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displaystyle (560), the (476), mathcal (184), sheaf (180), sheaves (136), and (125), for (98), open (78), this (73), are (68), that (67), with (65), #cohomology (61), space (60), functions (44), presheaf (42), theory (39), sections (38), edit (37), mathbb (37), topological (34), complex (34), spaces (33), set (33), from (32), there (32), sets (32), not (32), category (31), such (31), can (30), which (29), groups (27), example (26), functor (26), also (25), map (25), called (24), over (24), image (24), restriction (24), res (24), all (23), grothendieck (23), any (23), morphism (23), algebraic (22), continuous (22), geometry (21), abelian (21), then (21), its (20), given (20), morphisms (20), defined (19), constant (19), mathematics (18), topology (18), these (18), locally (18), modules (18), some (18), derived (17), holomorphic (17), coherent (17), general (17), denoted (17), above (17), presheaves (17), categories (16), étalé (16), data (16), between (16), natural (16), two (16), subseteq (16), manifolds (15), manifold (15), text (14), isbn (14), examples (14), theorem (14), point (14), but (14), where (14), another (14), subsets (14), gamma (14), was (13), doi (13), mathematical (13), direct (13), structure (13), zero (13), operatorname (13), function (13), subset (13), times (13), every (13), cap (13), using (12), other (12), see (12), duality (12), into (12), definition (12), section (12), one (12), rings (12), ringed (12), first (11), local (11), compact (11), construction (11), more (11), each (11), their (11), inverse (11), projective (11), let (11), maps (11), colon (11), global (10), only (10), functors (10), notion (10), mathbf (10), smooth (10), hom (10), associated (10), stalks (10), real (10), vector (10), smaller (10), 978 (9), second (9), pierre (9), differential (9), since (9), covering (9), main (9), following (9), whose (9), stalk (9), inclusion (9), respectively (9), rightarrow (9), valued (9), extension (9), wikipedia (8), pdf (8), serre (8), 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y two sections over x displaystyle x and y displaystyle y we cannot glue them uniquely let x r displaystyle x mathbb r be the real line and let f u displaystyle f u be the set of bounded continuous functions on u displaystyle u this is not a sheaf because it is not always possible to glue for example let u i displaystyle u_ i be the set of all x displaystyle x such that x i displaystyle x i the identity function f x x displaystyle f x x is bounded on each u i displaystyle u_ i consequently we get a section s i displaystyle s_ i on u i displaystyle u_ i however these sections do not glue because the function f displaystyle f is not bounded on the real line consequently f displaystyle f is a presheaf but not a sheaf in fact f displaystyle f is separated because it is a sub presheaf of the sheaf of continuous functions motivating sheaves from complex analytic spaces and algebraic geometry edit one of the historical motivations for sheaves have come from studying complex manifolds 4 complex analytic geometry 5 and scheme theory from algebraic geometry this is because in all of the previous cases we consider a topological space x displaystyle x together with a structure sheaf o displaystyle mathcal o giving it the structure of a complex manifold complex analytic space or scheme this perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces see below technical challenges with complex manifolds edit one of the main historical motivations for introducing sheaves was constructing a device which keeps track of holomorphic functions on complex manifolds for example on a compact complex manifold x displaystyle x like complex projective space or the vanishing locus in projective space of a homogeneous polynomial the only holomorphic functions f x c displaystyle f x to mathbb c are the constant functions 6 7 this means there exist two compact complex manifolds x x displaystyle x x which are not isomorphic but nevertheless their rings of global holomorphic functions denoted h x h x displaystyle mathcal h x mathcal h x are isomorphic contrast this with smooth manifolds where every manifold m displaystyle m can be embedded inside some r n displaystyle mathbb r n hence its ring of smooth functions c m displaystyle c infty m comes from restricting the smooth functions from c r n displaystyle c infty mathbb r n of which there exist plenty another complexity when considering the ring of holomorphic functions on a complex manifold x displaystyle x is given a small enough open set u x displaystyle u subseteq x the holomorphic functions will be isomorphic to h u h c n displaystyle mathcal h u cong mathcal h mathbb c n sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of x displaystyle x on arbitrary open subsets u x displaystyle u subseteq x this means as u displaystyle u becomes more complex topologically the ring h u displaystyle mathcal h u can be expressed from gluing the h u i displaystyle mathcal h u_ i note that sometimes this sheaf is denoted o displaystyle mathcal o or just o displaystyle mathcal o or even o x displaystyle mathcal o _ x when we want to emphasize the space the structure sheaf is associated to tracking submanifolds with sheaves edit another common example of sheaves can be constructed by considering a complex submanifold y x displaystyle y hookrightarrow x there is an associated sheaf o y displaystyle mathcal o _ y which takes an open subset u x displaystyle u subseteq x and gives the ring of holomorphic functions on u y displaystyle u cap y this kind of formalism was found to be extremely powerful and motivates a lot of homological algebra such as sheaf cohomology since an intersection theory can be built using these kinds of sheaves from the serre intersection formula operations with sheaves edit morphisms edit morphisms of sheaves are roughly speaking analogous to functions between them in contrast to a function between sets which is simply an assignment of outputs to inputs morphisms of sheaves are also required to be compatible with the local global structures of the underlying sheaves this idea is made precise in the following definition let f displaystyle mathcal f and g displaystyle mathcal g be two sheaves of sets respectively abelian groups rings etc on x displaystyle x a morphism φ f g displaystyle varphi mathcal f to mathcal g consists of a morphism φ u f u g u displaystyle varphi _ u mathcal f u to mathcal g u of sets respectively abelian groups rings etc for each open set u displaystyle u of x displaystyle x subject to the condition that this morphism is compatible with restrictions in other words for every open subset v displaystyle v of an open set u displaystyle u the following diagram is commutative f u φ u g u r v u r v u f v φ v g v displaystyle begin array rcl mathcal f u xrightarrow quad varphi _ u quad mathcal g u r_ v u biggl downarrow biggl downarrow r _ v u mathcal f v xrightarrow quad varphi _ v quad mathcal g v end array for example taking the derivative gives a morphism of sheaves on r displaystyle mathbb r d d x o r n o r n 1 displaystyle frac mathrm d mathrm d x colon mathcal o _ mathbb r n to mathcal o _ mathbb r n 1 indeed given an n displaystyle n times continuously differentiable function f u r displaystyle f u to mathbb r with u displaystyle u in r displaystyle mathbb r open the restriction to a smaller open subset v displaystyle v of its derivative equals the derivative of f v displaystyle f _ v with this notion of morphism sheaves of sets respectively abelian groups rings etc on a fixed topological space x displaystyle x form a category the general categorical notions of mono epi and isomorphisms can therefore be applied to sheaves in fact from the point of view of category theory the category of sheaves over a small category c displaystyle c with values in another category d displaystyle d is a full subcategory of the category of presheaves over c displaystyle c with values in d displaystyle d which is simply the category d c op displaystyle d c text op of contravariant functors from c displaystyle c to d displaystyle d with natural transformations between them as morphisms the notion of morphism defined above can simply be stated as φ displaystyle varphi being a natural transformation between the two sheaves seen as functors a morphism φ f g displaystyle varphi colon mathcal f rightarrow mathcal g of sheaves on x displaystyle x is an isomorphism respectively monomorphism if and only if for every open set u x displaystyle u subseteq x we have an isomorphism f u g u displaystyle mathcal f u approx mathcal g u which is natural with respect to the restriction maps these statements give examples of how to work with sheaves using local information but it s important to note that we cannot check if a morphism of sheaves is an epimorphism in the same manner indeed the statement that maps on the level of open sets φ u f u g u displaystyle varphi _ u colon mathcal f u rightarrow mathcal g u are not always surjective for epimorphisms of sheaves is equivalent to non exactness of the global sections functor or equivalently to non triviality of sheaf cohomology stalks of a sheaf edit main article stalk sheaf stalks and germs for a constant sheaf on a discrete two point space the stalk f x displaystyle mathcal f _ x of a sheaf f displaystyle mathcal f captures the properties of a sheaf around a point x x displaystyle x in x generalizing the germs of functions here around means that conceptually speaking one looks at smaller and smaller neighborhoods of the point of course no single neighborhood will be small enough which requires considering a limit of some sort more precisely the stalk is defined by f x lim u x f u displaystyle mathcal f _ x varinjlim _ u ni x mathcal f u the direct limit being over all open subsets of x displaystyle x containing the given point x displaystyle x in other words an element of the stalk is given by a section over some open neighborhood of x displaystyle x and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood the natural morphism f u f x displaystyle mathcal f u to mathcal f _ x takes a section s displaystyle s in f u displaystyle mathcal f u to its germ s x displaystyle s_ x at x displaystyle x this generalises the usual definition of a germ in many situations knowing the stalks of a sheaf is enough to control the sheaf itself for example whether or not a morphism of sheaves is a monomorphism epimorphism or isomorphism can be tested on the stalks in this sense a sheaf is determined by its stalks which are a local data by contrast the global information present in a sheaf i e the global sections i e the sections f x displaystyle mathcal f x on the whole space x displaystyle x typically carry less information for example for a compact complex manifold x displaystyle x the global sections of the sheaf of holomorphic functions are just c displaystyle mathbb c since any holomorphic function x c displaystyle x to mathbb c is constant by liouville s theorem 6 turning a presheaf into a sheaf edit it is frequently useful to take the data contained in a presheaf and to express it as a sheaf it turns out that there is a best possible way to do this it takes a presheaf f displaystyle mathcal f and produces a new sheaf a f displaystyle a mathcal f called the sheafification or sheaf associated to the presheaf f displaystyle mathcal f for example the sheafification of the constant presheaf see above is called the constant sheaf despite its name its sections are locally constant functions the sheaf a f displaystyle a mathcal f can be constructed using the étalé space e displaystyle e of f displaystyle mathcal f namely as the sheaf of sections of the map e x displaystyle e to x another construction of the sheaf a f displaystyle a mathcal f proceeds by means of a functor l displaystyle l from presheaves to presheaves that gradually improves the properties of a presheaf for any presheaf f displaystyle mathcal f l f displaystyle l mathcal f is a separated presheaf and for any separated presheaf f displaystyle mathcal f l f displaystyle l mathcal f is a sheaf the associated sheaf a f displaystyle a mathcal f is given by l l f displaystyle ll mathcal f 8 the idea that the sheaf a f displaystyle a mathcal f is the best possible approximation to f displaystyle mathcal f by a sheaf is made precise using the following universal property there is a natural morphism of presheaves i f a f displaystyle i colon mathcal f to a mathcal f so that for any sheaf g displaystyle mathcal g and any morphism of presheaves f f g displaystyle f colon mathcal f to mathcal g there is a unique morphism of sheaves f a f g displaystyle tilde f colon a mathcal f rightarrow mathcal g such that f f i displaystyle f tilde f i in fact a displaystyle a is the left adjoint functor to the inclusion functor or forgetful functor from the category of sheaves to the category of presheaves and i displaystyle i is the unit of the adjunction in this way the category of sheaves turns into a giraud subcategory of presheaves this categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves but not for kernels say subsheaves quotient sheaves edit if k displaystyle k is a subsheaf of a sheaf f displaystyle f of abelian groups then the quotient sheaf q displaystyle q is the sheaf associated to the presheaf u f u k u displaystyle u mapsto f u k u in other words the quotient sheaf fits into an exact sequence of sheaves of abelian groups 0 k f q 0 displaystyle 0 to k to f to q to 0 this is also called a sheaf extension let f g displaystyle f g be sheaves of abelian groups the set hom f g displaystyle operatorname hom f g of morphisms of sheaves from f displaystyle f to g displaystyle g forms an abelian group by the abelian group structure of g displaystyle g the sheaf hom of f displaystyle f and g displaystyle g denoted by h o m f g displaystyle mathcal hom f g is the sheaf of abelian groups u hom f u g u displaystyle u mapsto operatorname hom f _ u g _ u where f u displaystyle f _ u is the sheaf on u displaystyle u given by f u v f v displaystyle f _ u v f v note sheafification is not needed here the direct sum of f displaystyle f and g displaystyle g is the sheaf given by u f u g u displaystyle u mapsto f u oplus g u and the tensor product of f displaystyle f and g displaystyle g is the sheaf associated to the presheaf u f u g u displaystyle u mapsto f u otimes g u all of these operations extend to sheaves of modules over a sheaf of rings a displaystyle a the above is the special case when a displaystyle a is the constant sheaf z _ displaystyle underline mathbf z basic functoriality edit main article image functors for sheaves since the data of a pre sheaf depends on the open subsets of the base space sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them however given a continuous map f x y displaystyle f x to y between two topological spaces pushforward and pullback relate sheaves on x displaystyle x to those on y displaystyle y and vice versa direct image edit the pushforward also known as direct image of a sheaf f displaystyle mathcal f on x displaystyle x is the sheaf defined by f f v f f 1 v displaystyle f_ mathcal f v mathcal f f 1 v here v displaystyle v is an open subset of y displaystyle y so that its preimage is open in x displaystyle x by the continuity of f displaystyle f this construction recovers the skyscraper sheaf s x displaystyle s_ x mentioned above s x i s displaystyle s_ x i_ s where i x x displaystyle i x to x is the inclusion and s displaystyle s is regarded as a sheaf on the singleton by s s s displaystyle s s s emptyset emptyset for a map between locally compact spaces the direct image with compact support is a subsheaf of the direct image 9 by definition f f v displaystyle f_ mathcal f v consists of those s f f 1 v displaystyle s in mathcal f f 1 v whose support is mapped properly if f displaystyle f is proper itself then f f f f displaystyle f_ mathcal f f_ mathcal f but in general they disagree inverse image edit the pullback or inverse image goes the other way it produces a sheaf on x displaystyle x denoted f 1 g displaystyle f 1 mathcal g out of a sheaf g displaystyle mathcal g on y displaystyle y if f displaystyle f is the inclusion of an open subset then the inverse image is just a restriction i e it is given by f 1 g u g u displaystyle f 1 mathcal g u mathcal g u for an open u displaystyle u in x displaystyle x a sheaf f displaystyle mathcal f on some space x displaystyle x is called locally constant if x i i u i displaystyle x bigcup _ i in i u_ i by some open subsets u i displaystyle u_ i such that the restriction of f disp...
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