
prachodanaathmaka cohomology

beejaganita enangaludeyum kooduthal pothuvaaya skeemukaludeyum oru maattamillaatha gadakamaanu motivik cohomology. ithu uddeshyangalumaayi bandhapetta oru tharam cohomologyanu, koodaathe beejaganita chakrangalude chau ring oru prathyeka kesai ulppedunnu. beejaganita jyamithiyileyum samkhyaasampradaayathileyum ettavum aazhameriya chila prasnangal prachodanaathmaka samaroopatha manassilakkaanulla sramangalaanu.
motivik homology cohomology.
x" ennathu oru feeld "k" nuu mukalilulla parimithamaaya tharatthilulla oru schemyricutte. beejaganita jyaamithiyude oru pradhana lakshyam "x"-nte chau grouppukal kanakkaakkuka ennathaanu, kaaranam avaa "x"-nte ellah upavibhagangalekkurichum sakthamaaya vivarangal nalkunnu. x" ennathinte chau grouppukalkku topologyil boral-muur homology chila aupachaarika gunangalundu, ennaal chila karyangal kanunnilla. udaaharanathinu, "x" ennathinte oru clossed subskeem "z" nuu, choo grouppukalude krithyamaaya oru kramamundu, praadeshikavalkkarana kramam.
topologyil ithu oru neenda krithyamaaya shreniyude bhagamaayirikkum.
chau grouppukale oru valia kudumba grouppukalaayi (borrell-muur) motivik homology grouppukalaayi (bloach aadyam hier chau grouppukal ennu vilichirunnu) saamaanyavalkkarichaanu eee prashnam pariharichathu. athaayath, oru feeldil "k", poornnasamkhyakalaaya "i", "j" ennivayil parimithamaaya tharatthilulla oro schemine, namukk oru abelian group "h" i "(" x ", z (" j ")) undu, saadhaarana chau group prathyeka kesaanu.
x" enna skeeminte oru clossed subskeem "z"-nuu, motivik homology grouppukalkku krithyamaaya oru krithyamaaya praadeshikavalkkarana kramamundu, ithu choo grouppukalkkulla praadeshikavalkkarana shreniyil avasaanikkunnu
vaasthavathil, vovodsky nirmmicha naalu sidhaanthangalude oru kudumbamaanitha motivik cohomology, compact sapporttulla motivik cohomology, boral-muur motivik homology (mukalil paranjathupole), compact sapporttulla motivik homology. eee sidhaanthangalkku topologyile anubandha sidhaanthangalude niravadhi aupachaarika gunangalundu. udaaharanathinu, "h" i "(x, z (" j ")) enna prachodanaathmaka cohomology grouppukal oru feeldil parimithamaaya tharatthilulla" x "enna oro schemine oru bigraded ring roopappeduthunnu. x" ennathinte alavu "k" ennathinekkaal "n" enna alavil minusamaarnnathaanenkil, oru poincare duvality isomorphisamundu.
prathyekichum, "x" ennathu "k" nuu mukalilaayi minusamaarnnathaanenkil, "h" 2 "i" ("x", z ("i")) ennathinu "ch" i "(" x ") enna codimension chakrangalude choo group isomorphic aanu.
k" nuu mukalilulla "x" enna minusamarnna skeeminte motivik cohomology "h" i "(" x ", z (" j ")) ennathu" x "ennathil oru prathyeka samuchayathile sheeves z (j)-leeii gunakangalumaayi sarisky topologyil" x "-nte cohomologyanu. (nisnevichu topology upayogichu theliyikkaan chila gunangal eluppamaanu, pakshe ithu athe motivik cohomology grouppukal nalkunnu.) udaaharanathinu, "jeey" <0 nuu z (jeey) poojyamaanu, z (0) ennathu sthiramaaya sheef z aanu, koodaathe "x" muthal "g" m [−1] vareyulla vibhaagathil z (1) isomorphic aanu. ivide "g" m (gunana group) invertible regular functionukalude sheefine soochippikkunnu, shift [−1] ennaal eee sheefine digri 1 ll oru complexaayi kaanunnu ennaanu ardhamaakkunnathu.
motivik homology cohomology naalu pathippukale eathu abelian groupileyum gunakangal upayogichu nirvachikkaam. topologyil ullathupole, vyathyastha gunakangalulla sidhaanthangal saarvathrika gunaka sidhaanthathaal bandhappettirikkunnu.
mattu cohomology sidhaanthangalumaayulla bandhangal.
keey-sidhaanthavumaayulla bandham.
bloach, lichenbom, friedlander, suslin, levin enniva prakaram, topologyile athiya-hirsebrook spectral seekwansinu samaanamaayi oru feeldile oro minusamarnna skeem "x" num motivik cohomology muthal beejaganita k-sidhaantham vare oru spectral sequence undu
topologyil ullathupole, yukthikal upayogichu tensor cheythathinu sesham spectral sequence degenerate cheiyunnu. oru feeldil parimithamaaya tharatthilulla ekapaksheeyamaaya skeemukalkk (minusamaarnnathaayirikkanamine), motivik homology muthal jii-sidhaantham vare (vekttar bandilukalekkaal yojicha sheevukalude keey-sidhaantham) oru analog spectral sequence undu.
milnor keey-sidhaanthavumaayulla bandham.
motivik cohomology feeldukalkku ithinakam thanne sambannamaaya oru maattamillaatha sthaanam nalkunnu. (oru feeld "k" oru skeem speck ("k") nirnnayikkunnu, athinaayi motivik cohomology nirvachikkappedunnu.) "k" feeldukalkkulla motivik cohomology "h" i "(" k ", z (" j ")) pothuve manassilakkaan kazhiyaathathaanengilum," i "=" j "aayirikkumbol oru vivaranamunda
ivide "k" "j" m ("k") ennathu "k" yude "j" th milnor k-grouppaanu. oru feeldinte milnor keey-sidhaanthathe janarettarukalum bandhangalum vyakthamaayi nirvachikkunnathinaal, ithu "keey" yude motivik cohomology oru bhaagatthinte upayogapradamaaya vivaranamaan.
ettal cohomologyilekku mapp cheyyuka.
x" oru feeld "k" nuu mukalilulla oru minusamarnna skeem ayirikkatte, koodaathe "m" ennathu "k" ll invertable aaya oru positteev poornnasamkhyayaayirikkate. appol motivik cohomology muthal étale cohomology vareyulla oru swaabhaavika homomorphism (saikkil mapp) undu
ivide valathuvashathulla z/"m" ("j") ennaal étale sheef (μ "m") ̃ "j" ennaanu ardhamaakkunnathu, koodaathe "m" ennathu aikyathinte "m"-aam verukalaanu. ithu oru minusamarnna vaividhyathinte chau ringle ninnu ettal cohomology vare saikkil maappine saamaanyavalkkarikkunnu.
beejaganita jyaamithiyilo samkhya sidhaanthathilo pathiv lakshyam motivik cohomology kanakkaakkuka ennathaanu, athesamayam ettaale cohomology palappozhum manassilakkaan eluppamaanu. udaaharanathinu, adisthaana feeld "k" sankeernna samkhyakalaanenkil, étale cohomology ekavachana cohomologyumaayi (parimithamaaya gunakangalode) poruthappedunnu. bailinson-lichenbom anumaanam ennariyappedunna vovodsky theliyikkunna sakthamaaya oru phalam parayunnath pala motivik cohomology grouppukalum yadhaarthathil isomorphic tuu ettal cohomology grooppukalaanennu. ithu norm residue isomorphism sidhaanthathinte anantharafalamaanu. athaayath, bailinson-lichenbom anumaanam (vovodsky sidhaantham) parayunnath oru feeldil "k", "m" ennivaykku mukalilulla oru minusamarnna schemine "k" ll vipareethamaaya oru positteev poornnasamkhyaykku, saikkil mapp
ellah "jeey" ≤ "aii" num oru isomorphisamaanu, koodaathe ellah "jeey" ≤ "aii" −1 num kuthivaykkalaanu.
uddeshyangalumaayi bandhappettava.
k", commutative ring "r" ennee eathenkilum feeldukalkku, vovodsky "r", "dm (" k ";" r ") ennee gunakangalulla" k "nuu mukalilulla deriwade kattagari oof motives ennu vilikkunna oru" r "-leanier triangulated kaattagariye nirvachichu. x" ennathinekkaal "k" ennathinekkaal "x" enna oro skeem "x", m ("x") ennathinte motive ennum "x", mc ("x") ennathinte kompaakt support motive ennum vilikkunna dii. emm.-leeii rund objectukale nirnnayikkunnu; "x" ennathu "k" nuu mukalil properaanengil iva randum isomorphic aanu.
naalu tharam motivik homology motivik cohomology eee vibhaagathil roopeekaranangalude koottangalaayi uyarnnuvarunnu ennathaanu uthbhavicha vibhaagathinte oru adisthaana pooyint. ithu vivarikkunnathin, ellah poornnasamkhyakalkkum "jeey" ennathil "aarr" ("jeey") enna tettu uddeshyangal undennu aadyam shradhikkuka, athaayath projective spacinte uddeshyam tettu uddeshyangalude nerittulla koottamaanu
ivide "m" "m" [1] ennathu thrikonaakrithiyilulla vibhagamaaya dm ("k"; "r") leeii shift allengil "translation fungactor" soochippikkunnu. eee padangalil, motivik cohomology (udaaharanathinu) nalkunnathu
k" nuu mukalilulla parimithamaaya tharatthilulla "x" enna oro schemine.
r" enna gunakam yukthisahamaya samkhyakalaayirikkumbol, bailinson nadatthiya oru anumaanathinte aadhunika pathipp, dm (k; q) leeii compact objectukalude upavibhagam "k" nuu mukalilulla mishra uddeshyangalude vibhagamaaya abelian kattagari mm ("k")-nte parimithamaaya vakabheda vibhaagathin thulyamaanennu pravachikkunnu. prathyekichum, eee anumaanam soochippikkunnathu, sammishra uddeshyangalude vibhaagathil ext grouppukalaayi motivik cohomology grouppukale thirichariyaan kazhiyumennanu. ithu valare akaleyaan. krithyamaayi paranjaal, bailinsons anumaanam soochippikkunnathu bailinson-soul anumaanam "i" <0 nuu "h" i "(x, q (" j ")) poojyamaanu, ithu kurachu kesukalil maathrame ariyoo.
neremarichu, bailinson-soul anumaanathinte oru vakabhedam, grothandiyakkinte staanderd anumaanangalum chau uddeshyangalekkurichula mareyude anumaanangalum chernnu, "dm" ("keey"; kyu)-ll oru tii-gadanayude hridayamaayi oru abelian vibhaagam "emm" ("keey") nilanilkkunnathayi soochippikkunnu. emm" ("keey") leeii ektt grouppukale motivik cohomology upayogichu thirichariyaan kooduthal aavasyamaanu.
sankeernna samkhyakalkku "k" enna upa feeld, abelian vibhaagathile mishra uddeshyangalkkulla oru sthaanaarthiye noorry nirvachichittundu. prathekshikkunna gunangalulla oru vibhaagam "emm" ("keey") nilavilundenkil (prathyekichu "emm" ("keey") muthal kyu-vector speysukal vareyulla betti realisation fungactor vishwaasayogyamaanu), athu noriyude vibhaagathin thulyamayirikkanam.
ganitha jyaamithiyilekkulla prayogangal.
ell-functionukalude moolyangal.
x" oru nambar feeldil minusamarnna projective veriyettu ayirikkatte. oru poornnasankhya pooyintil "x" nte oru l-function aprathyakshamaakunnathinulla kramam anuyojyamaaya motivik cohomology grouppinte rankinu thulyamaanennu l-functionukalude moolyangalekkurichulla bloach-kaatto anumaanam pravachikkunnu. dilligne bailinson munkootty soochippicha oohangal ulppeduthikkondu ithu samkhyaasampradaayathinte kendraprashnangalilonna. birch-swinnarttan-dier anumaanam oru prathyeka kesaanu. kooduthal krithyamaayi paranjaal, regulettarmaarude kaaryatthil oru poornnasankhya pooyintil ell-functionte pradhana gunakam pravachikkukayum motivik cohomology uyaram jodiyaakkukayum cheiyunnu.
charithram.
choo grouppukalil ninnu beejaganita enangalkkulla kooduthal pothuvaaya motivik cohomology sidhaanthathilekkulla saadhyamaaya saamaanyavalkkaranathinte aadya vyakthamaaya adayaalam kvillante beejaganita keey-sidhaanthathinte (1973) nirvachanavum vikasanavumaayirunnu, ithu vekttar bandilukalude grothendic group "keey" 0 neey saamaanyavalkkarikkunnu. 1980 kalude thudakkathil, bailinson solum nireekshichathu, aadaaminte pravartthanangal beejaganita keey-sidhaanthathinte vibhajanam nalkunnu, yukthisahamayavayum koottukettukalum ippol motivik cohomology (yukthisahamaya gunakangalode) ennu vilikkunnu. bailinson lichenbaum prachodanaathmaka cohomology nilanilppum gunangalum pravachikkunna swaadheenamulla anumanangal nadatthi. avarude mikathum ennaal ellah oohangalum ippol theliyikkappettittund.
uyarnna chau grouppukalekkurichulla blochinte nirvachanam (1986) oru feeldile skeemukalkkulla motivik homology aadyathe avibhaajya gadakamaayirunnu (yukthisahamaayathin vipareethamaayi) "keey" (athinaal sugamamaya skeemukalude kaaryatthil motivik cohomology). x" ennathinte uyarnna chau grouppukalude nirvachanam, pratheekshicha alavilulla hyperplayinukalude (oru simplexinte mukhangalayi kaanapedunna) oru koottam koottatthe kandumuttunna affine spacinoppam "x" ennathinte gunanafalathil beejaganita chakrangal ulppedunna chau grouppukalude nirvachanatthinte swaabhaavika saamaanyavalkkaranamaanu.
oduvil, vovodsky (saslinumaayulla thante pravartthanatthe adisthaanamaakki) 2000-ll naalu tharam motivik homology motivik cohomology nirvachichu, oppam uddeshyangalude vakabhedavum. hanamura, levin ennivayum anubandha vibhaagangale nirvachichu.

