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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), 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eaf of differential operators on any topological space modules over the constant sheaf z _ displaystyle underline mathbf z are the same as sheaves of abelian groups in the sense above there is a different inverse image functor for sheaves of modules over sheaves of rings this functor is usually denoted f displaystyle f and it is distinct from f 1 displaystyle f 1 see inverse image functor finiteness conditions for sheaves of modules edit finiteness conditions for module over commutative rings give rise to similar finiteness conditions for sheaves of modules m displaystyle mathcal m is called finitely generated respectively finitely presented if for every point x displaystyle x of x displaystyle x there exists an open neighborhood u displaystyle u of x displaystyle x a natural number n displaystyle n possibly depending on u displaystyle u and a surjective morphism of sheaves o x n u m u displaystyle mathcal o _ x n _ u to mathcal m _ u respectively in addition a natural number m displaystyle m and an exact sequence o x m u o x n u m u 0 displaystyle mathcal o _ x m _ u to mathcal o _ x n _ u to mathcal m _ u to 0 paralleling the notion of a coherent module m displaystyle mathcal m is called a coherent sheaf if it is of finite type and if for every open set u displaystyle u and every morphism of sheaves ϕ o x n m displaystyle phi mathcal o _ x n to mathcal m not necessarily surjective the kernel of ϕ displaystyle phi is of finite type o x displaystyle mathcal o _ x is coherent if it is coherent as a module over itself like for modules coherence is in general a strictly stronger condition than finite presentation the oka coherence theorem states that the sheaf of holomorphic functions on a complex manifold is coherent the étalé space of a sheaf edit in the examples above it was noted that some sheaves occur naturally as sheaves of sections in fact all sheaves of sets can be represented as sheaves of sections of a topological space called the étalé space from the french word french pronunciation étalé meaning roughly spread out if f sh x displaystyle f in text sh x is a sheaf over x displaystyle x then the étalé space sometimes called the étale space of f displaystyle f is a topological space e displaystyle e together with a local homeomorphism π e x displaystyle pi e to x such that the sheaf of sections γ π displaystyle gamma pi of π displaystyle pi is f displaystyle f the space e displaystyle e is usually very strange and even if the sheaf f displaystyle f arises from a natural topological situation e displaystyle e may not have any clear topological interpretation for example if f displaystyle f is the sheaf of sections of a continuous function f y x displaystyle f y to x then e y displaystyle e y if and only if f displaystyle f is a local homeomorphism the étalé space e displaystyle e is constructed from the stalks of f displaystyle f over x displaystyle x as a set it is their disjoint union and π displaystyle pi is the obvious map that takes the value x displaystyle x on the stalk of f displaystyle f over x x displaystyle x in x the topology of e displaystyle e is defined as follows for each element s f u displaystyle s in f u and each x u displaystyle x in u we get a germ of s displaystyle s at x displaystyle x denoted s x displaystyle s _ x or s x displaystyle s_ x these germs determine points of e displaystyle e for any u displaystyle u and s f u displaystyle s in f u the union of these points for all x u displaystyle x in u is declared to be open in e displaystyle e notice that each stalk has the discrete topology as subspace topology a morphism between two sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps in the sense that every germ is mapped to a germ over the same point this makes the construction into a functor the construction above determines an equivalence of categories between the category of sheaves of sets on x displaystyle x and the category of étalé spaces over x displaystyle x the construction of an étalé space can also be applied to a presheaf in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf this construction makes all sheaves into representable functors on certain categories of topological spaces as above let f displaystyle f be a sheaf on x displaystyle x let e displaystyle e be its étalé space and let π e x displaystyle pi e to x be the natural projection consider the overcategory top x displaystyle text top x of topological spaces over x displaystyle x that is the category of topological spaces together with fixed continuous maps to x displaystyle x every object of this category is a continuous map f y x displaystyle f y to x and a morphism from y x displaystyle y to x to z x displaystyle z to x is a continuous map y z displaystyle y to z that commutes with the two maps to x displaystyle x there is a functor γ top x sets displaystyle gamma text top x to text sets sending an object f y x displaystyle f y to x to f 1 f y displaystyle f 1 f y for example if i u x displaystyle i u hookrightarrow x is the inclusion of an open subset then γ i f 1 f u f u γ f u displaystyle gamma i f 1 f u f u gamma f u and for the inclusion of a point i x x displaystyle i x hookrightarrow x then γ i f 1 f x f x displaystyle gamma i f 1 f x f _ x is the stalk of f displaystyle f at x displaystyle x there is a natural isomorphism f 1 f y hom t o p x f π displaystyle f 1 f y cong operatorname hom _ mathbf top x f pi which shows that π e x displaystyle pi e to x for the étalé space represents the functor γ displaystyle gamma e displaystyle e is constructed so that the projection map π displaystyle pi is a covering map in algebraic geometry the natural analog of a covering map is called an étale morphism despite its similarity to étalé the word étale etal has a different meaning in french it is possible to turn e displaystyle e into a scheme and π displaystyle pi into a morphism of schemes in such a way that π displaystyle pi retains the same universal property but π displaystyle pi is not in general an étale morphism because it is not quasi finite it is however formally étale the definition of sheaves by étalé spaces is older than the definition given earlier in the article it is still common in some areas of mathematics such as mathematical analysis sheaf cohomology edit main article sheaf cohomology in contexts where the open set u displaystyle u is fixed and the sheaf is regarded as a variable the set f u displaystyle f u is also often denoted γ u f displaystyle gamma u f as was noted above this functor does not preserve epimorphisms instead an epimorphism of sheaves f g displaystyle mathcal f to mathcal g is a map with the following property for any section g g u displaystyle g in mathcal g u there is a covering u u i i i displaystyle mathcal u u_ i _ i in i where u i i u i displaystyle u bigcup _ i in i u_ i of open subsets such that the restriction g u i displaystyle g _ u_ i are in the image of f u i displaystyle mathcal f u_ i however g displaystyle g itself need not be in the image of f u displaystyle mathcal f u a concrete example of this phenomenon is the exponential map o exp o displaystyle mathcal o stackrel exp to mathcal o times between the sheaf of holomorphic functions and non zero holomorphic functions this map is an epimorphism which amounts to saying that any non zero holomorphic function g displaystyle g on some open subset in c displaystyle mathbb c say admits a complex logarithm locally i e after restricting g displaystyle g to appropriate open subsets however g displaystyle g need not have a logarithm globally sheaf cohomology captures this phenomenon more precisely for an exact sequence of sheaves of abelian groups 0 f 1 f 2 f 3 0 displaystyle 0 to mathcal f _ 1 to mathcal f _ 2 to mathcal f _ 3 to 0 i e an epimorphism f 2 f 3 displaystyle mathcal f _ 2 to mathcal f _ 3 whose kernel is f 1 displaystyle mathcal f _ 1 there is a long exact sequence 0 γ u f 1 γ u f 2 γ u f 3 h 1 u f 1 h 1 u f 2 h 1 u f 3 h 2 u f 1 displaystyle 0 to gamma u mathcal f _ 1 to gamma u mathcal f _ 2 to gamma u mathcal f _ 3 to h 1 u mathcal f _ 1 to h 1 u mathcal f _ 2 to h 1 u mathcal f _ 3 to h 2 u mathcal f _ 1 to dots by means of this sequence the first cohomology group h 1 u f 1 displaystyle h 1 u mathcal f _ 1 is a measure for the non surjectivity of the map between sections of f 2 displaystyle mathcal f _ 2 and f 3 displaystyle mathcal f _ 3 there are several different ways of constructing sheaf cohomology grothendieck 1957 introduced them by defining sheaf cohomology as the derived functor of γ displaystyle gamma this method is theoretically satisfactory but being based on injective resolutions of little use in concrete computations godement resolutions are another general but practically inaccessible approach computing sheaf cohomology edit especially in the context of sheaves on manifolds sheaf cohomology can often be computed using resolutions by soft sheaves fine sheaves and flabby sheaves also known as flasque sheaves from the french flasque meaning flabby for example a partition of unity argument shows that the sheaf of smooth functions on a manifold is soft the higher cohomology groups h i u f displaystyle h i u mathcal f for i 0 displaystyle i 0 vanish for soft sheaves which gives a way of computing cohomology of other sheaves for example the de rham complex is a resolution of the constant sheaf r _ displaystyle underline mathbf r on any smooth manifold so the sheaf cohomology of r _ displaystyle underline mathbf r is equal to its de rham cohomology a different approach is by čech cohomology čech cohomology was the first cohomology theory developed for sheaves and it is well suited to concrete calculations such as computing the coherent sheaf cohomology of complex projective space p n displaystyle mathbb p n 11 it relates sections on open subsets of the space to cohomology classes on the space in most cases čech cohomology computes the same cohomology groups as the derived functor cohomology however for some pathological spaces čech cohomology will give the correct h 1 displaystyle h 1 but incorrect higher cohomology groups to get around this jean louis verdier developed hypercoverings hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space this flexibility is necessary in some applications such as the construction of pierre deligne s mixed hodge structures many other coherent sheaf cohomology groups are found using an embedding i x y displaystyle i x hookrightarrow y of a space x displaystyle x into a space with known cohomology such as p n displaystyle mathbb p n or some weighted projective space in this way the known sheaf cohomology groups on these ambient spaces can be related to the sheaves i f displaystyle i_ mathcal f giving h i y i f h i x f displaystyle h i y i_ mathcal f cong h i x mathcal f for example computing the coherent sheaf cohomology of projective plane curves is easily found one big theorem in this space is the hodge decomposition found using a spectral sequence associated to sheaf cohomology groups proved by deligne 12 13 essentially the e 1 displaystyle e_ 1 page with terms e 1 p q h p x ω x q displaystyle e_ 1 p q h p x omega _ x q the sheaf cohomology of a smooth projective variety x displaystyle x degenerates meaning e 1 e displaystyle e_ 1 e_ infty this gives the canonical hodge structure on the cohomology groups h k x c displaystyle h k x mathbb c it was later found these cohomology groups can be easily explicitly computed using griffiths residues see jacobian ideal these kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties the decomposition theorem paving the path for mixed hodge modules another clean approach to the computation of some cohomology groups is the borel bott weil theorem which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of lie groups this theorem can be used for example to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds in many cases there is a duality theory for sheaves that generalizes poincaré duality see grothendieck duality and verdier duality derived categories of sheaves edit the derived category of the category of sheaves of say abelian groups on some space x denoted here as d x displaystyle d x is the conceptual haven for sheaf cohomology by virtue of the following relation h n x f hom d x z f n displaystyle h n x mathcal f operatorname hom _ d x mathbf z mathcal f n the adjunction between f 1 displaystyle f 1 which is the left adjoint of f displaystyle f_ already on the level of sheaves of abelian groups gives rise to an adjunction f 1 d y d x r f displaystyle f 1 d y rightleftarrows d x rf_ for f x y displaystyle f x to y where r f displaystyle rf_ is the derived functor this latter functor encompasses the notion of sheaf cohomology since h n x f r n f f displaystyle h n x mathcal f r n f_ mathcal f for f x displaystyle f x to image functors for sheaves direct image f displaystyle f_ inverse image f displaystyle f direct image with compact support f displaystyle f_ exceptional inverse image r f displaystyle rf f f displaystyle f leftrightarrows f_ r f r f displaystyle r f_ leftrightarrows r f base change theorems v t e like f displaystyle f_ the direct image with compact support f displaystyle f_ can also be derived by virtue of the following isomorphism r f f displaystyle rf_ mathcal f parametrizes the cohomology with compact support of the fibers of f displaystyle f 14 r i f f y h c i f 1 y f displaystyle r i f_ mathcal f _ y h_ c i f 1 y mathcal f this isomorphism is an example of a base change theorem there is another adjunction r f d x d y f displaystyle rf_ d x rightleftarrows d y f unlike all the functors considered above the twisted or exceptional inverse image functor f displaystyle f is in general only defined on the level of derived categories i e the functor is not obtained as the derived functor of some functor between abelian categories if f x displaystyle f x to and x is a smooth orientable manifold of dimension n then 15 f r _ r _ n displaystyle f underline mathbf r cong underline mathbf r n this computation and the compatibility of the functors with duality see verdier duality can be used to obtain a high brow explanation of poincaré duality in the context of quasi coherent sheaves on schemes there is a similar duality known as coherent duality perverse sheaves are certain objects in d x displaystyle d x i e complexes of sheaves but not in general sheaves proper they are an important tool to study the geometry of singularities 16 derived categories of coherent sheaves and the grothendieck group edit another important application of derived cat...
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