In mathematics, crystallinecohomology is a Weil cohomology theory for schemes introduced by harvs txt ... over rings of Witt vector s over the base field. Crystallinecohomology is partly inspired ... de Rham cohomology introduced by Grothendieck 1963 . Roughly speaking, crystallinecohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystallinecohomology reduced mod p after taking into account higher Tor s . The idea of crystallinecohomology, roughly, is to replace the Zariski open .... Crystallinecohomology only works well for smooth proper schemes. Rigid cohomology extends ... adic L function s. Crystallinecohomology, from the point of view of number theory, fills a gap ... . Traditionally the preserve of ramification theory , crystallinecohomology converts this situation ... over a quadratic extension of the field of order p . Grothendieck s crystallinecohomology theory ... obvious that this cohomology is independent of the choice of lifting. The idea of crystallinecohomology .... Crystallinecohomology In characteristic p the most obvious analogue of the crystalline site ... power structure on J compatible with the one on W sub n sub . Crystallinecohomology of a scheme ... i X W n H i Cris X W n ,O math is the cohomology of the crystalline site of X W sub n sub with values in the sheaf of rings O O sub X Wn sub . A key point of the theory is that the crystallinecohomology ... i X W H i DR Z W quad H i Z, Omega Z W lim leftarrow H i Z, Omega Z W n math of the crystallinecohomology ... as the reduction mod p of its crystallinecohomology after taking higher Tor s into account ... Notes on crystallinecohomology publisher Princeton University Press isbn 978 0 691 08218 9 mr .... mr 0393034 year 1975 volume 29 chapter Report on crystallinecohomology pages 459 478 Citation last1 ... 1994 volume 55 chapter Crystallinecohomology pages 43 70 Citation last1 Kedlaya first1 Kiran S. editor1 ... more details
Coherent cohomologyCrystallinecohomology Cyclic cohomology Deligne cohomology Dirac cohomology tale ...In mathematics , specifically in algebraic topology , cohomology is a general term for a sequence of abelian group s defined from a chain complex co chain complex . That is, cohomology is defined as the abstract study of cochains , chain complex cocycle s, and coboundary coboundaries . Cohomology can ... algebraic structure than does homology mathematics homology . Cohomology arises from the algebraic ... , the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra . The terminology tends to mask the fact that in many applications cohomology , a contravariant ... with gives rise to a function F o on X . Cohomology groups often also have a natural product, the cup product , which gives them a ring mathematics ring structure. Because of this feature, cohomology ... that homology cannot. Definition For a topological space X , the cohomology group H sup n sup X ... in Ker sup n sup are n cocycles and elements in Im sup n 1 sup are n coboundaries . The cohomology groups with n &ge 1 are called the higher cohomology . History Although cohomology is fundamental ... duality theorem, contained the germ of the idea of cohomology, but this was not seen until later. There were various precursors to cohomology. In the mid 1920s, J.W. Alexander and Solomon Lefschetz ... naturally interpreted in terms of cohomology. In 1934, Lev Pontryagin proved the Pontryagin duality ... both introduced cohomology and tried to construct a cohomology product structure. In 1936 Norman Steenrod published a paper constructing ech cohomology by dualizing ech homology . From 1936 to 1938, Hassler Whitney and Eduard ech developed the cup product making cohomology into a graded ring ... the technical limitations, and gave the modern definition of singular homology and cohomology. In 1945, Eilenberg and Steenrod stated the Eilenberg Steenrod axioms axioms defining a homology or cohomology ... more details
File Table fructose.JPG thumb Crystalline fructose Crystalline fructose is a processed sweetener derived from Maize corn that is almost entirely fructose . It consists of at least 98 pure fructose, any remainder being water and trace minerals. It is used as a sweetener in the likes of beverages and yogurt s, where it substitutes for high fructose corn syrup HFCS and table sugar . Crystalline fructose is estimated to be about 20 percent sweeter than table sugar, ref name ADAFact1 cite web url http www.fructose.org pdf ADAFructosefactsheetfinal.pdf title Nutrition Fact Sheet Facts About Fructose publisher American Dietetic Association accessdate 2010 01 30 ref and 5 sweeter than HFCS. ref name LATimes1 Production See also High fructose corn syrup Production l1 Production of high fructose corn syrup Crystalline fructose is created from cornstarch , but other starch es such as rice and wheat can be used. ref cite web url http fructose.org facts.asp title Facts about Fructose date 2006 publisher Calorie Control Council accessdate 2010 01 30 ref In this method, corn is first milled to produce cornstarch, then processed to yield corn syrup which is almost entirely glucose . The glucose obtained ... is then allowed to crystallize out, and is finally dried and milled to produce crystalline fructose. Health effects There are studies for and against the health benefits of crystalline fructose. ref name LATimes1 cite news title Is crystalline fructose a better choice of sweetener? work Los Angeles ... accessdate 2010 01 30 first Elena last Conis ref Crystalline fructose is generally considered ... guest486771 sugar the bitter truth As of January 2010, the FDA has not designated crystalline ... fructose&sortColumn &rpt scogsListing accessdate 2010 02 10 ref Because crystalline fructose is sweeter ... benefit of crystalline fructose consumption is fueled primarily by the fact that fructose does have ... 2010 DEFAULTSORT Crystalline Fructose Category Sweeteners Ingredient stub ... more details
Infobox single Name Crystalline Cosmogony Cover CrystallineBj rk.png Caption Crystalline cover Artist Bj rk from Album Biophilia album Biophilia B side Tesla br Mawal Released June 28, 2011 small Crystalline ... Length 5 05 Crystalline br 4 51 Cosmogony Label One Little Indian Records One Little Indian , Polydor ... 10 24 ref Crystalline Certification Last single The Comet Song br 2010 This single Crystalline br Cosmogony ... Audio file Bj rk Crystalline audio sample.ogg Crystalline is a song by Icelandic artist Bj rk , released ... Crystalline et Cosmogony en cd et title Crystalline et Cosmogony en cd et vinyl publisher .... The release of the song was preceded by three teasers on the first one, entitled Road to Crystalline ... the internet on June 25, 2011. ref cite web url http www.bjorkspain.net 2011 06 25 escucha crystalline primer single de lo nuevo de bjork title Escucha Crystalline , primer single de lo nuevo de Bj rk ... via Facebook that they produced the song. Crystalline is a mostly electronic song, featuring a continuous ... last Cragg first Michael url http www.guardian.co.uk music musicblog 2011 jun 29 new music bjork crystalline title New music Bj rk Crystalline & 124 Music publisher guardian.co.uk date 2011 06 29 accessdate 2011 10 24 ref The lyrics to Crystalline talk about the process of crystallization in mineral ... thing and we re all taken care of. Crystalline music video File Crystallinevideo.jpg thumb left 200px Bj rk dancing in a sheer ball among many crystals in the Crystalline music video. The music video for Crystalline was recorded on May 26 and was directed by long time collaborator Michel Gondry . ref ... about these matters. ref cite web url http bjork.fr Le clip de Crystalline title Le clip de Crystalline publisher Bjork.fr date 2011 07 26 accessdate 2011 10 24 ref Apps File CosmogonyApp.jpg ... Biophilia Universe , which includes the shortcuts to the rest of the apps. Crystalline File Crystallineapp.png thumb 150px Still from the Crystalline game. The app for Crystalline consists in a video ... more details
structure , and carbon fiber , in which the crystalline anisotropy is so great that a good ... work destroys much of the crystalline order, and the new crystallites that arise with Annealing metallurgy ... on crystalline orientation, often needle or plate shaped. These particles align themselves as water ... Cullity Failures can correlate with the crystalline textures formed during fabrication or use of that component ... http www.ecole.ensicaen.fr chateign texture combined.pdf Combined Analysis DEFAULTSORT Texture Crystalline ... more details
In mathematics, rigid cohomology is a p adic cohomology theory introduced by harvtxt Berthelot 1986 . It extends crystallinecohomology to schemes that need not be proper or smooth, and extends Monsky Washnitzer cohomology to non affine varieties. References Citation last1 Berthelot first1 Pierre author1 link Pierre Berthelot mathematician title G om trie rigide et cohomologie des vari t s alg briques de caract ristique p url http www.numdam.org item?id MSMF 1986 2 23 R3 0 id MathSciNet id 865810 year 1986 journal M moires de la Soci t Math matique de France. Nouvelle S rie issn 0037 9484 issue 23 pages 7 32 Citation last1 Le Stum first1 Bernard title Rigid cohomology url http www.cambridge.org catalogue catalogue.asp?isbn 0521875242 publisher Cambridge University Press series Cambridge Tracts in Mathematics isbn 978 0 521 87524 0 id MathSciNet id 2358812 year 2007 volume 172 Citation last1 Tsuzuki first1 Nobuo title Rigid cohomology id MathSciNet id 2560145 year 2009 journal Mathematical Society of Japan. Sugaku Mathematics issn 0039 470X volume 61 issue 1 pages 64 82 Category Algebraic geometry Category Cohomology theories Category Homological algebra ... more details
contain the information provided by algebraic de Rham cohomology , and crystallinecohomology . In some sense motivic cohomology would be the mother of all cohomology theories in algebraic geometry the other cohomology theories would be specializations. Grothendieck gave a solution for Weil cohomology theory Weil cohomology theories over a field in 1967. This involved extending the category of smooth ...Motivic cohomology is a Cohomology cohomological theory in mathematics , the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so called standard conjectures on algebraic cycles , in algebraic geometry . It had a basis in category theory for drawing consequences from those conjectures Grothendieck and Enrico Bombieri showed the depth of this approach by deriving a conditional proof of the Weil conjectures by this route. The standard conjectures, however, resisted proof. This left the motive motif in French theory as having heuristic status. Jean Pierre Serre Serre , for example, preferred to work more concretely with a compatible system of adic representations , which at least conjecturally ... by means of its realisations in the tale cohomology theories with l adic coefficients, as l varied ... all varieties over the field, and a universal cohomology theory on mixed motives in the sense of Homological ... Hodge cycle s provided one technical fix. Beilinson s absolute Hodge cohomology provided a universal cohomology theory with rational coefficients and without any category of motives using algebraic K ... constructed a bigraded motivic cohomology theory math H p,q X math for algebraic varieties .... Voevodsky provided two constructions of motivic cohomology for algebraic varieties, via ... and C. Weibel title Lectures in Motivic Cohomology url http math.rutgers.edu weibel motiviclectures.html ... Cohomology Category Cohomology theories Category Topological methods of algebraic geometry fr Cohomologie ... more details
standard in algebraic geometry and complex manifold s. The particular needs of tale cohomology were more about reinterpreting sheaf in sheaf cohomology , than cohomology , given that the derived functor approach applied. Flat cohomology , crystallinecohomology and successors are also applications ...In mathematics , sheaf cohomology is the aspect of sheaf theory , concerned with sheaves of abelian group .... Its development was rapid in the years after 1950, when it was realised that sheaf cohomology ... or ranks of sheaf cohomology groups became a fresh source of geometric data, or gave rise to new interpretations of older work. Definitions The approach of ech cohomology Main ech cohomology The first version of sheaf cohomology to be defined was that based on ech cohomology , in which ... cohomology itself fails to have good properties, unless X itself is well behaved . This is not a difficulty ... theory manifests itself in the failure of the long exact sequence of cohomology group s associated ... definition clarified the status of sheaf cohomology of a topological space X with coefficients ... can be computed by applying the functor to any acyclic resolution and keeping the cohomology of the complex, there are a number of other ways to compute cohomology groups. Depending on the concrete situation, fine, flasque, soft or acyclic sheaves are used to calculate concrete cohomology groups see ... finite values on the cohomology groups in question. Therefore finiteness theorem s of two kinds are required. In theories such as coherent cohomology , where such theorems exist, the value of &chi ... theory to produce numerical results. Relationship with singular cohomology For a locally contractible topological space, the singular cohomology groups with coefficients in A agree with the sheaf cohomology ... on sheaves treats sheaf cohomology, for example Citation last1 Griffiths first1 Phillip author1 ..., i.e. referring to the Zariski topology Citation last1 Iversen first1 Birger title Cohomology of sheaves ... more details
In mathematics, the homology or cohomology of an algebra may refer to Banach algebra cohomology of a bimodule over a Banach algebra Cyclic homology of an associative algebra Group cohomology of a module over a group ring or a representation of a group Hochschild homology of a bimodule over an associative algebra Lie algebra cohomology of a module over a Lie algebra Supplemented algebra cohomology of a module over a supplemented associative algebra See also Ext functor Tor functor mathdab Category Homological algebra ... more details
In mathematics , specifically algebraic topology , the cohomology ring of a topological space X is a ring mathematics ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here cohomology is usually understood as singular cohomology , but the ring structure is also present in other theories such as de Rham cohomology . It is also functorial for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups H sup k sup X R on X with coefficients in a commutative ring R typically R is Z sub n sub , Z , Q , R , or C one can define the cup product , which takes the form math H k X R times H ell X R to H k ell X R . math The cup product gives a multiplication on the direct sum of modules direct sum of the cohomology groups math H bullet X R bigoplus k in mathbb N H k X R . math This multiplication turns H sup &bull sup X R into a ring. If fact, it is naturally an N graded ring with the nonnegative integer k serving as the degree. The cup product respects this grading. The cohomology ring is graded commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and & x2113 we have math alpha k smile beta ell 1 k ell beta ell smile alpha k . math A numerical invariant derived from the cohomology ring is the cup length , which means the maximum number of graded elements of degree &ge 1, which when multiplied give a non zero result. For example a complex projective space has cup length equal to its complex dimension . References Cite book first S. P. last Novikov title Topology I, General Survey publisher Springer Verlag year 1996 ISBN 7 03 016673 6 Category Homology theory ... more details
In mathematics , local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck . He developed it in seminars in 1961 at Harvard University , and 1961 2 at IHES . It was later written up as SGA2 . Applications to commutative algebra and hyperfunction theory followed. In the geometric form of the theory, sections &Gamma sub Y sub are considered of a sheaf mathematics sheaf F of abelian group s, on a topological space X , with support mathematics support in a closed subset Y . The derived functor s of &Gamma sub Y sub form local cohomology groups H sub Y sub sup i sup X , F There is a long exact sequence of sheaf cohomology linking the ordinary sheaf cohomology of X and of the open set U X Y , with the local cohomology groups. The initial applications were to analogues of the Lefschetz hyperplane theorem s. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety , except for some loss that can be controlled. These results applied to the algebraic fundamental group and to the Picard group . In commutative algebra for a commutative ring R and its spectrum of a ring spectrum Spec R as X , Y can be replaced by the closed subscheme defined by an ideal I of R . The sheaf F can be replaced by an R module mathematics module M , which gives a quasicoherent sheaf on Spec R . In this setting the depth of a module can be characterised over local ring s by the vanishing of local cohomology groups, and there is an analogue, the local duality theorem , of Serre duality , using Ext functors of R modules and a dualising module . References M. P. Brodman and R. Y. Sharp 1998 Local Cohomology An Algebraic Introduction with Geometric Applications R. Hartshorne 1967 Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. External links ... Cohomology theories Category Commutative algebra Category Duality theories ... more details
Wikify date September 2011 In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by harvtxt Spencer 1969 . References eom id S s130480 first Valentin last Lychagin Citation last1 Spencer first1 D. C. title Overdetermined systems of linear partial differential equations doi 10.1090 S0002 9904 1969 12129 4 mr 0242200 year 1969 journal Bulletin of the American Mathematical Society issn 0002 9904 volume 75 issue 2 pages 179 239 Category Sheaf theory Category Cohomology theories ... more details
In mathematics , the cohomology operation concept became central to algebraic topology , particularly homotopy theory , from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory , then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points the operations can be studied by combinatorial means and the effect of the operations is to yield an interesting bicommutant theory. The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod , who first defined the Pontryagin square , Postnikov square , and Steenrod square operations for singular cohomology , in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group . In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functor s, the derived functor s of Hom ... to come by. This connection established the deep interest of the cohomology operations for homotopy theory , and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints. Formal definition A cohomology ... Cohomology of CW complexes is representable functor representable by an Eilenberg MacLane space , so by the Yoneda lemma a cohomology operation of type math n,q, pi,G math is given by a homotopy class of maps math K pi,n to K G,q math . Using representable functor representability once again, the cohomology ... Citation last1 Mosher first1 Robert E. last2 Tangora first2 Martin C. title Cohomology operations ... title Cohomology operations url http books.google.com books?id CF3bt4oYZ2oC publisher Princeton ... volume 50 DEFAULTSORT Cohomology Operation Category Algebraic topology ... more details
DISPLAYTITLE L sup 2 sup cohomology In mathematics, L sup 2 sup cohomology is a cohomology theory for smooth non compact manifolds M with Riemannian metric . It is defined in the same way as de Rham cohomology except that one uses square integrable differential form s. The notion of square integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form . L sup 2 sup cohomology , which grew in part out of L sup 2 sub d bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker 1978 and Jeff Cheeger 1979 . It is closely related to intersection cohomology indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology. Another such result is the Zucker conjecture , which states that for a Hermitian locally symmetric variety the L sup 2 sup cohomology is isomorphic to the intersection cohomology with the middle perversity of its Baily&ndash Borel compactification Zucker 1982 . This was proved in different ways by Looijenga 1988 and by Saper and Stern 1990 . References Cite book publisher Soc. Math. France pages 43 72. Ast erisque, No. 32 33 last Atiyah first M. F title Colloque ..., 2103&ndash 2106. MathSciNet id 0530173 J. Cheeger, M. Goresky, R. MacPherson, L sup 2 sup cohomology ... goresky pdf zucker.pdf L sup 2 sup cohomology is intersection cohomology Frances Kirwan ..., Eduard L sup 2 sup cohomology of locally symmetric varieties. Compositio Math. 67 1988 , no. 1 ... Surveys in Mathematics year 2002 Saper, Leslie Stern, Mark L sub 2 sub cohomology of arithmetic ..., Steven, Hodge theory with degenerating coefficients L sup 2 sup cohomology in the Poincar metric. Annals of Math. 109 1979 , 415&ndash 476. Zucker, Steven, L sup 2 sup cohomology of warped products and arithmetic groups. Inventiones Math. 70 1982 , 169&ndash 218. DEFAULTSORT L Cohomology Category Cohomology theories Category Differential geometry Category Differential topology differential ... more details
In mathematics , equivariant cohomology is a theory from algebraic topology which applies to spaces with a group mathematics group group action action . It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory . Specifically, given a group math G math discrete or not , a topological space math X math and an action math G times X rightarrow X, math equivariant cohomology determines a graded ring math H GX, math the equivariant cohomology ring . If math G math is the trivial group , this is just the ordinary cohomology ring of math X math , whereas if math X math is contractible , it reduces to the group cohomology of math G math . Outline construction Equivariant cohomology can be constructed as the ordinary cohomology of a suitable space determined by math X math and math G math , called the homotopy orbit space math X hG math of math G math on math X math . The h distinguishes it from the ordinary orbit space math X G math . If math G math is the trivial group this space math X hG math will turn out to be just math X math itself, whereas if math X math is contractible the space will be a classifying space for math G math . Properties of the homotopy orbit space If math G times X rightarrow X math is a free action then math X hG sim X G. math If math G times X rightarrow X math is a trivial action then math X hG sim X times BG. math In particular as a special case of either of the above if math G math is trivial then math X hG sim X. math Construction of the homotopy orbit space The homotopy orbit space is a homotopically correct version of the orbit space the quotient of math X math by its math G math action in which math X math is first ... id e e036090 title Equivariant cohomology Cite journal last Tu first Loring W. title What Is . . . Equivariant Cohomology? journal AMS Notices volume 58 issue 03 pages 423 426 date March 2011 url http www.ams.org notices 201103 rtx110300423p.pdf Further reading Equivariant cohomology and equivariant ... more details
In mathematics , in particular in algebraic geometry and differential geometry , Dolbeault cohomology named after Pierre Dolbeault is an analog of de Rham cohomology for complex manifold s. Let M be a complex manifold. Then the Dolbeault cohomology groups H sup p , q sup M , C depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential form s of degree p , q . Construction of the cohomology groups Let &Omega sup p , q sup be the vector bundle of complex differential forms of degree p , q . In the article on complex differential form complex forms , the Dolbeault operator is defined as a differential operator on smooth sections math bar partial Gamma Omega p,q rightarrow Gamma Omega p,q 1 math Since math bar partial 2 0 math this operator has some associated cohomology . Specifically, define the cohomology to be the quotient space linear algebra quotient space math H p,q M, mathbb C frac hbox ker left bar partial Gamma Omega p,q ,M rightarrow Gamma Omega p,q 1 ,M right bar partial Gamma Omega p,q 1 . math Dolbeault cohomology of vector bundles If E is a holomorphic vector bundle on a complex manifold X , then one can define likewise a fine injective resolution resolution of the sheaf math mathcal O E math of holomorphic sections of E . This is therefore a recollection of the sheaf cohomology of math mathcal O E math . Dolbeault s theorem Dolbeault s theorem is a complex analog of de Rham s theorem . It asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf mathematics sheaf of holomorphic differential forms. Specifically, math H p,q M cong H q M, Omega p math where &Omega sup p sup is the sheaf of holomorphic p forms on M . Proof Let math mathcal F p,q math be the fine sheaf of math C infty ... breaks up into short exact sequences. The long exact sequences of cohomology corresponding ... Category Cohomology theories Category Complex manifolds Category Hodge theory de Dolbeault Kohomologie ... more details
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold . It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobian s. For introductory accounts of Deligne cohomology see harvtxt Brylinski 2008 loc section 1.5 , harvtxt Esnault Viehweg 1988 , and harvtxt Gomi 2009 loc section 2 . Definition The analytic Deligne complex Z p sub D, an sub on a complex analytic manifold X is math 0 rightarrow mathbf Z p rightarrow Omega 0 X rightarrow Omega 1 X rightarrow cdots rightarrow Omega X p 1 rightarrow 0 rightarrow dots math where Z p 2 i sup p sup Z . Depending on the context, math Omega X math is either the complex of smooth i.e., C sup sup differential form s or of holomorphic forms, respectively. The Deligne cohomology nowrap SubSup H D,an q X , Z p is the q th hypercohomology of the Deligne complex. Properties Deligne cohomology groups nowrap SubSup H D q X , Z p can be described geometrically, especially in low degrees. For p 0, it agrees with the q th singular cohomology group with Z coefficients , by definition. For q 2 and p 1, it is isomorphic to the group of isomorphism classes of smooth or holomorphic, depending on the context principal bundle principal C sup × sup bundles over X . For p q 2, it is the group of isomorphism classes of C sup × sup bundles with connection fiber bundle connection . For q ... on them harvtxt Gajer 1997 . Applications Deligne cohomology is used to formulate Beilinson ... volume 4 chapter Deligne Be linson cohomology chapterurl http www.uni due.de mat903 preprints ec ... of Deligne cohomology doi 10.1007 s002220050118 year 1997 journal Inventiones Mathematicae issn 0020 ... representations of smooth Deligne cohomology groups doi 10.1016 j.geomphys.2009.06.012 mr 2541824 ... arxiv math 0510187 Category Sheaf theory Category Homological algebra Category Cohomology theories ... more details
In mathematics , the tale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures . tale cohomology theory can be used to construct adic cohomology , which is an example of a Weil cohomology theory in algebraic geometry. This has many ... of finite groups of Lie type . History tale cohomology was suggested by harvtxt Grothendieck 1960 , using some suggestions by J. P. Serre , and was motivated by the attempt to construct a Weil cohomology ... de g om trie alg brique SGA 4 . Grothendieck used tale cohomology to prove some of the Weil conjectures ... hypothesis was proved by Pierre Deligne 1974 using adic cohomology. Further contact with classical ... cohomology in an extremely general setting, working with concepts such as Grothendieck topos es and Grothendieck ... cohomology theory. Grothendieck s use of these universes whose existence cannot be proved in ZFC led to some uninformed speculation that tale cohomology and its applications such as the proof of Fermat s last theorem needed axioms beyond ZFC. In practice tale cohomology is used mainly for constructible ... sets, and this can easily be done in ZFC and even in much weaker theories . tale cohomology ... algebraic varieties, invariants from algebraic topology such as the fundamental group and cohomology ... could be proved using such a cohomology theory. In the case of cohomology of coherent sheaves , Serre ... variety, and in the case of complex varieties this gives the same cohomology groups for coherent ... this does not work the cohomology groups defined using the Zariski topology are badly behaved. For example, Weil envisioned a cohomology theory for varieties over finite fields with similar power as the usual singular cohomology of topological spaces, but in fact, any constant sheaf on an irreducible variety has trivial cohomology all higher cohomology groups vanish . The reason that the Zariski ... more details
In mathematics , specifically algebraic topology , ech cohomology is a cohomology theory based on the intersection ... duality for the covering math mathcal U math . The idea of ech cohomology is that, if we ..., the ech cohomology of X is defined to be the simplicial homology simplicial cohomology ... the direct limit of the cohomology groups of the nerve over the system of all possible open ... U h A U h B U. math Cohomology The ech cohomology of math mathcal U math with values in math mathcal F math is defined to be the cohomology of the cochain complex math C textbf . mathcal U , mathcal F , delta math . Thus the q th ech cohomology is given by math check H q mathcal U , mathcal F H q ... cohomology of X is defined by considering Cover topology Refinement refinement s of open covers. If math mathcal V math is a refinement of math mathcal U math then there is a map in cohomology ... of abelian groups. The ech cohomology of X with values in F is defined as the direct limit math check H X, mathcal F varinjlim mathcal U check H mathcal U, mathcal F math of this system. The ech cohomology ... by A . A variant of ech cohomology, called numerable ech cohomology , is defined as above, except .... If X is paracompact and Hausdorff space Hausdorff , then numerable ech cohomology agrees with the usual ech cohomology. Relation to other cohomology theories If math X math is homotopy equivalent to a CW complex , then the ech cohomology math check H X A math is naturally isomorphic to the singular homology singular cohomology math H X A , math . If X is a differentiable manifold , then math check H X mathbb R math is also naturally isomorphic to the de Rham cohomology the article on de Rham cohomology provides a brief review of this isomorphism. For less well behaved spaces, ech cohomology differs from singular cohomology. For example if X is the topologist s sine curve closed topologist ... , then math check H mathcal U mathbb R math is isomorphic to the de Rham cohomology. If X is compact ... more details
In mathematics , elliptic cohomology is a cohomology theory in the sense of algebraic topology . It is related to elliptic curves and modular forms . History and motivation Historically, elliptic cohomology arose from the study of elliptic genus elliptic genera . It is known by Atiyah and Hirzebruch that if math S 1 math acts smoothly and non trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Edward Witten Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning math S 1 math actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. These got in turn by Witten related to conjectural index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Douglas Ravenel Ravenel ... space. Definitions and constructions Call a cohomology theory math A math even periodic if math A i ... are elliptic curve s. A cohomology theory A with math A 0 R math is called elliptic if it is even periodic ... construction of such elliptic cohomology theories uses the Landweber exact functor theorem . If the formal group laws of E is Landweber exact, one can define an elliptic cohomology theory on finite ... morphism flat . This gives then a presheaf of cohomology theories over the site of affine scheme ... elliptic cohomology theory by taking global sections has led to the construction of the topological ... of elliptic cohomology journal Mathematische Nachrichten volume 158 issue 1 pages 43 65 doi 10.1002 ... 1988 pages 1&ndash 10 isbn 3540194908 . Citation last Landweber first Peter S. chapter Elliptic cohomology ... first2 D. lastauthoramp yes last3 Stong first3 R. chapter Periodic cohomology theories defined ... of Elliptic Cohomology title Algebraic Topology The Abel Symposium 2007 editor1 last Baas editor1 first ... doi 10.1007 978 3 642 01200 6 . Category Algebraic topology Category Cohomology theories ... more details
In mathematics , Galois cohomology is the study of the group cohomology of Galois module s, that is, the application of homological algebra to module mathematics modules for Galois group s. A Galois group G associated to a field extension L K acts in a natural way on some abelian group s, for example those constructed directly from L , but also through other Galois representation s that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois invariant elements fails to be an exact functor . History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of idele class group s in algebraic number theory was one way to formulate class field theory , at the time in the process of ridding itself of connections to L function s. Galois cohomology makes no assumption that Galois groups are abelian groups, so that this was a non abelian theory. It was formulated abstractly as a theory of class formation s. Two developments of the 1960s turned the position around. Firstly, Galois cohomology appeared as the foundational layer of tale cohomology theory roughly speaking, the theory as it applies to zero dimensional schemes . Secondly, non abelian class field theory was launched as part of the Langlands philosophy . The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves . The normal basis theorem implies that the first cohomology group of the Additive group of a ring additive group of L will vanish this is a result on general field extensions, but was known in some form to Richard Dedekind . The corresponding ... cohomology. This was formulated by means of results in class field theory, such as Hasse s norm theorem ... link Jean Pierre Serre title Galois cohomology publisher Springer Verlag location Berlin, New York ... Cohomology theories Category Galois theory Category Homological algebra de Galoiskohomologie fr ... more details
about homology and cohomology of a group homology or cohomology groups of a space or other object Homology ... theory , as well as in applications to group theory proper, group cohomology is a way to study group ... acting on module mathematics modules and quotient module s is a motivation, but the cohomology .... Thus, the group cohomology of a group G can be thought of as, and is motivated by, the singular cohomology ... space . Thus, the group cohomology of math mathbb Z math can be thought of as the singular cohomology ... R mathbb P infty math . A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functorality, and how to change groups. The subject of group cohomology .... The first group cohomology H sup 1 sup G , N precisely measures the difference. The group cohomology ... n sup G , M , the n th cohomology group of G with coefficients in M . H sup 0 sup G , M is identified with M sup G sup . Long exact sequence of cohomology In practice, one often computes the cohomology ... Se1979 Serre 1979 ref Cochain complexes Rather than using the machinery of derived functors, the cohomology ... we have a cochain complex and we can compute cohomology. For n 0, define the group of n cocycles ... The functors Ext sup n sup and formal definition of group cohomology Yet another approach is to treat G modules as modules over the group ring G , which allows one to define group cohomology via Ext ... leftarrow operatorname Hom G F 0,M leftarrow operatorname Hom G mathbb Z,M . math The cohomology groups H sup sup G , M of G with coefficients in M are defined as the cohomology of the above cochain ... of group cohomology there is the following definition of group homology given a G module G ... math H n left G,M right math . Note that the superscript subscript convention for cohomology homology ... superscripts correspond to cohomology math H math and invariants math X G, math while subscripts ... of this chain complex, math H n G,M H n F otimes mathbb Z G M math for n 0. Group homology and cohomology ... more details
In mathematics , specifically in symplectic topology and algebraic geometry , a quantum cohomology ring mathematics ring is an extension of the ordinary cohomology ring of a closed manifold closed symplectic manifold . It comes in two versions, called small and big in general, the latter is more complicated ... of ordinary cohomology describes how subspaces of the manifold Intersection theory intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a fuzzy ..., quantum cohomology has important implications for enumerative geometry . It also connects to many ... math omega math . Novikov ring Various choices of coefficient ring for the quantum cohomology of X ... math . Alternative definitions are common. Small quantum cohomology Let math H X H X, mathbb Z mathrm torsion math be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients ... sums of the form math sum i a i otimes lambda i. math The small quantum cohomology is a graded R module with math deg a i otimes lambda i deg a i deg lambda i . math The ordinary cohomology ... math is generated as a math Lambda math module by math H X math . For any two cohomology classes ... A passing through the Poincar duals of a and b . So while the ordinary cohomology considers a and b to intersect only when they meet at one or more points, the quantum cohomology records a nonzero ... 0 X math is also the identity element for small quantum cohomology. The small quantum cup product is also ... on this connection. Big quantum cohomology There exists a neighborhood U of math 0 in H math such that math ... on H are called the big quantum cohomology . All of the genus math 0 math Gromov Witten invariants are recoverable from it in general, the same is not true of the simpler small quantum cohomology ... Salamon, Dietmar & Schwarz, Matthias 1996 . Symplectic Floer Donaldson theory and quantum cohomology .... ISBN 0 521 57086 7 Category Algebraic geometry Category Cohomology theories Category String theory ... more details
Infobox Disease Name Schnyder crystalline corneal dystrophy Image Schnyder corneal dystrophy 1.JPEG Caption Schnyder corneal dystrophy. Crystalline opacities are evident in the central cornea Courtesy Dr. G.N. Foulks DiseasesDB ICD10 ICD9 ICDO OMIM 121800 MedlinePlus eMedicineSubj article eMedicineTopic 1196212 MeshID Schnyder crystalline corneal dystrophy SCD is a rare form of corneal dystrophy human human corneal dystrophy . It is caused by heterozygous mutations in UBIAD1 gene ref Orr et al, PLoS One 2007 vol 2, e685 DOI 10.1371 journal.pone.0000685 PMID 17668063 ref ref Yellore et al, Molec Vision 2007 vol 13, 1777 1782 PMID 17960116 ref ref Weiss et al, IOVS 2007 vol 48, 5007 5012 DOI 10.1167 iovs.07 0845 PMID 17962451 ref . Cells in the cornea accumulate cholesterol and phosopholipid deposits leading to the opacity, in severe cases requiring corneal transplants. Abnormal cholesterol metabolism has been noted in other cell types of affected patients skin fibroblast s suggesting that this may be a systemic disorder with clinical manifestations limited to the cornea . Alternative names Crystalline stromal dystrophy Schnyder crystalline dystrophy sine crystals Hereditary crystalline stromal dystrophy of Schnyder Schnyder s crystalline corneal dystrophy Notes Reflist Human corneal dystrophy Category Diseases of the eye and adnexa ru ... more details
Information Description Left Pure crystalline boron. Right Amorphous brown boron as powder in a glass jar. 5 grams, jar diameter 4 cm. Source http images of elements.com boron.php Author User Jurii Jurii Date unknown Permission CC BY other versions File Boron.jpg 40px File Brown boron.jpg 40px Orphan image ... more details
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