (2.6.2). Set S = S/I. Then X is the image of the closed immersion Proj(S ) X associated to the canonical surjection S S. Corollary (2.9.3). In (2.9.2 (i)), if S 1 generates S +, then Φ (S(n) ) = S (n) for all n, and Φ (F(n)) = (Φ (F))(n) for every O X module sheaf F. Corollary (2.9.4). In (2.9.2 (ii)), the subscheme X is integral if and only if the ideal I is prime. [ If is clear from (2.4.4). Only if uses (I, 7.4.4).] Corollary (2.9.5). Let S be a graded A-algebra which is generated by S 1, M an A module, and u: M S 1 a surjective A module homomorphism, inducing u: S(M) S, where S(M) is the symmetric algebra of M. Then u induces a closed immersion of Proj(S) into Proj(S(M)). 7

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