The lemma below, called the projection formula, is very useful to compute a pushforward. Lemma 51 (Projection formula). Let f :(X, ) (Y, ) be a morphism of ringed OX ! OY spaces. Let be an -module, be a locally free -module of finite rank. Then there F OX E OY is a natural isomorphism

f ( f ⇤ ) f . ⇤ F⌦OX E ' ⇤F⌦OY E

Proof. Since f ⇤(f ) f ⇤f f ⇤ , there is a natural morphism f ⇤f f ⇤ ⇤F⌦E ' ⇤F⌦ E ⇤F⌦ E! f ⇤ .Sincef ⇤ and f are adjoint to each other, there is a corresponding natural F⌦ E ⇤ morphism ' : f f ( f ⇤ ). Note that being ' an isomorphism is a local ⇤F⌦E! ⇤ F⌦ E property, hence, we may replace Y by a suciently small open subset. Since f and the ⇤ tensor product commutes with the direct sum, we may assume furthermore that = Y . 1 E O The conclusion follows from the fact that f ⇤ Y = f Y f 1 X = X . O O ⌦ OY O O

Note that the category of -modules is an abelian category: we can talk about kernel, OX cokernel, and so on. In particular, we are able to consider (cochain) complexes of - OX modules, and do homological algebra. Fiberwisely, vector bundles on give vector OX spaces; and similarly, vector spaces also form an abelian category. However, when we focus on their global nature (= vector bundles themselves, or equivalently, locally free sheaves), they do not form an abelian category.

Example 52. Let X = P1 =ProjC[x, y], and let Z = V (x) X be a closed subset of ⇢ X. The ideal sheaf IZ is indeed locally free:

On an ane open subset Ux Spec C[y/x], which corresponds to the set of 1 ' points [x : y] P where x = 0, we have IZ (Ux)=C[y/x]sincethereisno 2 6 condition. On the other hand, on an ane open subset Uy Spec C[x/y], ' which corresponds to the set of points [x : y] P1 where y = 0, an element of 1 2 6 IZ (Uy) is a regular function on P [1 : 0] which has zero at Z = [0 : 1]. In x \{ } particular, IZ (Uy) C[x/y]. We conclude that IZ is locally free, since ' y · it can be covered by open (ane) subsets such that on IZ is free on each open subset.

We have a natural inclusion I , whose cokernel is isomorphic to which is Z !OX OZ not a locally free sheaf on . OX Hence, it is natural to extend the category of locally free sheaves by adding more objects so that the cokernels of maps of locally free sheaves appear as objects in the enlarged category. In most places, we are in the (locally) Noetherian setting, so that the finite rank case will behave much better. Therefore, it is also reasonable to extend the category of locally free sheaves of finite rank in this manner. In general, there are three finiteness conditions for an A-module M:

(1) finitely generated, there is a surjection A n M 0 for some integer ! ! n;

21 Topic 1 : Sheaf

(2) finitely presented, so that the finite number of generators have only finitely many relations, i.e., there is a right exact sequence A m ! A n M 0 for some integer m, n; ! ! (3) coherent, M is finitely generated, and any homomorphism A n M ! (not necessarily surjective) has the finitely generated kernel.

When A is a Noetherian ring, all these conditions are equivalent. Since we will only work over Noetherian schemes (= schemes which can be covered by a finite number of open ane subschemes Spec Ai where Ai are Noetherian), or even better: ane/projective varieties over a field (of finite dimension), we do not need to distinguish them.

Definition 53. Let (X, ) be a ringed space. An -module is quasi-coherent it OX OX F has a local presentation, that is, for every point x X, there is an open neighborhood 2 U X and a right exact sequence ✓ I J 0 OX |U !OX |U !F|U ! for some (possibly infinite) set I and J. In other words, a quasi-coherent sheaf is locally a cokernel of (locally) free sheaves. An -module is coherent if it is quasi-coherent, and locally coherent, that is, it OX F satisfies the following properties:

(i) is of finite type, i.e., for every point x X, there is an open neigh- F n2 borhood U X and a surjection 0 for some integer ✓ OX |U !F|U ! n; (ii) for any open subset U X, any integer n, and any morphism ' : n ✓ ,thekernelker' is of finite type. OX |U !F|U As we discussed above, the notion becomes much simpler for noetherian schemes. We start by defining the sheaf of modules M on Spec A associated to an A-module M.

Definition 54. Let A be a ring and letfM be an A-module. The sheaf associated to M, denoted by M,isdefinedby D(s) M 7! s f for each s A and the distinguished open subset D(s) Spec A, together with the 2 ✓ natural restriction maps.

In particular, the stalk at p Spec A is isomorphic to the localized module (M) = M , 2 p p and (Spec A, M)=M. We first see that the map M M is functorial: f f 7! Proposition 55. Let A, B be rings, ' : B A be a ring homomorphism, and let f ! f :SpecA Spec B be the corresponding morphism of spectra. Then, ! (1) the map M M gives an exact, fully faithful functor from the category of A-modules 7! to the category of -modules; OSpec A f

22 (2) if M and N are two A-modules, then (M A N)⇠ M N; ⌦ ' ⌦OSpec A (3) if M afamilyofA-modules, then ( M ) M ;f e { i} i ⇠ ' i

(4) for any A-module M, we have f (M) (BM)⇠, wheref BM means M considered as ⇤ ' B-modules; f (5) for any B-module N, we have f (N) (N A) . ⇤ ' ⌦B ⇠ Proof. Note that the map M M givese a functor since a homomorphism of A-modules 7! M N commutes with the localization M N for each s A. It is exact, since ! s ! s 2 it commutes with every localizationf at a prime ideal p A, and the exactness can be ⇢ measured at the stalks. It is also fully faithful:

the functor ⇠ gives a natural map HomA(M,N) Hom (M,N); • ! OSpec A the global section functor (Spec A, ) gives the inverse map Hom (M,N) • fOSpece A ! HomA(M,N). f e In particular, we embed the category of A-modules into the category of -modules. OSpec A All the other statement follow from the definitions and statements in terms of modules.

Definition 56. Let (X, ) be a scheme. An -module is quasi-coherent if X OX OX F can be covered by ane open subsets U =SpecA such that M for some A - i i F|Ui ' i i module M for each i. is coherent if furthermore each M can be taken to be a finitely i F i generated Ai-module. f

Example 57. (1) Let X be any scheme. The structure sheaf is quasi-coherent and OX coherent.

(2) Let X =SpecA be an ane scheme, Y =SpecA/I be the closed subscheme of X defined by an ideal I A, together with a closed immersion i : Y X. The sheaf ✓ ! i Y is a quasi-coherent (and coherent) X -module, which is isomorphic to (^A/I). ⇤O O (3) Let Y be a closed subscheme of a scheme X with the inclusion i : Y, X.The ! sheaf = i 1 is not a quasi-coherent sheaf on Y in general. In fact, it needs OX |Y OX not to be an -module. OY n (4) Let X = A =Speck[x1, ,xn]. Let be the constant sheaf associated to the ··· K group K = k(x , ,x ) which is the function field (= the field of rational functions) 1 ··· n of X.Then is a quasi-coherent -module, but not coherent unless n = 0. K OX Maybe the following proposition seems to be too strong, and in fact, we have to check carefully that their global property comes from their localizations. The idea of proof is almost the same as using the “partition of unity” when we define the structure sheaf of a spectrum, so we will skip its proof. OSpec A

23 Topic 1 : Sheaf

Proposition 58. Let X be a scheme. An -module is quasi-coherent if and only OX F if for every ane open subset U =SpecA of X, there is an A-module M such that M.IfX is noetherian, then is coherent if and only if furthermore we can take F|U ' F M as a finitely generated A-module. f We mentioned several times that the global section functor X (X, ) is left exact. 7! The sheaf cohomology we will see later measures how (X, ) is di↵erent from the exact functor. One distinguished example we seen is a flasque sheaf, that is, a sheaf such that every restriction map is surjective. When is flasque, and we have a short exact F 0 sequence 0 0, then we check that the induced maps on global !F0 !F!F00 ! sections again form a short exact sequence

0 (X, 0) (X, ) (X, 00) 0. ! F ! F ! F ! Naively speaking, a flasque sheaf forces that the global section functor becomes “exact”. Hence, it is the “simplest” sheaf in this viewpoint. We also give a remark that the ane schemes and quasi-coherent sheaves are “simple”, following a similar manner. In other words, they will play a role of basic building blocks.

Proposition 59. Let X =SpecA be an ane scheme, and let 0 0 !F0 !F!F00 ! be a short exact sequence of -modules. Assume that is quasi-coherent. Then the OX F 0 sequence 0 (X, 0) (X, ) (X, 00) 0 ! F ! F ! F ! is exact.

Proof. Only need to check that the last map is surjective. Let s (X, ). Since 00 2 F 00 is surjective, for any point x X there is an open neighborhood D(f) X F!F00 2 ✓ such that s lifts to a section s (D(f)). Choose one and fix f A.Wefirst 00|D(f) 2F 2 claim that a multiple f N s is the image of a global section ⇣ (X, ) for some N>0. 00 2 F Since X =SpecA is quasi-compact, we may cover X with a finitely many D(gi)’s such that for each i, s lifts to a section s (D(g )). On the intersection 00|D(gi) i 2F i D(f) D(g )=D(fg ), both sections s and s map to the same s . \ i i |D(fgi) i|D(fgi) 00|D(fgi) In particular, s := (s s ) (D(fg )) if we regard as a subsheaf of i0 |D(fgi) i|D(fgi) 2F0 i F 0 . F Now is quasi-coherent, there is an A-module M such that = M . In particular, F 0 0 F 0 0 (D(fg )) = M . Hence, s is represented as a quotient t /(fg )ni where t M . 0 i fg0 i i0 i i i 0 F n 2 n Thus a section obtained by a multiplication f i s0 M 0 extends to af section ti/(gi) i i 2 fgi 2 0(D(gi)) = Mg0i . By taking a large enough n and a suitable choice of ti, we may assume F n n n n that f si0 extends to a section ti/gi 0(D(gi)). Let ⇠i := f si+ti/gi (D(gi)). Note 2F n n 2nF n that on D(fgi) we see that the two sections ⇠i = f (si + ti/(fgi) )=f (si + si0 )=f s coincide. On every intersection D(gi) D(gj)=D(gigj), note that the images of ⇠i and ⇠j in 00 n \ n F are same as f s00, since each ti/gi part will map to 0, and si will maps to s00 on D(gi). In other words, ⇠ ⇠ (D(g g )) = M .Since⇠ and ⇠ are equal (to i|D(gigj ) j|D(gigj ) 2F0 i j g0igj i j f ns ) on D(fg g ), it means that f mij (⇠ ⇠ ) = 0 for some m > 0 |D(fgigj ) i j i|D(gigj ) j|D(gigj ) ij

24 when we regard it as an element of the module Mg0igj . We also take m large enough if m m necessary, we may assume that the sections f ⇠i and f ⇠j coincide on the intersection D(g g ). The sections f m⇠ will glue together, and form a global section ⇣ (X, ). i j i 2 F Its image in equals to f n+ms . F 00 00 Now we cover X by a finitely many f , ,f so that s (f ) lifts to a section of over 1 ··· r 00|D i F D(fi) for each i. By the claim, we can find a single integer N>0 and global sections N ⇣i (X, ) such that ⇣i is a lifting of fi s00. Since the open sets D(fi)coverX,in 2 F N N r N particular, the ideal (f1 , ,fr ) = (1) is the unit ideal. We write 1 = i=1 aifi for ··· r some ai A, and we define ⇣ = i=1 ai⇣i (X, ). Then the image of ⇣ in 00 equals r 2 N 2 F P F to aif s = s , as desired. i=1 i 00 00 P WeP check that the category of quasi-coherent (respectively, coherent) -modules is OX good enough to do homological algebra.

Proposition 60. Let X be a noetherian scheme. The kernel, image, and cokernel of any morphism of quasi-coherent (resp., coherent) sheaves are quasi-coherent (resp., coherent). Any extension of quasi-coherent (resp., coherent) sheaves are quasi-coherent (resp., coherent).

Proof. The question is local, that is, it is enough to show that the property holds for (suciently small) open subsets. In particular, we may assume that X =SpecA is ane, and A is noetherian. The kernel, image, and cokernel can be successfully translated in terms of A-modules. Let 0 0 be an exact sequence of -modules, where and are !F0 !F!F00 ! OX F 0 F 00 quasi-coherent (resp., coherent). Since the induced map on global sections 0 M := ! 0 (X, ) M := (X, ) M := (X, ) 0 is exact, we have the following F 0 ! F ! 00 F 00 ! commutative diagram

0 / M 0 / M / M 00 / 0

f f g ✏ ✏ ✏ 0 / / / / 0. F 0 F F 00 Now the 5-lemma implies that the middle arrow is also an isomorphism.

We also leave a statement for pushforwards and pullbacks without a proof:

Proposition 61. Let f : X Y be a morphism of noetherian schemes. ! (1) If is a quasi-coherent (resp., coherent) -module, then the pullback f is a G OY ⇤G quasi-coherent (resp., coherent) -module. OX

(2) If is a quasi-coherent X -module, then the pushforward f is a quasi-coherent F O ⇤F -module. OY (3) When f is projective(e.g., both X and Y are projective varieties), a pushforward f of a coherent X -module is a coherent Y -module. ⇤F O F O

25 Topic 1 : Sheaf

Example 62. When X =SpecA is an ane scheme, we discussed that the ane schemes of the form Y =SpecA/I for some ideal I A are the closed subschemes of X. # ⇢ In particular, the morphism i : X i Y associated to the inclusion i : Y, X is O ! ⇤O ! equivalent to the data consisted of the ring quotient map A A/I and its localizations. ! In general, by patching local ane open pieces, a closed subscheme Y of a scheme X is defined as a morphism i : Y, X of locally ringed spaces such that the morphism # ! # of sheaves i : X i Y is surjective. The kernel of i , denoted by IY , is called O ! ⇤O the ideal sheaf of Y (Check: compare with the definition of an ideal sheaf in previous lectures). When X is a noetherian scheme, any ideal sheaf is coherent. Conversely, any (quasi-)coherent sheaf of ideals on X is the ideal sheaf of a uniquely determined closed subscheme of X.

Example 63. Let X be a scheme, be a coherent sheaf on X. As we seen above, E _ := om ( , X ) is called the dual of . It is easy to check that _ is also a E H OX E O E E coherent sheaf on X. One can check that there is a natural morphism to its double dual ( ) .If E! E_ _ E is a vector bundle of finite rank, it is an isomorphism. In general, it needs not to be an isomorphism. For instance, let X =Speck[x], is a sheaf associated to the k[x]- E module k k[x]/(x). Since there is no nontrivial k[x]-module homomorphism from ' k[x]/(x) k[x], the dual is zero. A coherent sheaf for which the double dual ! E_ E morphism is an isomorphism is called a reflexive sheaf. A locally free sheaf of finite rank is immediately reflexive, however, the converse is not true in general. The canonical map is called the trace map. E⌦E_ !OX Exercise 64 (Geometric Nakayama’s lemma). Let X be a scheme, be a coherent sheaf. F Let U X be an open neighborhood of a point P X, and let s , ,s (U)be ⇢ 2 1 ··· n 2F sections such that their images s , ,s := ( /m ) generate the 1|P ··· n|P 2F|P FP ⌦ OX,P X,P geometric fiber . Show that there is an ane open neighborhood P Spec A U F|P 2 ✓ such that s , ,s generate in the following sense: 1|Spec A ··· n|Spec A F|Spec A (i) s , ,s generate as an A-module; 1|Spec A ··· n|Spec A F|Spec A (ii) for any Q Spec A, s , ,s generate the stalk as an - 2 1 ··· n FQ OX,Q module.

As a special case, we want to associate a (quasi-)coherent sheaf on Proj S from a graded S-module over a graded ring S. As in the case of Spec, it comes from a module structure over a ring, however, we should choose only degree 0 elements as similar as we construct Proj S.

Definition 65. Let S be a graded ring and let M be a graded S-module. We define the sheaf associated to M on Proj S, denoted by M, by assigning M(D+(f)) = M(f), where M M denotes the group of degree 0 elements in the localized module M . (f) ✓ f f One can check that M is a quasi-coherent -module.f If S is noetherianf and M is OProj S finitely generated, then M is coherent. f f 26 On Proj S, we have the notion of “twisting”, which does not change the ring S itself but translate only the grading structure on S.

Definition 66. Let S = S be a graded ring, and let M be any graded S-module. For d any n Z,wedefineitsn-twist M(n)by 2

M(n)d := Md+n.

It is clear that M(n) is a graded S-module.

Definition 67. Let S be a graded ring, X =ProjS, and let n Z be an integer. We 2 define the sheaf (n):=S](n). The sheaf (1) is called the twisting sheaf of Serre. OX OX For any sheaf of X -module, we denote by (n) the twisted sheaf X (n). F O F F⌦OX O Proposition 68. Let S be a graded ring and let X =ProjS.AssumethatS is generated by S1 as an S0-algebra.

(1) The sheaf (n) is an invertible sheaf on X. OX (2) For any graded S-module M, we have M(n) M^(n). In particular, (m) ' OX ⌦ (n) (m + n). OX 'OX f Proof. Since S is generated by S1 as an S0-algebra, X is covered by the distinguished open sets D (f)wheref S . Consider the restriction (n) .SinceD (f) + 2 1 OX |D+(f) + ' Spec S(f), this is isomorphic to S^(n)(f).ItisafreeS(f)-module of rank 1 since there is an isomorphism sending a degree 0 element s S to a degree n element f ns S , 2 (f) 2 f where the later one is a degree 0 element in S(n) . Hence, (n) is locally free of rank (f) OX 1. The second statement follows from the fact that (M N) = M N .Since ⌦S (f) (f) ⌦S(f) (f) D (f),f S covers X, we are done. + 2 1 The twisting operation allows us to recover a graded S-module from a sheaf of - OProj S modules. Note that we made it just by taking the global section of a sheaf of modules on an ane scheme. On the other hand, on Proj, taking the global section only recovers the degree 0 elements, not for the whole graded module. Hence, we have to adjust its grading and collect all the elements of various degrees.

Definition 69. Let S be a graded ring, X =ProjS, and let be an -module. We F OX define the graded S-module associated to to be ( )= n (X, (n)). If s Sd F ⇤ F Z F 2 is a degree d element, it gives a global section s (X, (d))2 in a natural way. Hence, 2 OX L for any t (X, (n)), we define the product s t (X, (n + d)) by taking the image 2 F · 2 F of s t under the natural isomorphism (n) (d) (n + d). ⌦ F ⌦OX 'F Proposition 70. Let A be a ring, S = A[x , ,x ],r 1 be a polynomial ring over 0 ··· r A, and let X =ProjS be a projective r-space over A. Then ( X ) S. ⇤ O '

27 Topic 1 : Sheaf

Proof. Note that a global section t (X, (n)) is equivalent to a finite collection of 2 OX sections t (n)(D (x )) = S(n) which agree on the intersection D (x x ). Since i 2OX + i (xi) + i j the localization maps S S , S S are injective, all these rings are subrings of ! xi xi ! xixj S0 = Sx0 xr . In particular, an element of ( X ) is a finite collection of sections ti Sx ··· ⇤ O 2 i whose images in S0 are same. Since any homogeneous element in S0 can be uniquely m0 mr expressed by a product x x f(x0, ,xr), where mi Z and f is a homogeneous 0 ··· r ··· 2 polynomial not divisible by any of xi. It is an element of Sxi if and only if the exponent r mj is nonnegative for j = i. Hence, ( X )= i=0 Sxi = S. 6 ⇤ O

Caution. If S is a graded ring which is not aT polynomial ring, then ( X ) does not ⇤ O coincide with S in general. Let = M for some graded S-module M. In general, it is not true that the module of F twisted sections ( ) and the original S-module M coincide. However, in most cases, ⇤ F they definef the same sheaf of -modules. OProj S

Proposition 71. Let S be a graded ring, which is finitely generated by S1 as an S0- algebra. Let X =ProjS, and let be a quasi-coherent sheaf on X. Then there is a F natural isomorphism : ( )⇠ . ⇤ F !F d Proof. Let f S1. A section of ( )⇠ on D+(f) is represented by a fraction m/f , 2 ⇤ F where m (X, (d)) for some d 0. We may think 1/f d as a section of ( d), 2 F OX defined on D (f), hence, we obtain a section m f d (D (f), ). This defines . + ⌦ 2 + F Now let be quasi-coherent. To show that is an isomorphism, it is sucient to F identify the module ( )(f) with the sections (D+(f)). ⇤ F F Note that f is a global section of an invertible sheaf (1). Since S is finitely generated OX by elements in S , we may choose finitely many elements f , ,f S such that X 1 0 ··· r 2 1 is covered by open ane subsets D (f ), ,D (f ). Similar as in the ane case, any + 0 ··· + r section t (D (f), ) admits an extension from f nt (D (f), (n)) to a global 2 + F 2 + F section s of (n) for a suciently large n (using the partition of unity given by F⌦OX powers of f ). In other words, a section t (D (f), ) can be expressed as s/f n,a i 2 + F section of ( )⇠ on D+(f). We conclude that (D+(f)) ( )(f) as desired. ⇤ F F ' ⇤ F Corollary 72. Let A be a ring, S = A[x , ,x ], and let X =ProjS.IfY is a closed 0 ··· r subscheme of X = Pr , then there is a homogeneous ideal I S such that Y is a closed A ✓ subscheme determined by I.

Proof. Let IY be the ideal sheaf of Y on X. Since the global section functor is left exact, and the twisting functor is exact, the graded module (IY ) is a submodule of ⇤ ( X )=S. In particular, I := (IY ) is a homogeneous ideal of S. Now I induces ⇤ O ⇤ a closed subscheme V (I) Proj S/I, whose sheaf of ideals will be I.SinceI is ' Y quasi-coherent, we have I IY . In fact, I =(IY ) is the largest ideal in S defining ' ⇤ Y . e e Exercise 73. Let X be a scheme, and A be a quasi-coherent sheaf of -algebras, that OX is, a sheaf of rings which is a quasi-coherent -module simultaneously. OX

28 (1) Show that there is a unique scheme Y and a morphism f : Y X such that for ! every ane open U X, f 1(U) Spec A (U), and for every inclusion V, U of ✓ ' ! open anes of X, the morphism f 1(V ) , f 1(U) corresponds to the restriction ! homomorphism A (U) A (V ). The scheme Y is called the global Spec, or the ! relative Spec, and denoted by Spec A .

(2) Let be a locally free sheaf of rank r on a scheme X. Take A =Sym be the E E_ sheaf associated to the presheaf U Sym( (U)). Show that this is a sheaf of 7! E_ -algebra. OX (3) Check that the morphism Spec A X gives a vector bundle of rank r.Thisis ! called the total space of a locally free sheaf of a finite rank, or the vector bundle E associated to a locally free sheaf . E Exercise 74. Let X be a noetherian scheme, and S be a quasi-coherent sheaf of graded X -algebras, that is, S d 0 Sd where Sd is the homogeneous part of degree O ' d. Assume that S0 = , and S1 is a coherent -module, and that S is locally OX L OX generated by S as an -algebra. Complete the details of the following construction. 1 OX (1) For each open ane open subset U =SpecA X,letS (U) be the graded A- ✓ algebra (U, S U ). We have a natural morphism ⇡U :ProjS (U) U. Check that | ! 1 this is compatible with a further localization, that is, Proj S (U ) ⇡ (U )where f ' U f U =SpecA U for some element f A. f f ✓ 2 1 1 (2) Let U, V be two ane open subsets of X. Check that ⇡ (U V ) ⇡ (U V ). U \ ' V \ In particular, the schemes Proj S (U) glue together and form a scheme Proj S , together with a morphism ⇡ : Proj S X. ! (3) Check that the invertible sheaves (1) on each Proj S (U) are also compatible under O the above construction. They give rise to an invertible sheaf (1) on Proj S . O (4) Let be a locally free sheaf of rank (r + 1) on X. Take S = Symd be the E d 0 E_ symmetric algebra. Check that this is a quasi-coherent sheaf of -algebras. We L OX denote by P( ):=Proj S be the projective space bundle, together with a natural E projection ⇡ : P( ) X.If is free over a suciently small open ane subset E !1 rE U Spec A,then⇡ (U) P =ProjA[x0, ,xr+1]. ' ' A ··· d (5) Let Y be a closed subscheme of X, and let I be the ideal sheaf. Take S = d 0 I , where I 0 = . Check that S is a quasi-coherent sheaf of -algebras. The OX OX L scheme X := Proj S , together with a natural morphism ⇡ : X X, is called I ! the blowing-up of X with respect to I . Compute it when X is an ane n-space, and Y ise the origin of X. e

Remark 75. Using the above constructions, one may define “aneness” and “projec- tiveness” of a morphism of schemes. A morphism ⇡ : X Y is ane if there is an !

29 Topic 1 : Sheaf isomorphism X ' / Spec A ⇡ Y z for some quasi-coherent sheaf A of -algebras. Similarly, A morphism ⇡ : X Y is OX ! projecive if there is an isomorphism

X ' / Proj S ⇡ Y z for some quasi-coherent sheaf S of graded -algebras, finitely generated by the degree OX 1 part.

(N+1) Example 76. Let V = C be a vector space of dimension N + 1 with basis e0, ,eN . Its projectivization PV := Proj Sym(V _) becomes a projective N-space ··· over C. If we denote x0, ,xN be the dual basis, then PV =ProjS,whereS = ··· C[x0, ,xN ] be the polynomial ring with variables x0, ,xN . This space is a scheme- ··· ··· theoretical analogue of the quotient space V 0 /C⇥, which parametrizes the set of lines \{ } passing through the origin 0 V . Hence, a point P =[a0 : : aN ] PV corresponds 2 ··· 2 to a line passing through the origin 0 V and the point (a , ,a ) V .Onemay 2 0 ··· N 2 consider the incidence correspondence

:= ([a0 : : aN ],v) PV V v = c(a0, ,aN ) for some c C . I { ··· 2 ⇥ | ··· 2 } One can check that this is a line bundle, together with the first projection pr1 : PV , I! and is called the tautological line bundle on the projective space. is not the twisting L L sheaf (1) of Serre, since it does not have any nonzero global section: O If then, a section s : PV PV V is a map which sends a point x PV !I⇢ ⇥ 2 to a pair (x, s(x)) where s : PV V is a regular map. Since a projective ! space PV is complete, and thanks to the open mapping theorem, s must be a constant map. In particular, v := s(x) must be a point which lies on every line determined by a choice of a point x PV ,hence,v must be 0. 2 On the other hand, ( 1) = ( (1)) as in the following way: L'O O _

Let Ui = D+(xi). trivializes over Ui by a local trivialization Ui : Ui C 1 I ⇥ ! pr1 (Ui)bysending x x x ([x : : x ],c)= [x : : x ], c 0 , ,c= c i , ,c N . 0 ··· N 0 ··· N x ··· x ··· x ✓ ✓ i i i ◆◆ Hence, the transition function on D (x ) D (x )is: T = x /x .One + i \ + j ij i j can immediately see that this is the multiplicative inverse of the transition function for (1), in particular, O(1) = O( 1). O L' _

30 One can check that if we do not take the symmetric algebra over a dual invertible sheaf in the construction of global Spec, namely, take a global Spec construction applied into Sym (1), then the sheaf of sections we will obtain is not (1), but ( 1). O O O

31 Topic 2 – Line bundles, sections, and divisors

Line bundles, or equivalently, invertible sheaves are one of the most important objects in algebraic geometry. A ringed space (X, ) can be regarded as a triple (X, , ), OX OX OX together with a choice of a line bundle. In the last section, we learned one of the most important line bundles in projective geometry, namely, the twisting sheaf of Serre (1) OX on a projective variety (scheme) X. We want to look X as a triple (X, , (1)). This OX OX idea leads us to projective geometry inside a projective space, in other words, X as a projective variety embedded in a certain projective space. We will see a systematic way to study line bundles by observing its global sections. We will also study divisors, in the both sense of Weil and Cartier, and how they are related to line bundles.

1 Global sections of coherent sheaves

We will call a scheme X is projective over Spec A if it is isomorphic to Proj S for some graded ring S,whereS0 = A and S is finitely generated by S1 as an A-algebra. In particular, S is a quotient of a polynomial ring over A by a homogeneous ideal in S.

Definition 77. Let X is a scheme over a ring A, that means, there is a morphism X Spec A. An invertible sheaf is very ample if there is an immersion i : X Pr ! L ! A for some r, such that i⇤ r (1) . Note that a morphism i : X Z is an immersion OPA 'L ! if it gives an isomorphism of X with an open subscheme of a closed subscheme of Z. We say is generated by global sections, or globally generated if there is a family of F sections s (X, ) such that the images of s generate the stalk as an - i 2 F i FP OX,P module for every point P X. Note that is globally generated if and only if can 2 F F be written as a quotient of a free sheaf, that is, there is a surjection

0. OX !F! M Hence, an A-scheme X is projective if and only if it is “proper” (universally closed), and there is a very ample invertible sheaf on X. L Example 78. (1) Let A be a ring, S = A[x ,x , ,x ], and let X =ProjS be the 0 1 ··· n projective n-space over A. The twisting sheaf of Serre X (1) is very ample since the n O identity map id : X P is an immersion such that id⇤ n (1) = n (1) = X (1). ! A OPA OPA O

(2) Let X be as above. The invertible sheaf X (d),d > 0 is very ample. Here we O r give a naive explanation. We first take the d-uple embedding vd : X, P ,where ! A

32 n+d r = 1. Note that X is isomorphic to a closed subset of Pr , it only suces d A to show that v⇤ r (1) X (d). Since the sections of r (1) are spanned by dOPA 'O OPA coordinate functions of the projective r-space, whose pullbacks on X are monomials of degree d in x ,x , ,x . In particular, they generate (X, (d)). 0 1 ··· n OX (d) (d) (d) Precisely, one can show that X Proj S where Sn := S . Clearly, S is ' nd isomorphic to a quotient of a polynomial ring A[Yi0,i1, ,in ], (i0 + i1 + + in)=d ··· ··· by an ideal I generated by the elements of the form

Yi0,i1, ,in Yj0,j1, ,jn Yk0,k1, ,kn Yl0,l1, ,ln , ··· ··· ··· ··· where (i , ,i )+(j , ,j )=(k , ,k )+(l , ,l ). In particular, the sheaf 0 ··· n 0 ··· n 0 ··· n 0 ··· n (1) on Proj S(d) equals to (d). O OX (3) A very ample invertible sheaf is globally generated. Indeed, by the definition, we r may regard X as an open subscheme of a closed subscheme Z of PA via an immersion i. Let S = A[x , ,x ], and let Z =ProjS/I for some homogeneous ideal I S. 0 ··· r ✓ Since (1) is generated by the images of x , ,x S in S /I , their restrictions OZ 0 ··· r 2 1 1 1 on X also generate the sheaf i . Hence, by definition, the restrictions of images ⇤OProj S of the coordinate functions will generate a very ample invertible sheaf . L (4) Any quasi-coherent sheaf on an ane scheme is globally generated. If = M for F an A-module M, then any set of generators of M as an A-module will generate . F f Exercise 79. Let , be coherent sheaves on X. Suppose that is globally generated, F G F and there is a surjection 0. Show that is also globally generated. F!G! G

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