Notes 1—Sheaves MAT4215 — Vår 2015 Sheaves

Warning: Version prone to errors. Version 0.07—lastupdate:3/28/1510:46:24AM The concept of a sheaf was conceived in the German camp for prisoners of war called Oflag XVII where French officers taken captive during the fighting in France in the spring 1940 were imprisoned. Among them was the mathematician and lieutenant Jean Leray. In the camp he gave a course in algebraic topology (!!) during which he introduced some version of the theory of sheaves. He was aimed at calculating the cohomology of a total space of a fibration in terms of invariants of the base and the fibres (and naturally the fibration). To achieve this in addition to the concept of sheaves, he invented spectral sequences. After the war Henri Cartan and Jean Pierre Serre developed the theory further, and finally the theory was brought to the state as we know it today by Alexandre Grothendieck.

Definition of presheaves Let X be a topological space. A presheaf (preknippe) of abelian groups on X consists of two sets of data: Sections over open sets: For each open set U X an abelian group F (U),also ⇤ ✓ written (U, F ). The elements of F (U) are called sections (seksjoner) of F over U.

Restriction maps: For every inclusion V U of open sets in X agrouphomo- ⇤ ✓ morphism ⇢U : F (U) F (V ) subjected to the conditions V ! ⇢U = ⇢V ⇢U W W V for all sequences W V U of inclusions of open sets in X. The maps ⇢U are ✓ ✓ V called restriction maps (restriksjonsavbildninger) and if s is a section over U,the U restriction ⇢ (s) is often written as s V . V | The notion of a presheaf is not confined to presheaves of abelian groups. One may speak about presheaves of sets, rings, vector spaces or whatever you want: Indeed, for any category C one may define presheaves with values in C. The definition goes just like for abelian groups, the only difference being that one requires the “spaces” of sections F (U) over open sets U be objects in the category C and not abelian groups, and of course all the restriction maps are required to be morphisms in C.Onemayphrasethis definition purely in categorical terms by introducing the small category openX of open sets in X whose objects are the open sets, and the morphisms are the inclusion maps between open sets. With that definition up our sleeve, a presheaf with values in the category C is just a contravariant functor

F : open C X ! —1— Notes 1—Sheaves MAT4215 — Vår 2015

We are certainly going to meet sheaves with a lot more structure than mere the structure of abelian groups—e.g., like sheaves of rings—but they will all have an un- derlying abelian group, so we start with those. That said, sheaves of sets play a great role in mathematics, and in algebraic geometry, so we should not completely wipe them under the rug. Most results we establish for sheaves of abelian groups can be proved mutatis mutandis for sheaves of sets as well as long as it can be formulated in terms of sets. Hartshorn includes a third axiom when defining a presheaf. He requires that F ( )= ; 0 , but most other texts do not include that axiom. It follows from the sheaf axioms { } that F ( )=0whenever F is a sheaf, see the comment further down. ; Definition of sheaves A sheaf (knippe) of abelian groups F on X is a presheaf of abelian groups on X satisfying the two following requirements:

⇤ Locality axiom: Let Ui i I be an open cover of the open set U and let s be a { } 2 section of F over U.Iftherestrictionsofs to Ui all vanish, i.e., one has s U =0 | i for all i,thens =0.

⇤ Gluing axiom: Let Ui i I be an open cover of the open set U.Givensectionssi { } 2 over Ui matching on the intersections Uij = Ui Uj, i.e., si U U = sj U U ,then \ | i\ j | i\ j there is a section s of F over U satisfying s U = si. | i The locality condition says that sections are determined locally; that is, if two sections coincide on all open sets of a covering, they are equal. The gluing axiom says that locally given sections (i.e., si over Ui) matching where they can match (i.e., over the intersections Ui Uj)canbepatchedtogethertoaglobalsection(i.e., asection \ over U = i Ui). This of course applies to functions of any type on X,andindeed, sheaves of functions are simple and familiar examples of sheaves. S An alternative formulation of the sheaf axioms There is a nice alternative way of formulating the two sheaf axioms. They are equivalent to the following sequences all being exact:

↵ ⇢ 0 / F (U) / F (Ui) / F (Ui Uj) (1) i i,j \ Q Q where as usual Ui i I is an open cover of the open set U,andthemaps↵ and ⇢ { } 2 are defined by ↵(s)=(s U )i I and ⇢((si)i I )=(si U U sj U U )i, j, and where the | i 2 2 | i\ j | i\ j indices of the second product run over I I. The locality axiom for the cover Ui is ⇥ { } equivalent to ↵ being injective and the gluing axiom to Im ↵ =Ker⇢. Of course in the definition of the map ⇢ where we take difference between the rest- rictions of the sections si working with sheaves of abelian groups is essential. However, when working with sheaves not being sheaves of abelian groups, e.g., sheaves of sets,

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the map ⇢ must be replaced with two maps, ⇢01 and ⇢2 where

⇢1((si)i I )=(si U U )i, j 2 | i\ j ⇢2((si)i I )=(sj U U )i, j. 2 | i\ j We underline that the indices i, j run through the product I I.Inthiscasethe ⇥ sequence ( 1)takestheform

⇢1 ↵ / / F (U) i F (Ui) / i,j F (Ui Uj), (2) ⇢2 \ Q Q and to say that it is exact amounts to saying that ↵ is injective and that if ⇢1()=⇢2() if and only if lies in the image of ↵.Onesaysthat↵ is the equalizer (ekvilisatoren) of the the to maps ⇢1 and ⇢2. First and familiar examples Take X = R and let C be the sheaf whose sections over an open set U is the ring of continuous real valued functions on U,andthere- striction maps are just the good old restriction of functions. Then C is a sheaf of rings (functions can be added and multiplied), both the sheaves axioms are obviously satisfied. You should convince yourself that this is true. For a second familiar example let X C be any open set. On X one has the sheaf ✓ OX of holomorphic functions. That is, for any open U X the sections (U, OX ) is ✓ the ring of holomorphic (i.e., complex analytic) functions on U.Onecanrelaxthe condition of holomorphy to get the larger sheaf KX of meromorphic function in X.It contains OX ,andthesectionsoveranopenU are the meromorphic functions on U. In a similar way, one can get smaller sheafs contained in OX by imposing vanishing conditions on the functions. For example if a X is any point, one has the sheaf 2 denoted OX ( a) of holomorphic functions vanishing at a.Asthenameindicatesthe sections of OX ( a) over U are holomorphic functions in U,andifa U,onerequires 2 additionally that they should vanish at a. Convince yourself that this indeed is a sheaf. A third example, which at least is familiar to students having followed the course Algebraic Geometry I, is highly relevant for us. This time X is a variety over the algebraically closed field k,andOX is the sheaf whose sections over the Zariski-open set U are the regular functions in U. Convince yourself that this indeed is a sheaf of rings. The empty set again The empty set is always around, and to develop a theory properly one has to deal with it. This is not always clear how to do, and from time to time loud arguments among mathematicians erupt. For critics of academia this is gefundenes Fressen: Loud arguments about nothing! It follows from the sheaf axioms that F ( )=0when F is an abelian sheaf. May be the best thing is just to believe ; it, but anyhow here here follows the argument: Use the sequence above for the empty covering of the empty set (You don’t need any open sets to cover nothing), and then use that the empty product is 0! Problem ￿.￿. If F is a sheaf of sets, what is your guess at F ( ) being? X ;

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Problem ￿.￿. Let X C be an open set, and assume a1,...,an are distinct points in ✓ X and n1,...,nr be natural numbers. Define (U, F ) to be the set of those function f (U, OX ) that vanishes to an order at least ni at ai when ai U.ShowthatF is 2 2 asheafofrings. X

Problem ￿.￿. Let X C be an open set, and assume a1,...,an are distinct points in X ✓ and n1,...,nr be natural numbers. Define (U, F ) to be the set of those meromorphic functions f (U, KX ) holomorphic away from the ai’s and having a pole order 2 bounded by ni at ai.ShowthatF is a sheaf. Is it a sheaf of rings. X Problem ￿.￿. Let X = R.Ifonedefines(U, B) to be the ring of continuous and bounded functions on the open set U R.ShowthatB is a presheaf but not a sheaf. ✓ X

Maps between sheaves A map (or morphism)betweentopresheaves,say: F G,isafamilyofmaps ! between sections of F and G over open sets compatible with the restrictions. The data defining the map consists of one map U : F (U) G(U) for each open set, and the ! compatibility requirement reads: (s )= (s) V |V U |V holding for any two open sets V U and every section s of F over U.Ifthesheavesare ✓ sheaves of abelian groups, the maps are to be homomorphism, and for sheaf of rings they must be ring homomorphisms, etc. For fans of diagrams: The requirement amounts to the following diagrams—one for each inclusion U V of open sets—being commutative: ✓ F (U) U / G(V )

U U ⇢V ⇢V ✏ ✏ F (V ) / G(V ) V For fans of categories: If one considers the sheaves F and G as contravariant functors on the category openX , a map between them is what is called a natural transformation (naturlig transformasjon) between the two functors. In this way the abelian sheaves on X form a category AbShX whose objects are the abelian sheaves on X and the morphism the maps between them, the composition of two maps of sheaves is defined in the obvious way as the composition of the maps on sections. As usual, a map between F and G is an isomorphism (isomorfi) if it has a two-sided inverse, i.e., amap : G F such that =idG and =idF . ! Familiar examples In the case X = R let C r be the subsheaf of C consisting of functions being r times continuously differentiable (check that this a subsheaf ). The r r 1 differential operator D = d/dx defines a map D : C C . ! In the same vain, the differential operator is a map D : OX OX , where as pre- ! viously X C is an open set. ✓ —4— Notes 1—Sheaves MAT4215 — Vår 2015

Problem ￿.￿. Let X C be an open set. Show that if one defines (U, A )= f ✓ { 2 (U, OX ) Df =0 one obtains a subsheaf A of OX .ShowthatifU is a connected | } open subset of X,oneshas(U, A )=C.Ingeneralforanotnecessarilyconnected U (U, )= ⇡ U set ,showthat A ⇡0U C where the product is taken over the set 0 of connected components of U. X Q Stalks and germs Given a presheaf F of abelian groups on X.Witheverypointx F there is 2 associated an abelian group Fx called the stalk (stilk) of F at x. The elements of Fx are called germs of sections (seksjonskimer eller bare kimer) near x.

The definition goes as follows: We begin with the disjoint union x U F (U) whose elements we think of as pairs (s, U) where U is an open neighbourhood2 of x and s is ` asectionofF over U.Wewanttoidentifysectionsthatcoincidenearx;thatis,we declare (s, U) and (s0,U0) to be equivalent, and write (s, U) (s0,U0),ifthereisan ⇠ open V U U 0 with x V such that s and s0 coincide on V ;thatisonehas ✓ \ 2

s = s0 . |V |V This is clearly a reflexive and symmetric relation. And it is transitive as well. Indeed, if (s, U) (s0,U0) and (s0,U0) (s00,U00),onemayfindopenneigbourhoodsV U U 0 ⇠ ⇠ ✓ \ and V 0 U 0 U 00 of x over which s and s0,respectivelys0 and s00, coincide. Clearly s ✓ \ and s00 then coincide over the intersection V 0 V ,andtherelation is an equivalence \ ⇠ relation. The stalk Fx at x is by definition the set of equivalence classes. In case F is an abelian sheaf, the stalks Fx are all abelian groups. This is not a priori obvious, since sections over different open sets can not ve added, however if (s, U) and (s0,U0) are given, the restrictions s V and s0 V to any open V U U 0 can | | ✓ \ be added, and this suffices to define an abelian group structure on the stalks.

Problem ￿.￿. Fill in all the details in the argument for Fx being an abelian group just given. X

The germ of a section For any neighbourhood U of x there is a map F (U) Fx ! sending a section s to the equivalence class where the pair (s, U) belongs. This class is called the germ (kimen) of s at x,andacommonnotationforitiss(x). The map is a homomorphism of abelian groups (rings,. . . ) as one easily verifies. Clearly one has s(x)=s V (x) for any other open neighbourhood V of x contained in U—or expressed | in the lingo of diagrams—the following diagram commutes:

F (U) / F < x U ⇢V ✏ F (V ).

When working with sheaves and stalks, it is important to remember the three following properties, the third one is easily deduced from the two first.

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⇤ The germ s(x) of s vanishes if and only s vanishes on a neighbourhood of x, i.e., there is an open neighbourhood U of x with s U =0 | ⇤ All elements of the stalk Fx are germs, i.e., of the shape s(x) for some section s over an open neighbourhood of x.

⇤ The abelian sheaf F is the zero sheaf if and only if all stalks are zero, i.e., Fx =0 for all x X. 2 Functoriality Amap: F G between two presheaves F and G induces for every ! point x X amapx : Fx Gx between the stalks. Indeed, one may send a pair 2 ! (s, U) to the pair (U (s),U),andsince behaves well with respect to restrictions, this assignment is compatible with the equivalence relations; that is, if (s, U) and (s0,U0) are equivalent and s and s0 coincide on V ,onehas

(s) = (s )= (s0 )= (s0) . U |V V |V V |V U 0 |V Obviously ( )x = x x and (idF ) =id(Fx),sotheassignment x is a x ! functor from the category of abelian sheaves to the category of abelian groups. A familiar example We take a closer look at the case when X = C and the sheaf is the sheaf OX of holomorphic functions. In this case the germ of an analytic function at point a is represented by its Taylor series; indeed, two holomorphic functions having the same Taylor series are equal. And so the stalk OX,a is the ring of power series in z a having a positive radius of convergence. Subsheaves and saturation of subpresheaves If F is a presheaf on X,asubpresheaf (underpreknippe) G is a presheaf such that G(U) F (U) for every open U,andsuchthattherestrictionmapsofG are induced ✓ by those of F .IfF and G are sheaves, of course G is called a subsheaf (underknippe). Let F be a sheaf on X and G F asubpresheaf.Wesaythatasections of F over U ✓ locally lies (ligger lokalt i) in G if for some open covering Ui i I one has s U G(Ui) { } 2 | i 2 for each i. We define a subpresheaf G of F by letting the sections of G over U be the sections of F over U that locally lies in G. This of course contains G. The saturation G is by definition, or at least almost by definition, a sheaf. Given a set of patching data for G; that is, an open covering Ui i I of an open set U and sections si over Ui matching on { } 2 intersections, the si’s can be glued together in F since F is a sheaf, and since they are born to locally lie in G,thepatchlieslocallyinG as well and is a section of G over U. The sheaf G is the smallest subsheaf of F containing G,andwebaptizeitthesheaf saturation of G in F .IfG already is a sheaf, then of course we don’t get anything new, and G = G.

Kernels and images Let : F G be a map between two abelian sheaves on X. The kernel (kjernen) ! Ker to is a subsheaf of F whose sections over U are just Ker U , i.e., the sections

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in F (U) mapping to zero under U : F (U) G(U). ! The requirement in the definition is compatible with the restriction maps since V (s V )=U (s) V ,foranysections over the open set U and any open V U. Thus | | ✓ we have defined a subpresheaf of F which, indeed, is a subsheaf: If si i are gluing { } data for the kernel, one may glue the si’s to a section s of F over U.Onehas(s) U = | i (s U )=(si)=0,andfromthelocalityaxiomforG it follows that (s)=0.We | i leave it to the student to verify that the stalk (Ker )x of Ker at x equals Ker x.We have proven

Lemma ￿.￿ Let : F G be a homomorphism of abelian sheaves. The kernel Ker ! is a subsheaf of F having the two properties

⇤ Taking the kernel commutes with taking sections: (U, Ker )=KerU ,

⇤ Forming the kernel commutes with forming stalks: (Ker )x =Kerx. One says that the map is injective (injektiv) if Ker =0. This is, in view of the previous lemma, equivalent to the condition Ker x =0for all x, i.e., that all x are injective. One often expresses this in a slightly imprecise manner by saying that is injective on all stalks. When it comes to images the situation is not as nice as for kernels. One defines the image presheaf contained in G by letting the sections over U be equal to Im U .

However this is not necessarily a sheaf. If si = Ui (ti) are gluing data for the image presheaf there is no reason for the ti’s to match on the intersections Uij even if the si’s do; the differences ti U tj U may very well be non-zero sections of the kernel of U ! | ij | ij ij To remedy this, we simply let Im be the saturation of the image presheaf, i.e., the smallest subsheaf containing the images. Thus, forming the image of a map does not always commute with taking sections, but as we shall verify in the upcoming lemma, forming images commutes with forming stalks. One has

Lemma ￿.￿ Let : F G be a homomorphism of abelian sheaves. The image Im ! is a subsheaf of G.

For all open subsets U of X one has Im U (U, Im ). ⇤ ✓

For all x X one has (Im )x =Imx. ⇤ 2

Proof: We only has to verify the last statement, so let t(x) Im x and pick an 2 s(x) Fx with x(s(x)) = t(x). This means that over some open neighbourhood V , 2 one has V (s)=t,andt is a section of Im over V . This shows that Im x (Im )x. ✓ The other way around, if t is a section of G over an open U containing x locally lying in image presheaf, the restriction t V lies in Im V for some smaller neighbourhood ✓ V of x,hencethegermt(x) lies in Im x. o

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The map : F G is said to be surjective if the image sheaf Im equals G. This is ! equivalent to all the stalk-maps x being surjective (one says is surjective on stalks), but it does not imply that all maps U on sections are surjective. However, one has Proposition ￿.￿ Let : F G be a homomorphism of abelian sheaves. Then following ! three conditions are equivalent

⇤ The map is an isomorphism.

For all x X the stalk-map x : Fx Gx is an isomorphism, ⇤ 2 ! ⇤ One has Ker =0and Im = G,

For all open subsets U X the map on sections U : F (U) G(U) is an iso- ⇤ ✓ ! morphism. Proof: Most of the implications are straight forward from what we have done so far and are left to student. We comment just on the two most the salient points. Firstly, assume that all the stalk maps x : Fx Gx are isomorphism, and let us ! deduce from this that all the maps U : F (U) G(U) on sections are isomorphisms. ! It is clear that U is injective since forming kernels commute with taking sections as in lemma ￿.￿ on page 7.Soassumethats G(U).Foreachx U there is a germ 2 2 t(x) induced by a section tx of F over some open neighbourhood Ux of x satisfying x(t(x)) = s(x).Letsx = s U . | x After possibly having shrunken the neighbourhood Ux,onehasUx (tx)=sx. The sx’s match on the intersections Ux Ux —they are all restrictions of the section s—and \ 0 therefore the tx’s match as well because Ux is injective as we just observed. Hence, the tx’s patch together to a section t of F that must map to s since it does so locally. 1 Secondly, if all the U ’s are isomorphisms, we have all the inverse maps U at our disposal. They commute with restrictions since the maps U do. Indeed, from U U U 1 1 U 1 V ⇢V = ⇢V U one obtains ⇢V U = V ⇢V and the U ’s thus define a map 1 : G F of sheaves, which of course, is inverse to .Weconcludethat is an ! isomorphism. o Exact sequences of abelian sheaves. Given a sequence

i 2 i 1 i i+1 ... / Fi 1 / Fi / Fi+1 / ... of abelian sheaves. We say that the sequence is exact (eksakt) at stage i if Ker i = Im i 1. The short exact sequences are the ones one most frequently encounter. They are sequences shaped like

1 2 0 / F1 / F2 / F3 / 0 (3) that are exact at each stage. This is just another and very convenient way of simul-

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taneously saying that 1 is injective, that 2 is surjective and that Im 1 =Ker2. Forming the sections of the short exact sequence (3)oversomeopensetU one finds the sequence 0 / (U, F1) / (U, F2) / (U, F3) (4) which is exact at the three leftmost stages, but the map (U, F2) / (U, F3) to the right is not necessarily surjective (although it of course can be). One way of phrasing this is to say that taking sections over an open set U—that is (U, )—is a left exact ⇤ functor (venstreeksakt funktor), which is not exact in general. The defect of this lacking surjevtivity is a fundamental problem in every part of mathematics where sheaf theory is used, and to cope with it one has cohomology! Problem ￿.￿. Show that the sequence (4)iexact. X

Example A classical example is the following. Let X = C and recall the sheaf OX of holomorphic functions and the map D : OX OX sending f(z) to the derivative f 0(z). ! There is an exact sequence

/ / D / / 0 C OX OX 0.

This hinges on the two following facts. Firstly, a function whose derivative vanishes identically is locally constant; hence the kernel Ker D equals the constant sheaf CX . Secondly, in small open disks any holomorphic function has a primitive function, i.e., n is a derivative, e.g., if f(z)= n 0 an(z a) in a small disk around a,thefunction 1 n+1 g(z)= n 0 an(n +1) (z a) has f(z) as derivative. However, taking sections P over open sets U we merely obtain the sequence P / / DU / 0 (U, C) (U, OX ) (U, OX ).

Whether DU is surjective or not, depends on the topology of U.IfU is simply connected, one deduces from Cauchy’s integral theorem that every function holomorphic in U is a derivative, so in that case DU is surjective. On the other hand, if U not simply 1 connected, DU is not surjective; e.g., if U = C 0 ,thefunctionz is not a derivative \{ } in U. The exponential function gives another example with the same flavor. The non- vanishing functions holomorphic functions in an open set U X form a multiplicative ✓ group, an there is a sheaf OX⇤ with these groups as sections. For any f holomorphic in U the exponential exp f(z) is a section of OX⇤ .Hencethereisanexactsequence

/ / exp / / 0 Z OX OX⇤ 1, since non-vanishing functions locally have logarithms. However unless U is simply con- nected, expU is not surjective. In that case there will always be some non-vanishing function in U without a logarithm defined throughout U.

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Another example This an example from the algebraic geometry of varieties. Recall that for any variety X over the algebraically closed field k one has the sheaf OX of regular functions on X.Foranypointx X denote by k(x) the skyscraper sheaf (see 2 definition page 11 below) whose only non-zero stalk is the field k located at x. There is amapofsheavesevx : OX k(x) sending a function that is regular in a neighbourhood ! of x to the value it takes at x. This maps sits in the exact sequence of sheaves

evx 0 / Ix / OX / k(x) / 0, where Ix by definition is the kernel of evx (the sections of Ix are the functions vanishing at x). Taking two distinct points x and y in X we find a similar exact sequence

evx,y 0 / I / O / k(x) k(y) / 0, x,y X where evx,y acts on a function f (U, OX ) following the rules: 2 0 if x, y / U, 2 f(x) if x U but y/U, evx,y(f)=8 2 2 >f(y) if y U but x/U, <> 2 2 f(x)+f(y) if x, y U. > 2 1 > If for example X = P (or: any other complete variety), there are no global regular functions on X other than the constants, and hence (X, OX )=k.Butofcourse, (U, k(x) k(y)) = k k,sothemapevx,y can not be surjective on global sections. A family of examples—Godement sheaves We treat these Godement sheaves to show the versatility and the generality of the notion of sheaves. The Godement sheaves play a role in the theoretical development of sheaf theory, but x hardly show up in the daily work of most mathematicians working in algebraic geometry. Let X any topological space. Assume that we for each point x X are given an 2 abelian group Ax. The groups Ax can be chosen in a completely free way, at random if you want. The choice of these groups gives rise to a sheaf A on X whose sections over an open set U X are given as ✓

(U, A )= Ax, x U Y2 and whose restriction maps are defined as the natural projections

⇢U : A A , V x ! x x U x V Y2 Y2 where V U is any pair of open subsets of X. The restriction map just “throws away” ✓ the components at points in U not lying in V . The two sheaf axioms are satisfied:

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The locality condition holds since if the family Ui i of open sets covers U,any { } point x0 U lies in some Ui0 ,soifs =(ax)x U (U, A ) is a section, the component 2 2 2 ax0 survives in the projection onto (Ui, A )= x U Ax.Henceif Ui =0for all i,it 2 i | follows that =0. Q The gluing condition holds: Assume given an open cover Ui i of U and sections { } si Ui matching on the intersections Ui Uj. The matching conditions mean that the 2 \ component of si at a point x is the same whatever i is as long as x Ui.Henceweget 2 asections of A over U by using this common component as the component of s at x. Obviously s U = i. | i The sheaf A is sometimes called the Godement sheaf of the selection Ax . The { } construction is not confined to abelian groups, but works for any category where general products exist; like sets, rings,. . . . So what is the stalk of A at a point x? Don’t ask that question! The answer is simple but the group is fairly complicated. Skyscraper sheaves. This is a very special instant of a Godement sheaf where all the abelian groups Ax are zero except for one. So we fix just one abelian group A,and we choose a closed point x X. The corresponding Godement sheaf is denoted A(x) 2 and is called a skyscraper sheaf (skyskraperknippe). The sections are described by

A if x U, (U, Ax)= 2 (0 otherwise .

Contrary to the general case, in this case the stalks are easy to describe. One verifies easily that they are zero everywhere except at x, where the stalk equals A,indeed,if y = x,theny lies in the open set X x over which all sections of A(x) vanish. 6 \{ } In most spaces concerning us there are plenty of non-closed points. For such points one still have the Godement sheaf A(x),buttheargumentabovedoesnotwork,and the description of the stalks are somehow more complicated. One would hardly call A(x) askyscrapersheaf. Problem ￿.￿. Assume that x is not closed and let Z = x be the closure of the { } singleton x .ShowthatthestalksofA(x) are A(x)y =0if y Z and A(x)y = A for { } 2 points y belonging to Z X Slightly generalizing the construction of a skyscraper sheaf, one may form the Gode- ment sheaf A defined by a finite set of distinct closed points x1,...,xr and correspon- ding abelian groups A1,...,Ar. Then one sees, having in mind that an empty direct U (U, )= A sum is zero, that the sections of A over an open set is given as A xi U i. The stalks of A are 2 L 0 when x = xi for all i Ax = 6 (Ai when x = xi. One is tempted to call such a sheaf a barcode sheaf (barcodeknippe)!

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The Godement sheaf associated with a presheaf Assume F is a given abelian presheaf on X. The stalks Fx of F is a selection of abelian groups indexed by points in X as good as any, and we may form the corresponding Godement sheaf which we denote by god F . The sections of god F are given by

(U, god F )= Fx, (5) x U Y2 and the restriction maps are the projections like for any Godement sheaf. There is an obvious and canonical map F : F god F sending a section s of F ! over U to the element (s(x))x U of the product in (5). This construction is functorial in 2 F ,forif: F G is a map, one has the stalkwise maps x : Fx Gx,andbytaking ! ! appropriate products of these, we obtain a map god :godF god G.Bysomethis ! sheaf is called the sheaf of discontinuous sections of F .OnsectionsoverU it has the effect

god (s(x))x U = x(s(x)) x U 2 2 and there is a commutative diagram

 F F / god F (6)

god ✏ ✏ G / god G. G

It is the matter of an easy verification that god =god god and that god idF = idgod F . Flabby sheaves AGodementsheafA corresponding to a selection of abelian groups Ax trivially has the property that all the restriction maps are surjective; indeed, they { } are projections between to products. In general, a sheaf F on X with this property —that is, the maps ⇢U : F (U) F (V ) are surjective—is called flabby (et fett knip- V ! pe1).Phrased differently, F is flabby if every section of F over an open V can be exten- ded to a section over any bigger open set U. The main importance of flabby sheaves is the following:

Lemma ￿.￿ Given a short exact sequence of abelian sheaves on X

0 / F1 / F2 / F3 / 0. (7)

If the sheaf F1 is flabby, then for every open set U the following sequence of obtained

1Det er meg bekjent ingen standard norsk terminologi for flabby, så dette er forslag. Ifølge Ord- nett.no har flabby betydningene: slapp, fet og slapp, løs i kjøttet, kvapsete.Denfranskebetegnelsener flasque some oversettes med: slapp, løs (i fisken), slakk.Derforfett knippe og ikke et kvapsete knippe!

—12— Notes 1—Sheaves MAT4215 — Vår 2015 by taking sections of (7)overU is exact:

0 / (U, F1) / (U, F2) / (U, F3) / 0.

Proof: The only thing needing a proof is that U is surjective. Take a section s 2 (U, F3) over any open U.Wearesearchingforasectiont (U, F2) with (t)=s. 2 To begin with let us show that if for an open V U the restriction s V can be lifted ✓ | to some section t0 of F2 over V ,theneitherV = U,orthereisanopensubsetW of U strictly bigger than V over which s W can be lifted to a section of F2. | Assume that x U but x/V .Since(7)exact,thereisanopenneighbourhood 2 2 W of x over which s W can be lifted to a t00.OntheintersectionV W both t0 V W | \ | [ and t00 V W map to s V W . Hence the difference t0 V W tV00 W is a section of F1 over | \ | \ | [ \ V W ,andasF1 is flabby it can be extended to a section ⌧ of F2 over V . Then t0 + ⌧ \ and t00 mach on V W and can be glued to a section t over V W ,thatobviously [ [ maps to s V W since ⌧ maps to zero. | [ We conclude by taking V maximal among the open subsets of U over which s can be lifted to a section of F2;thereisonebyZornslemma.Ife.g., U is quasicompact, ordinary induction on the number of sets in the covering will do. o A peculiar examples Of course if Z is a discrete subset of X,notnecessarilyfinite, one may associate a Godement sheaf A with any family Az z Z of abelian groups { } 2 indexed by Z (letting Ax =0if x/Z)andthesectionsoveranopensetU is by 2 definition of a Godement sheaf given by

(U, Ax)= Az z Z U 2Y\ Just as in the finite case the stalks will be

Ax = Ax

. However if Z has accumulation points this is no more true. Assume for simplicity that zi is a sequence converging to z.Twosequencesofelementsai and a0 is said to { } i define the same tail if there is some N such that ai = ai0 for i N. Then the stalk at Ai z will be the space of tails from the Ai.thatis / . i ⇠ Sheafification Q Given any presheaf F on X. There is a canonical way of defining a sheaf F + that in some sense is the sheaf that best approximates F .WhatpreventsthepresheafF from being a sheaf is the failure of one or both of the sheaves axioms. To remedy this one must factor out all sections of F whose germs are everywhere zero, and one has to enrich F by adding enough sections to unrestrictedly be able to glue. AniceandcanonicalwayofdoingthisisbyusingtheGodementsheafgod F associated with F .Recallthecanonicalmap: F god F that sends a section s of F ! —13— Notes 1—Sheaves MAT4215 — Vår 2015

over an open U to the sequence of germs (sx)x U x U Fx =(U, god F ). This map 2 2 2 clearly kills the doomed sections, i.e., those whose germs all vanish. We let F + be the Q image sheaf of this canonical map, which then factors through a map : F F +. The ! sections of F + over an open U are characterized by locally coming from F . Lemma ￿.￿ The sheafification F + depends functorially on F .IfF is a sheaf, it holds true that F = F +. Proof: Assume that : F G is a map between two presheaves. Let s be section of ! god F over some open set U,thatiss comes locally from F .Inotherwordsthereisa covering Ui of U and sections s0 of F over Ui with s U = F (s0 ).Henceby(6)one { } i | i i has god (s )=god( (s0 )) =  ((s0 )). |Ui F i G i We see that god (s) lies locally in G,andgod takes F + into G+. There is a commu- tative diagram F / F + / god F

+ god ✏ ✏ ✏ G / G+ / god G

In case F is a sheaf, the map F maps F invectively into god F and F =ImF is its own saturation, hence F = F +.

o

Proposition ￿.￿ Given a presheaf F on X.ThenthesheafF + and the natural map F F + enjoys the universal property that any map F G where G is sheaf, factors ! ! through F + in a unique way. This characteries F + up to unique isomorphisms. Proof: If G in the diagram above is a sheaf, the map G G+ is an isomorphism and ! + takes values in G and provides the wanted factorization. The uniqueness statement is a general feature of solutions to universal problems. o

Constant sheaves For any space X and any abelian group A there is a so called constant sheaf denoted by AX on X. There is an obvious constant presheaf whose group of sections over any nonempty open set U equals A and equals 0 if U = . This is not a sheaf, since if U U 0 ; \ is a disjoint union any choice of elements a, a0 A will give sections over U and U 0 2 respectively, and they match on the empty intersection! But if a = a0 they can not be 6 glued. However, the sheafification of this constant presheaf is there, and that is what we call the constant sheaf with value A and denote by AX . The sheaf AX is not quite worthy of the name constant sheaf as is is not quite constant. One has

(U, AC )= A, ⇡ U Y0 where ⇡0U denotes the set of connected components of the open set U.

—14— Notes 1—Sheaves MAT4215 — Vår 2015

Quotient sheaves and cokernels The main application of the sheafification process we just described is to show that one may form quotient sheaves. So assume that F G are abelian sheaves, and ✓ define a presheaf whose sections over U is the quotient G(U)/F (U). The restriction maps of F and G respects the inclusions F (U) G(U) and hence pass to the quotients ✓ G(U)/F (U). We use these maps as the restriction maps for the quotient presheaf. The quotient sheaf (kvotientknippet) F/G is by definition the sheafification of the quotient presheaf. The quotient sits in the exact sequence

0 / F / G / F/G / 0 .

The cokernel (kokjernen) Coker of the map : F G of abelian sheaves is then just ! the quotient sheaf G/Im ,itlivesintheexactsequence

0 / Ker / F / G / Coker / 0

The push forward of a sheaf Let X and Y be two topological spaces with a continuous map f : X Y between ! them. Assume that F is an abelian sheaf on X. This allows us to define an abelian sheaf f F on Y in by specifying the sections of f F over the open set U Y to be: ⇤ ⇤ ✓ 1 (U, f F )=(f U, F ), ⇤ and the restriction maps are those of F . The sheaf f F is called the push forward sheaf or the direct image (det direkte ⇤ bildet) of F .Itisstraightforwardtoseethatf F is a sheaf and not merely a presheaf. ⇤ Indeed, if Ui i is an open covering of U asetofpatchingdataconsistsofsection { } 1 si (Ui; f F )=(f U, F ). That they match on the intersection means that they 2 ⇤ 1 1 coincide in (Ui Uj,f F )=(f Ui f Uj,F),andthereforecanbegluedtoa \ ⇤ \ section in (f U, F )=(U, f ) since F is a sheaf. The locality axiom is as easy to ⇤ check. If you want, the push forward sheaf f F is just the restriction of F to the subcate- ⇤ gory of the open sets in X which are inverse images of open sets in Y . This is a functorial construction:

Lemma ￿.￿ If g : X Y and f : Y Z are continuous maps between topological ! ! spaces, and F is a sheaf on X,onehas

(f g) F = f (g F ). ⇤ ⇤ ⇤ Remark, this is indeed an equality, nor merely an isomorphism.

Proof: Agoodandeasyexerciseinmanipulatingsections. o

—15— Notes 1—Sheaves MAT4215 — Vår 2015

Lemma ￿.￿ The functor f is left exact. That is: Given an exact sequence of abelian ⇤ sheaves on X 0 / F1 / F2 / F3 / 0 then the following sequence is exact

0 / f F1 / f F2 / f F3. ⇤ ⇤ ⇤ Proof: As f F are sections of F over inverse images of opens in Y , the lemma follows ⇤ readily from the exactness of the sequence (4)onpage9 above. o

Problem ￿.￿. Let x X be a closed point and let i: x X be the inclusion map. 2 { }! Let A be an abelian group. Show that the skyscraper sheaf A(x) equals i A (where A ⇤ also denotes the constant sheaf on the one point space x !) Hint: This shows that { } deliberations around sections over the empty set is not completely stupid. X Problem ￿.￿￿. Denote by aonepointset.Letf : X be the one and {⇤} !{⇤} only map. Show that f F =(X, F) (where strictly speaking (X, F) stands for the ⇤ constant sheaf on ). X {⇤}

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