Our notes on ample sheaves are essentially just a translation of EGA II 4 2 while the notes on ample families are based on Illusies SGA 6 Expos57524e II 2 3 and parts of Thomason Trobaughs Higher Algebraic Theory of Schemes and of Derived Categorie ID: 35514 Download Pdf

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Our notes on ample sheaves are essentially just a translation of EGA II 4 2 while the notes on ample families are based on Illusies SGA 6 Expos57524e II 2 3 and parts of Thomason Trobaughs Higher Algebraic Theory of Schemes and of Derived Categorie

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Ample Sheaves and Ample Families Daniel Murfet October 5, 2006 In this short note we recall the deﬁnition of an ample sheaf and an ample family of sheaves. Our notes on ample sheaves are essentially just a translation of EGA II 4 2, while the notes on ample families are based on Illusie’s SGA 6 Expose II 2 3 and parts of Thomason & Trobaugh’s “Higher Algebraic -Theory of Schemes and of Derived Categories”. Contents 1 Ample Sheaves 2 Ample Families 1 Ample Sheaves Let be a scheme and an invertible sheaf. Given a global section Γ( X, ) the set germ f / is

open ( MOS Lemma 29 ). The inclusion is aﬃne RAS Lemma 6 ) and in particular if is an aﬃne scheme then is itself aﬃne. Given a sequence of global sections ,...,f the open sets cover if and only if the generate MOS Lemma 32 ). Lemma 1. Let X, be a quasi-compact ringed space and a sheaf of modules of ﬁnite type. If is generated by global sections then it can be generated by a ﬁnite number of global sections. Proof. See ( MOS Deﬁnition 2 ) for the deﬁnition of a sheaf of modules of ﬁnite type . Let be a nonempty family of global sections of

which generate. Let Λ be the set of all ﬁnite subsets of and for each Λ let be the submodule of generated by the belonging to . This is a direct family of submodules and clearly = lim , so it follows from ( MOS Lemma 57 ) that for some . In other words, can be generated by a ﬁnite number of global sections. Lemma 2. Let X, be a quasi-noetherian ringed space, a sheaf of modules on and a direct family of submodules of . If is quasi-compact then U, Γ( U, Proof. By ( COS Proposition 23 ) the functor Γ( U, ) : Mod −→O Mod preserves direct limits, and

since Mod ) and Ab are both grothendieck abelian the direct limit of a direct family of subobjects is just their categorical union (that is, their internal sum). One can also give a direct proof along the lines of ( SGR Lemma 12 ). Given a scheme and a ﬁxed invertible sheaf , we deﬁne ) = for any This notation does not reﬂect the sheaf , but this is unlikely to ever cause any confusion. Note that if is a sheaf of modules of ﬁnite type then the same is true of ) for any Proposition 3. Let be a concentrated scheme and an invertible sheaf on . The following conditions

are equivalent:

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(a) The open sets for all Γ( X, with n > form a basis for (b) There are sections Γ( X, with n > such that the form an aﬃne basis for (c) There are sections Γ( X, with n > such that the form an aﬃne cover for (d) For any quasi-coherent sheaf and n > let denote the submodule of generated by elements of Γ( X, )) . Then is the sum of the submodules for n > (d’) The property for every quasi-coherent sheaf of ideals on (e) For any quasi-coherent sheaf of ﬁnite type, there exists N > such that for all the sheaf is generated by

global sections. (f) For any quasi-coherent sheaf of ﬁnite type, there exist integers n > and k > such that is a quotient of ⊗O (f’) The property for every quasi-coherent sheaf of ideals of ﬁnite type. Proof. ) Given a point ﬁnd an aﬃne open neighborhood and Γ( X, with . The inclusion is aﬃne so must be aﬃne. ( ) is trivial. ) The invertible sheaf is ﬂat, so ) is a submodule of )( Let global sections Γ( X, ) for various 1 be given, such that the are an aﬃne open cover of . Fix one of these global sections Γ( X, ). Any

quasi-coherent sheaf on an aﬃne scheme is generated by its global sections, so can be generated by global sections. Since is concentrated we can apply (H, II.5 14) (see also EGAI 9 1) to see that every element of Γ( ) corresponds under the canonical isomorphism km )( km ) to a section of the form for some m > 0 and Γ( X, km )). In other words, every element of Γ( ) belongs to km km ) for some m > 0. It follows that is the sum of the submodules ,n > 0, as required. ( ) is trivial. ( ) Given an open set and let be the ideal sheaf of the closed set SI Deﬁnition 1 ).

This is quasi-coherent and moreover Supp so X,x . We can therefore ﬁnd n > 0 and Γ( X, )) such that = 0 (meaning germ f / ). Since this can be considered as a global section in Γ( X, ). By construction so the proof is complete (we have since outside ). ) Since is quasi-compact we can ﬁnd a ﬁnite number of Γ( X, ) such that the are aﬃne and cover . By ( MOS Lemma 38 ) we may as well assume all these powers are equal to a single integer k > 0. For each the restriction can by Lemma be generated by a ﬁnite number of global sections ij , and by EGAI

9 1 there is ij 0 such that ij ij can be lifted to a global section ij of km ij ). Once again we can assume the ij do not vary with or , so they are all equal to some ﬁxed 0. The germ of at any point of is a unit, so it follows that the global sections ij Γ( X, km )) generate km ). In fact the argument shows that km ) is generated by global sections for any The sheaves (1) ,..., 1) are also quasi-coherent of ﬁnite type, so we can apply the same process to these sheaves and increase if necessary to work for all of them simultaneously. That is, for the sheaves km km + 1) ,...,

km 1) are generated by global sections. Clearly then ) is generated by global sections for all km as required. ( ) and ( ) are trivial.

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) Suppose that ( ) holds for all quasi-coherent sheaves of ideals of ﬁnite type, and let be a quasi-coherent sheaf of ideals. By ( MOS Corollary 64 ) we can write as the sum of its quasi-coherent submodules of ﬁnite type. It follows that for any n > ) = and hence by Lemma we have β,n . Twisting back we deduce ) = β,n and summing over shows that is the sum of its submodules ). So it suﬃces to prove ) for ideals of

ﬁnite type. But in this case by ( ) there is n > 0 such that ) is generated by a ﬁnite number of global sections, so ) = )( ) = and is trivially the sum of all its submodules ) for n > 0. Deﬁnition 1. Let be a concentrated scheme and an invertible sheaf on . We say that is ample if it satisﬁes the equivalent conditions of Proposition . This property is stable under isomorphism. It is worth checking that the present deﬁnition of an ample sheaf agrees with the one given in Hartshorne, which occurs in our notes as ( PM Deﬁnition 6 ). Lemma 4. Let be a

noetherian scheme and an invertible sheaf on . Then is ample if and only if for every coherent sheaf there is N > such that is generated by global sections for Proof. Suppose that is ample in the sense of Deﬁnition . Any coherent sheaf is quasi- coherent of ﬁnite type because it is locally ﬁnitely presented ( MOS Lemma 34 ), so Proposition ) gives the desired property. Conversely, suppose that is ample in the sense of Hartshorne and let be a quasi-coherent sheaf of ideals. This is trivially coherent, so it is easy to check that condition ( ) of Proposition is

satisﬁed for the sheaf . Hence is ample in the new sense, and the proof is complete. Example 1. Here are some examples of ample invertible sheaves: On an aﬃne scheme any invertible sheaf is ample, because any quasi-coherent sheaf is gen- erated by its global sections. Let where is a ﬁeld. Up to isomorphism the only invertible sheaves on are the twisting sheaves ) for DIV Corollary 47 ), and of these it is precisely the ones with ` > 0 that are ample ( PM Example 2 ). Any quasi-projective scheme over a noetherian ring has an ample invertible sheaf ( BU Lemma 17 ). 2 Ample

Families Those schemes which admit ample invertible sheaves have many good properties. By generalising the notion of an ample sheaf to an ample family of sheaves, we can extend many of these good properties to a wider class of schemes. For the duration of the next proof, if we are given a family of invertible sheaves then we write α,n ) for the sheaf Proposition 5. Let be a concentrated scheme and a nonempty family of invertible sheaves on . The following conditions are equivalent: (a) The open sets for all Γ( X, with ,n > form a basis for (b) There is a family of sections Γ( X,

with ,n > such that the form an aﬃne basis for

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(c) There is a family of sections Γ( X, with ,n > such that the form an aﬃne cover for (d) For any quasi-coherent sheaf and ,n > let α,n denote the submodule of α,n generated by the elements of Γ( X, α,n )) . Then is the sum of the submodules α,n α, for ,n > (d’) The property for every quasi-coherent sheaf of ideals on (e) For any quasi-coherent sheaf of ﬁnite type there exist integers ,k and morphisms α,n such that for every some α,x is surjective. (e’) For any

quasi-coherent sheaf of ﬁnite type there exist integers ,k such that is a quotient of ⊗O (e”) The property for every quasi-coherent sheaf of ideals of ﬁnite type. Proof. The implications ( ) and ( 00 ) are either trivial or follow in exactly the same way as in the proof of Proposition ) Since is quasi-compact we can ﬁnd a ﬁnite number of Γ( X, ) such that the are aﬃne and cover . We can assume the are all equal to a ﬁxed integer t > 0. Fix an index Λ occurring among the and argue as in Proposition part ( ) to see that we can ﬁnd

an integer 0 together with a ﬁnite number of global sections of α,tm ) which generate this sheaf over every open set for which . In other words, we have an integer 0 and a morphism α,tm which is an epimorphism on stalks for every with . Setting tm we have ). Twisting back we have a morphism ⊗O with the same property, and so the induced morphism out of the coproduct over all Λ is an epimorphism, which shows that ) and completes the proof. Deﬁnition 2. Let be a concentrated scheme and a nonempty family of invertible sheaves on . We say that this is an ample

family if it satisﬁes the equivalent conditions of Proposition . Clearly a single invertible sheaf is ample if and only if it is an ample family. Deﬁnition 3. We say that a concentrated scheme is divisorial if there exists an ample family of invertible sheaves on . In particular a scheme admitting an ample invertible sheaf is divisorial. Remark 1. Recall from ( DIV Deﬁnition 12 ) the deﬁnition of a locally factorial scheme . We refer the reader to SGA for the proof of the following result: a separated noetherian scheme which is locally factorial is divisorial. In

particular a regular separated noetherian scheme is divisorial, and therefore so is any nonsingular variety over a ﬁeld. Remark 2. Let be a concentrated scheme and an ample family of invertible sheaves. It follows from Proposition ) that we can ﬁnd a ﬁnite subset ,..., which is also an ample family. The advantage of having a ﬁnite family is that we can cover with open sets on which the are simultaneously free. Hence an arbitrary coproduct of tensor powers of the will be a locally free sheaf (locally ﬁnitely free if the coproduct is ﬁnite). A famous

result of Serre says that on a projective scheme over a noetherian ring, every coherent sheaf is a quotient of a ﬁnite direct sum of twisting sheaves ). This result is crucial in the calculation of cohomology on a projective scheme, because it allows us to reduce to these twisting sheaves which are very well-behaved. On an arbitrary scheme with an ample family of invertible sheaves we have a similar result. Proposition 6. Let be a divisorial scheme. Then (a) For any quasi-coherent sheaf there is an epimorphism with locally free.

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(b) For any quasi-coherent sheaf of

ﬁnite type there is an epimorphism with locally ﬁnitely free. If is a ﬁnite ample family of invertible sheaves then in both cases may be taken to be a coproduct of tensor powers of sheaves in the ample family. Proof. We can by Remark ﬁnd a ﬁnite ample family . If is a quasi-coherent sheaf of ﬁnite type then it is immediate from Proposition ) that is a quotient of a ﬁnite coproduct of (negative) tensor powers of sheaves from the ample family. This is by Remark a locally ﬁnitely free sheaf, which proves ( ). We can by ( MOS Corollary 64 )

write any quasi-coherent sheaf as the sum of all its quasi- coherent submodules of ﬁnite type. If we write each as a quotient of a locally free sheaf using ( ), then it is clear that the canonical morphism is an epimorphism. We can assume that is a coproduct of tensor powers of the sheaves in the ample family, which ensures that is locally free and proves ( ). Corollary 7. If is a divisorial scheme and a quasi-coherent sheaf then there is an epimor- phism with quasi-coherent and ﬂat. If is a scheme then the abelian category Mod ) is generated by the sheaves corre- sponding to

open subsets MRS Corollary 31 ). If is concentrated then Qco ) is grothendieck abelian, and it is generated by a representative set of quasi-coherent sheaves of ﬁ- nite type ( MOS Proposition 66 ). This is a very large and impersonal set of generators, which is improved on on the next result. Lemma 8. Let be a concentrated scheme and an ample family of invertible sheaves. The following set of quasi-coherent sheaves ,n generates Qco Proof. Let be a nonzero morphism of quasi-coherent sheaves. By Proposition there is an epimorphism where is a coproduct of objects from . We deduce a

morphism for some α,n with = 0, as required. Actually when you study the proof of Proposition it is clear that the set ,n< actually generates Qco ), as the non-negative tensor powers are not necessary.

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