
aarr pravartthanangal

ganithashaasthrathil, alexander grothendiyakkinte perilulla grothendiyakkinte aarr pravartthanangal, homological algibrayile oru aupachaarikathayaanu, ithu aarr-fungactor aupachaarikatha ennum ariyappedunnu. skeemukalude roopeekaranathil ninnu uyarnnuvarunna ettal cohomology bandhangalil ninnaan ithu yadhaarthathil uthbhavichath. x", "y" ennivayekkurichulla cohomologyumaayi bandhapetta pala praadhamika vasthuthakalum cheria ennam svayamsidhaanthangalude aupachaarika anantharafalangalaayirunnu ennathaanu adisthaana ulkkaazhcha. eee sidhaanthangal pala kesukalilum yadhaartha sandarbhavumaayi poornnamaayum bandhamillaathavayaanu, athinaal aupachaarikamaaya anantharafalangalum nilanilkkunnu. aarr ooppareshans aupachaarikatha beejaganita enangalekkurichulla "dii"-moduleukal, praadeshikamaayi othukkamulla topological edangalile sheevukal, uddeshyangal enniva polulla sandarbhangalil baadhakamaanennu theliyikkappettittund.
ooppareshanukal.
aarr pravarthanangalaanu pravartthanangal. saadhaaranayaayi iva deriwade vibhagangal thammilulla fungectorukalaanu, athinaal yadhaarthathil idathu, valath deriwade fungectorukalaanu.
functor formula _ 2 umm formula _ 1 umm oru adjayint functor jody roopappeduthunnu, formula _ 3 umm formula _ 4 umm pole. athupole, aantharika tensor uthppannam aantharika hominode chernnu avasheshikkunnu.
ettal cohomology aarr pravartthanangal.
skeemukalude roopeekaranamaakatte. morphism "eph" niravadhi fangaktarukale prerippikkunnu. prathyekichum, ithu "x", "y" ennivayile sheevukalude vibhaagangalkkidayil "f" *, "f" * enniva nalkunnu, koodaathe ithu fungectorinu "f" nalkunnu! sariyaaya pinthunayulla nerittulla chithram. rf" enna vibhaagathil ninnu uruthirinjathaanu! oru sariyaaya adjayint "eph" ennu sammathikkunnu!. avasaanamaayi, abelian sheevukalumaayi pravartthikkumbol, oru tensor uthppanna fungactor oru aantharika hoome fungactor undu, iva adjayint aanu. aarr pravartthanangal uruthirinja vibhaagathile anubandha fangaktarukalaanu,,,,,,,,,.
9-adic tortion sheeves enna formulayude oru vibhaagathilekku nammal svayam parimithappeduthunnunnu karuthuka, avide formula 9 "x", "y" ennivayude swabhaavasavisheshathakalumaayi sahapradhaanamaan. ezz. jii. eiii 4aii. aii. yill, grothendiac artinum theliyichathu "eph" aapekshika alavil "dii" yude minusamaarnnathaanenkil, "elf" * ennathu isomorphic aanennum, avide "dii" th vipareeta tettu twistine soochippikkukayum digri bye maattathe soochippikkukayum cheiyunnu. koodaathe, "f" verthirichathum parimithamaaya tharathilullathumaanennu karuthuka. skeemukalude mattoru roopeekaranam aanenkil, "x" nte adisthaana maattathe "g" kondu soochippikkukayaanenkil, "f", "g" enniva "f", "g" ennivayude adisthaana maattangale yathaakramam "g", "f" ennivayaal soochippikkukayaanenkil, swaabhaavika isomorphisangal nilavilunda
x", "n" ennee padangalilninnu uruthirinja eathenkilum vasthukkalaaya "m" ennivaykku "f" verthirichathum parimithamaaya tharathilullathumaanennu veendum anumaanichaal, "y" enna padatthilninnu uruthirinja eathenkilum vasthuvin swaabhaavika isomorphisangal nilavilunda
i" ennathu "z"-neey "s"-lake poornnamaaya thuranna immershanodoppam "j" ennathilekku adaykkunna oru immershanaanengil, athil ninnu uruthirinja vibhaagathil oru prathyeka trikonamunda
aadyathe rund bhoopadangal yathaakramam kaundittum yoonittum aakunnu. z", "s" enniva sthiramaanenkil, oru isomorphism undu
tensor uthppanna pravartthanangalude unitukal evideyaanu, avaa evideyaanu (eathu tharam formula _ 9-aadik tortion sheeves parigananayilaan ennathine aashrayich ithu vyathyaasappedunnu).
s" regular aanenkil, "k" ennathu "s" ennathil ninnu uruthirinja vibhaagathil oru invertable objectaanengil, "d" x "neey fangaktaraayi nirvachikkuka. thudarnnu, "x" ennathil ninnu uruthirinja vibhaagathile "m", "m" ennee vasthukkalkk, canonical maappukal
avaa isomorphisangalaanu. avasaanamaayi, "s"-skeemukalude roopeekaranamaanengil, "m", "n" enniva "x", "y" ennee vibhagangalil ninnu uruthirinja vasthukkalaanenkil, swaabhaavika isomorphisangal undu

