invertable sheef

ganithashaasthrathil, modyoolukalude sheevukalude tensor ulppannavumaayi bandhappett vipareethamaaya oru ring cheytha sthalathulla oru sheefaanu invertible sheef. ithu oru line bundle enna topological aashayathinte beejaganita jyamithiyil thulyamaanu. cartier divaisarukalumaayulla avayude edapedal kaaranam, beejaganita enangalekkurichulla padanatthil avaa oru pradhana punk vahikkunnu. 
nirvachanam.
("x", "o" x ") oru ring space ayirikkatte. o" x "-modyoolukalude sheevukalude isomorphism klaasukal" o "x"-modyoolukalude tensor prodakttinte pravarthanatthinu keezhil oru monoid roopappeduthunnu. eee pravartthanatthinte aidantitti ghadakam "o" x "aanu. eee monoidinte vipareeta gadakangalaanu vipareeta sheevukal. prathyekichum, "l" ennathu "o" "x"-modyoolukalude oru sheef aanenkil, inipparayunna thulyamaaya vyavasthakalil eathenkilum thripthippeduthukayaanel "l" neey invertible ennu vilikkunnu
rank onninte praadeshikamaayi soujanyamaaya oro shef invertable aanu. x" praadeshikamaayi valanja sthalamaanengil, praadeshikamaayi rank vann illaathathaanengil maathrame "l" invertable aakoo. eee vastutha kaaranam, invertable sheevukal line bandilukalumaayi adutha bandhamullavayaan, avaa randum chilappol samyojippikkappedunna ghattathilekku.
udaaharanagal.
x" oru affine skeem ayirikkatte. appol "x"-ll oru invertable sheef ennathu "r"-nuu mukalilulla oru rank vann projective modyoolumaayi bandhapetta sheef aanu. udaaharanathinu, ithil beejaganita samkhya feeldukalude bhinnasamkhyakal ulppedunnu, kaaranam iva samkhya feeldinte poornnasamkhyakalude valayangalil oru projective modyoolukalaanu.
picard group.
pothuve, "x"-ll invertible sheevukalude isomorphism klaasukal thanne tensor uthppannathin keezhil oru abelian group roopeekarikkunnu. eee group anuyojyamaaya klaas groupine saamaanyavalkkarikkunnu. pothuve, athu ezhuthappettirikkunnu
picard functor "pick" cheythukondu. beejaganita vakrathinte jacobian vaividhyathinte sidhaanthavum ithil ulppedunnathinaal, eee fangaktarinekkurichulla patanam beejaganita jyaamithiyile oru pradhana prasnamaanu.
x" leeii daattayiloode invertible sheevukalude nerittulla nirmmaanam cartier devisor enna aashayathilekku nayikkunnu.

