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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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ace 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 categories of sheaves is with the derived category of coherent sheaves on a scheme x displaystyle x denoted d c o h x displaystyle d_ coh x this was used by grothendieck in his development of intersection theory 17 using derived categories and k theory that the intersection product of subschemes y 1 y 2 displaystyle y_ 1 y_ 2 is represented in k theory as y 1 y 2 o y 1 o x l o y 2 k coh x displaystyle y_ 1 cdot y_ 2 mathcal o _ y_ 1 otimes _ mathcal o _ x mathbf l mathcal o _ y_ 2 in k text coh x where o y i displaystyle mathcal o _ y_ i are coherent sheaves defined by the o x displaystyle mathcal o _ x modules given by their structure sheaves sites and topoi edit main articles grothendieck topology and grothendieck topos andré weil s weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields that would give an analogue of the riemann hypothesis the cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf c _ displaystyle underline mathbf c in the euclidean topology which suggests defining a weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf but the only classical topology on such a variety is the zariski topology and the zariski topology has very few open sets so few that the cohomology of any zariski constant sheaf on an irreducible variety vanishes except in degree zero alexandre grothendieck solved this problem by introducing grothendieck topologies which axiomatize the notion of covering grothendieck s insight was that the definition of a sheaf depends only on the open sets of a topological space not on the individual points once he had axiomatized the notion of covering open sets could be replaced by other objects a presheaf takes each one of these objects to data just as before and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering this allowed grothendieck to define étale cohomology and ℓ adic cohomology which eventually were used to prove the weil conjectures a category with a grothendieck topology is called a site a category of sheaves on a site is called a topos or a grothendieck topos the notion of a topos was later abstracted by william lawvere and miles tierney to define an elementary topos which has connections to mathematical logic history edit this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed january 2023 learn how and when to remove this message the first origins of sheaf theory are hard to pin down they may be co extensive with the idea of analytic continuation clarification needed it took about 15 years for a recognisable free standing theory of sheaves to emerge from the foundational work on cohomology 1936 eduard čech introduces the nerve construction for associating a simplicial complex to an open covering 1938 hassler whitney gives a modern definition of cohomology summarizing the work since j w alexander and kolmogorov first defined cochains 1943 norman steenrod publishes on homology with local coefficients 18 1945 jean leray publishes work carried out as a prisoner of war motivated by proving fixed point theorems for application to pde theory it is the start of sheaf theory and spectral sequences 19 1947 henri cartan reproves the de rham theorem by sheaf methods in correspondence with andré weil see de rham weil theorem leray gives a sheaf definition in his courses via closed sets the later carapaces 1948 the cartan seminar writes up sheaf theory for the first time 1950 the second edition sheaf theory from the cartan seminar the sheaf space espace étalé definition is used with stalkwise structure supports are introduced and cohomology with supports continuous mappings give rise to spectral sequences at the same time kiyoshi oka introduces an idea adjacent to that of a sheaf of ideals in several complex variables 1951 the cartan seminar proves theorems a and b based on oka s work 1953 the finiteness theorem for coherent sheaves in the analytic theory is proved by cartan and jean pierre serre 20 as is serre duality 1954 serre s paper faisceaux algébriques cohérents 21 published in 1955 introduces sheaves into algebraic geometry these ideas are immediately exploited by friedrich hirzebruch who writes a major 1956 book on topological methods 1955 alexander grothendieck in lectures in kansas defines abelian category and presheaf and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces as derived functors 1956 oscar zariski s report algebraic sheaf theory 22 1957 grothendieck s tohoku paper 23 rewrites homological algebra he proves grothendieck duality i e serre duality for possibly singular algebraic varieties 1957 onwards grothendieck extends sheaf theory in line with the needs of algebraic geometry introducing schemes and general sheaves on them local cohomology derived categories with verdier and grothendieck topologies there emerges also his influential schematic idea of six operations in homological algebra 1958 roger godement s book on sheaf theory is published at around this time mikio sato proposes his hyperfunctions which will turn out to have sheaf theoretic nature at this point sheaves had become a mainstream part of mathematics with use by no means restricted to algebraic topology it was later discovered that the logic in categories of sheaves is intuitionistic logic this observation is now often referred to as kripke joyal semantics but probably should be attributed to a number of authors see also edit coherent sheaf gerbe stack mathematics sheaf of spectra perverse sheaf presheaf of spaces constructible sheaf de rham s theorem notes edit eisenbud david harris joe 6 april 2006 the geometry of schemes gtm new york ny springer pp 11 18 isbn 978 0 387 22639 2 tennison b r 1975 sheaf theory cambridge university press mr 0404390 bredon 1997 chapter v 1 demailly jean pierre complex analytic and differential geometry pdf archived pdf from the original on 28 august 2020 cartan henri variétés analytiques complexes et cohomologie pdf archived pdf from the original on 8 october 2020 1 2 differential geometry holomorphic functions on a complex compact manifold are only constants mathematics stack exchange retrieved 2020 10 07 hawley newton s 1950 a theorem on compact complex manifolds annals of mathematics 52 3 637 641 doi 10 2307 1969438 jstor 1969438 sga 4 ii 3 0 5 iversen 1986 chapter vii ramanan 2005 hartshorne 1977 theorem iii 5 1 deligne pierre 1971 théorie de hodge ii publications mathématiques de l ihés 40 5 57 doi 10 1007 bf02684692 s2cid 118967613 deligne pierre 1974 théorie de hodge iii publications mathématiques de l ihés 44 5 77 doi 10 1007 bf02685881 s2cid 189777706 iversen 1986 chapter vii theorem 1 4 kashiwara schapira 1994 chapter iii 3 1 de cataldo migliorini 2010 grothendieck formalisme des intersections sur les schema algebriques propres steenrod n e 1943 homology with local coefficients annals of mathematics 44 4 610 627 doi 10 2307 1969099 jstor 1969099 dieudonné jean 1989 a history of algebraic and differential topology 1900 1960 birkhäuser pp 123 141 isbn 978 0 8176 3388 2 cartan henri serre jean pierre 1953 un théorème de finitude concernant les variétés analytiques compactes comptes rendus hebdomadaires des séances de l académie des sciences de paris 237 128 130 zbl 0050 17701 serre jean pierre 1955 faisceaux algébriques cohérents pdf annals of mathematics 61 2 197 278 jstor 1969915 mr 0068874 zariski oscar 1956 scientific report on the second summer institute several complex variables part iii algebraic sheaf theory bulletin of the american mathematical society 62 2 117 141 doi 10 1090 s0002 9904 1956 10018 9 issn 0002 9904 grothendieck alexander 1957 sur quelques points d algèbre homologique the tohoku mathematical journal second series 9 2 119 221 doi 10 2748 tmj 1178244839 issn 0040 8735 mr 0102537 references edit bredon glen e 1997 sheaf theory graduate texts in mathematics vol 170 2nd ed springer isbn 978 0 387 94905 5 mr 1481706 oriented towards conventional topological applications de cataldo andrea mark migliorini luca 2010 what is a perverse sheaf pdf notices of the american mathematical society 57 5 632 4 arxiv 1004 2983 bibcode 2010arxiv1004 2983d mr 2664042 godement roger 2006 1973 topologie algébrique et théorie des faisceaux paris hermann isbn 2705612521 mr 0345092 grothendieck alexander 1957 sur quelques points d algèbre homologique the tohoku mathematical journal second series 9 2 119 221 doi 10 2748 tmj 1178244839 issn 0040 8735 mr 0102537 hirzebruch friedrich 1995 topological methods in algebraic geometry classics in mathematics springer isbn 978 3 540 58663 0 mr 1335917 updated edition of a classic using enough sheaf theory to show its power iversen birger 1986 cohomology of sheaves universitext springer doi 10 1007 978 3 642 82783 9 isbn 3 540 16389 1 mr 0842190 kashiwara masaki schapira pierre 1994 sheaves on manifolds grundlehren der mathematischen wissenschaften fundamental principles of mathematical sciences vol 292 springer isbn 978 3 540 51861 7 mr 1299726 advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces mac lane saunders moerdijk ieke 1994 sheaves in geometry and logic a first introduction to topos theory universitext springer verlag isbn 978 0 387 97710 2 mr 1300636 category theory and toposes emphasised martin william t chern shiing shen zariski oscar 1956 scientific report on the second summer institute several complex variables bulletin of the american mathematical society 62 2 79 141 doi 10 1090 s0002 9904 1956 10013 x issn 0002 9904 mr 0077995 ramanan s 2005 global calculus graduate studies in mathematics vol 65 american mathematical society doi 10 1090 gsm 065 isbn 0 8218 3702 8 mr 2104612 seebach j arthur seebach linda a steen lynn a 1970 ...
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