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1 Presheaves and sheaves “The concept of a sheaf provides a systematic way of keeping track of local algebraic data on a topological ” – Hartshorne, ‘Algebraic ’

Figure 1: A jig-saw puzzle is consisted of small pieces and rules for patching them together

Let us recall our old memory. Let X RN be an n-dimensional topological . ⇢ One e↵ective way to study X is not only considering X itself, but also observing contin- uous functions to R. Since an atlas (U↵,'↵) defines X, a continuous f can be { } seen as a collection of continuous functions on smaller open (U ,f ) .Indeed, { ↵ |U↵ } we may assume that each U↵ is an n-dimensional Euclidean ball, which is a fundamental object in classical , and f is a real-valued . When we are |U↵ interested in smooth , one may substitute X by a smooth manifold and observe smooth functions. It is straightforward to assign local data from additional structures such as: continuous functions, smooth functions, or holomorphic functions, etc. In every cases, such data can be restricted to smaller open sets in a natural way. Conversely, suppose that we have a collection of local data which comes from a single (X, f : X R). In order to make it sense, local pieces should recover the original ! (X, f : X R) by gluing them together. Hence, a right notion for the ‘playground’ ! should contain the following data:

the topology of X, that is, the of all open subsets of X; •

3 Topic 1 : Sheaf

the collection of a local structure we want to observe, for instance, the set of local • continuous functions on each U X; ✓ a notion of “restriction” when V U is a further open subset, for instance, sending • ✓ fU : U R to fU V : V R; ! | ! a notion of “gluing” when we have two local data compatible on their intersection. • It is hard to pin down who invented the sheaf theory. It seems reasonable to the connec- tion between the extension problem (such as analytic continuations). Such an idea was mostly developed by several people working in during 30’s and 40’s, including . The work when he was in a prisoner camp is essential to bring out the modern sheaf theory and the development of spectral sequences. Serre brought the sheaf theory to in 50’s, and then it becomes a fundamental language in this world. Sheaves have various applications in algebraic topology and in algebraic geometry. First, geometric structures like vector bundles on a smooth manifold (or an ) can be expressed in terms of sheaves of modules on the given space. Also, sheaves provide (a hint of) a theory generalizing the usual cohomology such as singular cohomology. Sheaves are designed in a much more general and abstract way, however, in this course, we only deal with a few simpler cases. Definition 1. Let X be a . A presheaf of abelian groups on X is a F contravariant from Top(X)toAb. Note that the Top(X) is consisted of the objects as the open subsets of X and the are the inclusion maps between open subsets. Therefore, is a collection F of the data (1) abelian (U) for each open subset U X; F ✓ (2) a (homo) of abelian groups ⇢ : (U) (V ) for every UV F !F inclusion V U of open subsets of X ✓ satisfying the following conditions (a) ( ) = 0, the empty set assigns a terminal object; F ; (b) ⇢ : (U) (U) is the identity ; UU F !F (c) ⇢ = ⇢ ⇢ where W V U are three open subsets. UW VW UV ✓ ✓ Of course, we may define a presheaf of rings, or of sets just by replacing the words “abelian groups” by “rings”, or “sets”. The abelian groups (U), also denoted by F (U, ), is called the sections of the presheaf over the U. The morphisms F F ⇢ are called restriction maps.Wewrites := ⇢ (s)wheres (U). UV |V UV 2F Roughly speaking, a sheaf is a presheaf whose sections are determined by further local data, as we discussed above. Definition 2. A presheaf is a sheaf if it satisfies the following further conditions: F

4 (3) if U is an open covering of an open subset U X, and if s, t (U) { i} ✓ 2F are sections such that s = t for all i,thens = t; |Ui |Ui (4) if U is an open covering of an open subset U X, and if we have { i} ✓ sections s (U ) for each i such that they agree on the intersections: i 2F i si U U = sj U U for every pair i, j, then there is a s (U) | i\ j | i\ j 2F such that s = s for each i. |Ui i Example 3. Let X be a variety over a field k. For each open subset U X,let (U) ✓ OX be the of regular functions from U to k, and for each V U,let⇢ : (U) ✓ UV OX ! (V ) be the usual restriction map. Then is a sheaf of rings on X, which is called OX O the sheaf of regular functions on X, or the structure sheaf of X (when we regard X as a locally , or a ). Let Y X be a closed subset of X. For each open set U X, we assign U I (U) ✓ ✓ 7! Y where I (U) (U) be the ideal of regular functions vanish at every point of Y U. Y ✓OX \ This forms a sheaf IY , and called the sheaf of ideals of Y , or the of Y . Example 4. One can define the sheaf of continuous functions on any topological space, or the sheaf of di↵erentiable functions on a di↵erentiable manifold, or the sheaf of holo- morphic functions on a . Example 5. Let A be an . We define the A on X as follows. Give A the discrete topology, and for each U X,letA (U) be the group of all ✓ continuous maps from U to A.WehaveA (U) A for every connected open subset U. ' Example 6. Let A be an abelian group, and P X be a point. We define a sheaf 2 i (A) on X as follows: i (A)(U)=A if P U and i (A)(U) = 0 otherwise. i (A)is P P 2 P P called the skyscraper sheaf. Now we will see more definitions to play with sheaves. They are not quite di↵erent from the basic notions in commutative algebra and . Definition 7. Let be a presheaf on X, and let P X be a point. We define the F 2 of at P be the direct of the groups (U) for every open subset U containing FP F F P , via the restriction maps. An element of is represented by a pair U, s ,whereU is an open neighborhood of FP h i P , and s (U). Two pairs U, s and V,t defines the same element in if and only 2F h i h i FP if there is a smaller open subset W U V such that s = t . Hence, the elements ✓ \ |W |W of the stalk are germs of sections at the point P . FP Exercise 8. What are the stalks of the previous examples? Definition 9. Let and are presheaves on X.Amorphism ' : is a collection F G F!G of morphisms of abelian groups '(U): (U) (U) for each open set U X such F !G ✓ that the diagram '(U) (U) / (U) F G

⇢UVF ⇢UVG ✏ '(V ) ✏ (V ) / (V ) F G

5 Topic 1 : Sheaf commutes for any inclusion of open sets V U,where⇢ and ⇢ denotes the restriction ✓ F G maps in and . A morphism of sheaves follows the same definition. An F G is a morphism which admits a two-sided inverse. A morphism ' : of presheaves on X induces a morphism ' : on the F!G P FP !GP stalks, for any P X. The following proposition shows the local nature of a sheaf. 2 Proposition 10. Let ' : be a morphism of sheaves on X. ' is an isomorphism F!G if and only if the induced map ' is an isomorphism for every P X. P 2 Proof. It is enough to show that ' is an isomorphism when all the induced maps on the stalks are . To do this, we claim that each '(U): (U) (U) is an F !G isomorphism where U X is an open subset. Suppose the claim is true. Let be the ✓ 1 inverse defined by (U)='(U) .Since

'( (⇢UVG (t))) = ⇢UVG (t)=⇢UVG ('( (t))) = '(⇢UVF ( (t))) for any t (U) and an open subset V U, we conclude that (⇢G (t)) = ⇢ ( (t)) 2G ✓ UV UVF commutes. First we show that '(U) is injective. Let s (U) be an element such that '(s)=0 2F in (U). For every point P U, the image '(s) of '(s) in the stalk is zero. Since G 2 P GP ' is injective, we have s =0in for every P X. It means, there exist an open P P FP 2 neighborhood P W U such that s = 0. Since W gives a covering of U,we 2 P ✓ |WP { P } conclude that s =0inU from the sheaf property. Next, we show that '(U) is surjective. Let t (U) be a section, and t be its 2G P at P U. Since each ' is an isomorphism, we have s such that ' (s )=t . 2 P P 2FP P P P Let s is represented by a section s(P ) on an open neighborhood P W U.Then P 2 P ✓ '(s(P )) and t are two sections in W whose germs at P coincide. Hence, replacing |WP P W by a smaller neighborhood if necessary, we may assume that '(s(P )) = t in P |WP (WP ). For any P, Q X, '(s(P )) W W = t W W = '(s(Q)) W W coincide on G 2 | P \ Q | P \ Q | P \ Q the intersection. Since ' is injective, we have s(P ) W W = s(Q) W W . Hence, by | P \ Q | P \ Q the sheaf property, there is a section s (U) such that s = s(P ) for each P X. 2F |WP 2 Again by the sheaf property for , we conclude that '(s)=t. G Caution. (1) The proposition only holds for a morphism of sheaves, not for presheaves. (2) It is not true that ' is an isomorphism when and are isomorphic for each FP GP P X. Isomorphisms on the stalks must be induced from a single morphism '. 2 For instance, let X = P, Q with the discrete topology, and let A be an abelian group. { } We take as a presheaf defined by (U)=A for any nonempty open subset U X, F F ⇢ and let as the constant sheaf. Let ' : be a morphism of presheaves defined by G F!G '( P )=id ,'( Q )=id ,'(X)= { } A { } A A where (a)=(a, a) A A. Clearly, they are not isomorphic, but the induced maps A 2 on all the stalks are isomorphisms.

6 We now define the kernel, cokernel, and image of a morphism of (pre)sheaves. Definition 11. Let ' : be a morphism of presheaves. We define the presheaf ker- F!G nel, presheaf cokernel, presheaf image of ' to be the presheaves given by U ker '(U), 7! U coker '(U), and U im '(U), respectively. 7! 7! Caution. Let ' : is a morphism of sheaves. The presheaf kernel of ' is a sheaf, F!G but the presheaf cokernel and image are not sheaves in general. Definition–Theorem 12. Given a presheaf , there is a sheaf + and a morphism F F ✓ : +, with the that for any sheaf and a morphism ' : , F!F G F!G there is a unique morphism : + such that ' = ✓. The pair ( +,✓)isunique F !G F up to unique isomorphism. + is called the sheaf associated to the presheaf , or the F F sheafification of . F Proof. For any open set U X,let +(U) be the set of functions s+ : U ✓ F ! P U FP such that 2 S (i) for each P U, s+(P ) , and 2 2FP (ii) for each P U, there is an open neighborhood P V U and an element 2 2 ✓ t (V ) such that for every Q V , the germ t of t at Q is equal to s+(Q). 2F 2 Q Then + with the natural restriction maps is a sheaf. All the other conditions are F clear, so let us only check the gluing property. Let U be an open covering of an open { i} set U X. Suppose that we have functions s+ : U compatible on the i i P Ui P ✓ ! + 2 F intersections. It is straightforward that there is a function s : U P U P such that + + + S ! 2 F s = s and s (P ) for each P U. Let P Ui U. By the condition (ii), |Ui i 2FP 2 2 ✓ S there is an open neighborhood P V U and an element t (V ) such that the 2 i ✓ i i 2F i germ of t at Q V is equal to s+(Q)=s+ (Q)=s+(Q). Thus we have s+ +(U). i 2 i i |Ui 2F It is also clear that the natural map

s (U) s+ : U 2F 7! ! FP P U ! [2 by s+(P )=s defines a morphism ✓ : +. P F!F Now let be a sheaf and ' : be a morphism (of presheaves). We define G F!G : + as follows. Let U X be an open set, and let s+ +(U). For each F !G ✓ 2F P U, there is an open neighborhood P V U and an element t(P ) (V ) whose 2 2 P ✓ 2F P germ at Q is equal to s+(Q) for every Q V . We have sections '(t(P )) (V ) for 2 P 2G P each P X. Since two germs ' (s+(Q)) and '(t(P )) coincide in for every Q V , 2 Q Q GQ 2 P we replace V by a smaller open subset if necessary, and may assume that '(t(P )) P { } are compatible on the intersections. Since is a sheaf, they glue together and form a G section (s+). This defines a morphism : + . It is straightforward that this F !G equals to '(s)whens+ = ✓(s). The uniqueness of the pair ( +,✓) follows from the universal property. F Remark 13. Note that the induced map on stalks ✓ : + is an isomorphism P FP !FP for each P X.When itself is a sheaf, then is isomorphic to +. 2 F F F

7 Topic 1 : Sheaf

2 More definitions and basic properties

In several literatures, one can find the same process via the notion of espace ´etal´e. Let F be a presheaf on X.Wedefinetheespace ´etal´e as the topological space Sp´e( ) together F with a projection ⇡ :Sp´e( ) X in a following way. As a set, Sp´e( )= be F ! F P X FP the disjoint union of all the stalks. The projection is defined in a trivial way: ⇡2(s):=P ` when s . Let s (U) be a section over an open set U. Then we have an element 2FP 2F s : U Sp´e( )bysendingP s , its germ at P .Since⇡ s = id , one may consider ! F 7! P U s as a “(topological) section” of ⇡ over U. The topology of Sp´e( ) is given by the F strongest topology such that all the maps s : U Sp´e( ) are continuous for any choice ! F of open sets U X and s (U). Via this notion, +(U) is identified with the set of ✓ 2F F continuous sections of Sp´e( )overU. F Definition 14. Let be a sheaf on X, and let s (U) be a section on an open set F 2F U X.Thesupport of s is defined to be the set P U s =0 . We immediately see ✓ { 2 | P 6 } that the set Supp(s) U is closed; if we take a point P U Supp(s) in the complement, ✓ 2 \ then there is an open neighborhood P V U such that s = 0. The of is 2 ✓ |V F defined to be the set P X =0 . { 2 |FP 6 } Definition 15. Let , be sheaves, and ' : be a morphism of sheaves. A F G F!G subsheaf of a sheaf is a sheaf such that for every open set U X, (U) (U) F 0 F ✓ F 0 ✓F is a subgroup, and the restriction maps are induced by those of .Thekernel of ', F denoted by ker ', is the presheaf kernel of ' (which is a sheaf). A morphism ' : F!G is injective if ker ' = 0. The image of ', denoted by im ', is the sheaf associated to the presheaf image of '. By the universal property, there is a natural injective map im ' . A morphism ' : is surjective if im ' = .Thecokernel of ', !G F!G G denoted by coker ', is the sheaf associated to the presheaf cokernel of '.Thesheaf of local morphisms of into (or sheaf hom for short), denoted by om( , ), is defined F G H F G by U Hom( U U ). 7! F| !G|'i 1 'i Asequence i 1 i i+1 of sheaves is exact if ker 'i =im'i 1 at ···!F ! F ! F !··· each stage. Let be a subsheaf of .Thequotient sheaf / is the sheaf associated to the F 0 F F F 0 presheaf U (U)/ (U). 7! F F 0 Remark 16. A morphism ' : of sheaves is injective if and only if the map on F!G sections '(U): (U) (U) is injective for each U X. However, the corresponding F !G ✓ statement for the surjective morphisms is not true. For instance, let X = S1 R2 with the usual topology, be the sheaf of all R-valued ⇢ F continuous functions, and let be the sheaf of all S1 = R/Z-valued continuous functions. G A natural morphism is surjective on any small open neighborhood, however, there F!G 1 is no global R-valued continuous function from S which maps to idS1 . We give another example here. Let X = C 0 be the punctured complex plane, be \{ } F the sheaf of holomorphic functions, and let be the sheaf of nowhere-zero holomorphic G functions. Let ' : be the exponential map '(f)=ef . On a small open F!G neighborhood, we may take the log branch, and hence ' is a surjective morphism of

8 sheaves. However, it is impossible to take a log branch on the whole punctured complex plane, in particular, there is no global f : X C such that ! ef (z)=z for every z X. 2 Definition 17. A sheaf is called a flasque sheaf if the restriction map (U) (V ) F F !F is surjective for every inclusion V U of open subsets. ✓ Let us briefly observe just one major property of flasque sheaves, which will be useful in ' the remaining course. Let 0 0 be a short of sheaves. !F0 !F!F00 ! In general, the (global) section functor (U, ) is left exact but not exact: the map (U) (U) needs not to be surjective. On the other hand, when is flasque, then F !F00 F 0 we have a short exact sequence of abelian groups 0 (U) (U) (U) 0 for !F0 !F !F00 ! any open set U X. A sketch of proof is as follows: ✓

Let s00 00(U). Since the induced map on stalks P : P P00 is surjective, 2F 1 F !F we may take s (s ) for each P X. Let s is represented by a pair P 2 P P00 2 P (V ,s(P )) where P V U be an open neighborhood and s(P ) (V ) P 2 P ✓ 2F P such that (s(P )) = s . The problem is that s(P ) and s(Q) needs not 00|VP to be the same on the intersection V V . On the other hand, followed P \ Q from the left exactness, there is a section s (PQ) (V V ) such that 0 2F0 P \ Q '(s0(PQ)) = s(P ) V V s(Q) V V .Since 0 is flasque, there is a section | P \ Q | P \ Q F s0(Q) 0(VQ) such that s0(Q) V V = s0(PQ) for each Q U.Wemay 2F | P \ Q 2 replace s(Q) (V )bys(Q)+'(s (Q)) for every Q U to correct the 2F Q 0 2 di↵erence.

So far, we discussed sheaves on a single topological space X. We also need to define some operations on sheaves with two topological spaces X and Y , together with a continuous map f : X Y . ! Definition 18. Let f : X Y be a continuous map between two topological spaces, ! and let be a sheaf on X, be a sheaf on Y .Wedefinethedirect image sheaf f F 1 G ⇤F on Y by (f )(V )= (f (V )) for each open set V Y .Wedefinetheinverse image ⇤F F ✓ sheaf f 1 on X to be the sheaf associated to the presheaf U lim (V ). Note G 7! V f(U) G 1 ! ◆ that the stalk is easy to compute: (f )P = f(P ). Also note that the functor f ( ) G G ⇤ is left exact, and the functor f 1( ) is exact. When Z X is a subset, and i : Z X be the inclusion map, then := i 1 is ⇢ ! F|Z F called the restriction of onto Z. F Remark 19. When we deal with a morphism f : X Y of locally ringed spaces, ! we often play with sheaves of -modules. Since f 1 does not have an - OY G OX structure in general, one defines another functor f and the sheaf f which is di↵erent ⇤ ⇤G from f 1 . G We finish by two fundamental propositions.

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