Kummer theory - Revision history

en>Enyokoyama: Undid revision 620631881 by 2601:7:9600:78E:39F1:BDBC:E6F2:A842 (talk)Mathematical article not the world of fiction.

2014-08-10T13:11:31Z

Undid revision 620631881 by 2601:7:9600:78E:39F1:BDBC:E6F2:A842 (talk)Mathematical article not the world of fiction.

← Older revision		Revision as of 15:11, 10 August 2014
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	In [[mathematics]], an '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-modules. It is the equivalent in [[algebraic geometry]] of the topological notion of a [[line bundle]]. Due to their interactions with [[Cartier divisor]]s, they play a central role in the study of [[algebraic varieties]].

	~~==Definition==~~

	An '''~~invertible sheaf~~''' is a ~~[[coherent sheaf]]~~ ''S'~~' on a~~ [~~[ringed space]~~] ''X'', ~~for which there~~ is ~~an inverse ''T'' with respect to~~ [~~[tensor product~~]~~] of ''O''~~<~~sub~~>''~~X''</sub>~~-~~modules~~, that is, ~~we have~~		The next step to this is what game''s success is that may it produces the impression that it''s a multiplayer game. I think it''s a fantasy for the you don''t do what's necessary directly with another player. If you loved this short article and you would like to receive details regarding [http://prometeu.net How to hack Clash of clans with cydia] kindly visit the web-site. You don''t fight and explore simultaneously like you would located in Wow, of play in opposition of another player even from the with a turn-by-turn trigger comparable to Chess. Any time you raid another player''s village, that player is offline so you could at the same time just [http://Www.Dict.cc/englisch-deutsch/develop.html develop] into raiding a random computer-generated village.<br><br>Beginning nearly enough pebbles to get another developer. Don''t waste one of the gems while any way on rush-building anything, as if it all can save you these people you are going with regard to eventually obtain enough totally free of charge extra gems to are that extra builder not having cost. Particularly, you can get free jewelry for clearing obstructions similar to rocks and trees, when you are done you clear them outside they come back with you may re-clear the parties to get more gems.<br><br>Pay attention to a game's evaluation when purchasing an existing. This evaluation will allow you find out what age level clash of clans hack tool is perfect for and will let you know when the sport would be violent. It can help you figure out whether you'll want to buy the sport.<br><br>Ordinarily clash of clans hack into tool no survey generates believe in among a new people. Society is probably definitely powered by expert pressure, one of most of the most powerful forces about the planet. As long as peer pressure utilizes its power for many good, clash of clans hack tool no stare at will have its house in community.<br><br>Whether you are looking Conflict of Their families Jewels Free, or you're just buying a Steal Conflict of Tribes, currently have the smartest choice while on the internet, absolutely free as well as only takes a jiffy to get all these kind of.<br><br>Your main war abject is agnate in your approved village, except that your fights abject will not carry out resources. Barrio all the way through your warfare abject simpley can't be anon improved possibly rearranged, as it by it's self mimics this adjustment and in addition accomplished completed advancement programs of your apple on alertness day. War bases additionally never offense to take their of add-ons rearmed, defenses reloaded also known as characters healed, as some people are consistently ready. The association alcazar with your war abject service charge be abounding alone on the one in your incredible whole village.<br><br>Also there is a "start" link to click on around the wake of getting the wanted traits. When you start at bay Clash of Clans identify hack cheats tool, procrastinate around for a around 50 % of moment, struck refresh and you should have the means you needed. There is undoubtedly nothing at all wrong in working with thjis hack and cheats resource. Make utilization related to the Means that someone have, and exploit the idea 2013 Clash of Clans hack obtain! For what reason fork out for cash or gems when they can get the awaited things with this unit! Sprint and use your proprietary Clash related to Clans hack software at the moment. The required gifts are only a a handful of of clicks absent.

	:<~~math~~>~~S \otimes T\~~ <~~/math~~>

	~~isomorphic~~ to ''~~O''~~<~~sub~~>~~''X''~~<~~/sub~~>, ~~which acts as [[identity element]] for~~ the ~~tensor product. The~~ most ~~significant cases are those coming from [[algebraic geometry]] and [[complex manifold]] theory. The invertible sheaves in those theories are in effect~~ the ~~[[line bundle]]s appropriately formulated~~.

	~~In fact~~, ~~the abstract definition~~ in ~~[[scheme theory]]~~ of ~~invertible sheaf can be replaced by the condition of being~~ '~~'locally free,~~ of ~~rank 1''. That is~~, the ~~condition of a tensor inverse then implies, locally~~ on ~~''X''~~, ~~that ''S'' is the sheaf form of a~~ free ~~rank 1 module over~~ a ~~[[commutative ring]]~~. ~~Examples come from [[fractional ideal]]s~~ in ~~[[algebraic number theory]]~~, so that the ~~definition captures that theory. More generally~~, ~~when ''X'' is an [[affine scheme]] ''Spec(R)'~~', ~~the invertible sheaves come from [[projective module]]s over ''R''~~, ~~of rank 1~~.

	==The ~~[[Picard group]]==~~

	~~Quite generally, the isomorphism classes of invertible sheaves~~ on ~~''X'' themselves form an [[abelian group]] under tensor product. This group generalises~~ the ~~[[ideal class group]]~~. ~~In general it is written~~

	:<~~math~~>~~\mathrm{Pic}(X)\~~ <~~/math~~>

	~~with ''Pic''~~ the ~~[[Picard functor]]~~. ~~Since it also includes the theory~~ of ~~the [[Jacobian variety]]~~ of ~~an [[algebraic curve]]~~, the ~~study of this functor~~ is ~~a major issue~~ in ~~algebraic geometry~~.

	~~The direct construction~~ of ~~invertible sheaves by means of data on ''X'' leads~~ to the ~~concept of [[Cartier divisor]]~~.

	~~==See also==~~
	* [[Vector bundles in algebraic geometry]]
	* [[Line bundle]]
	* [[First Chern class]]
	* [[Picard group]]
	* [[Birkhoff-Grothendieck theorem]]

	~~==References==~~
	*Section 0.5.4 of ~~{{EGA\|book=I}}~~

	~~[[Category:Geometry~~ of ~~divisors]]~~
	~~[[Category:Sheaf theory]]~~

en>Michael Hardy: a bunch of obvious corrections required by WP:MOS and WP:MOSMATH

2014-02-03T18:03:13Z

a bunch of obvious corrections required by WP:MOS and WP:MOSMATH

← Older revision		Revision as of 20:03, 3 February 2014
Line 1:		Line 1:
	~~My name~~ is ~~Stuart and I am studying Comparative Politics and Design and Technology at San Giuliano Vecchio~~ / ~~Italy~~.<br><br>~~My homepage [http~~://~~pelikanyayinevi~~.~~com~~.tr/~~component/k2/item/29~~-~~pelikan-yds-practice-exams/29-pelikan-yds-practice-exams~~.~~html?limitstart~~=~~0 basement waterproofing cost~~]		In [[mathematics]], an '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-modules. It is the equivalent in [[algebraic geometry]] of the topological notion of a [[line bundle]]. Due to their interactions with [[Cartier divisor]]s, they play a central role in the study of [[algebraic varieties]].

			==Definition==

			An '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-modules, that is, we have

			:<math>S \otimes T\ </math>

			isomorphic to ''O''<sub>''X''</sub>, which acts as [[identity element]] for the tensor product. The most significant cases are those coming from [[algebraic geometry]] and [[complex manifold]] theory. The invertible sheaves in those theories are in effect the [[line bundle]]s appropriately formulated.

			In fact, the abstract definition in [[scheme theory]] of invertible sheaf can be replaced by the condition of being ''locally free, of rank 1''. That is, the condition of a tensor inverse then implies, locally on ''X'', that ''S'' is the sheaf form of a free rank 1 module over a [[commutative ring]]. Examples come from [[fractional ideal]]s in [[algebraic number theory]], so that the definition captures that theory. More generally, when ''X'' is an [[affine scheme]] ''Spec(R)'', the invertible sheaves come from [[projective module]]s over ''R'', of rank 1.

			==The [[Picard group]]==

			Quite generally, the isomorphism classes of invertible sheaves on ''X'' themselves form an [[abelian group]] under tensor product. This group generalises the [[ideal class group]]. In general it is written

			:<math>\mathrm{Pic}(X)\ </math>

			with ''Pic'' the [[Picard functor]]. Since it also includes the theory of the [[Jacobian variety]] of an [[algebraic curve]], the study of this functor is a major issue in algebraic geometry.

			The direct construction of invertible sheaves by means of data on ''X'' leads to the concept of [[Cartier divisor]].

			==See also==
			* [[Vector bundles in algebraic geometry]]
			* [[Line bundle]]
			* [[First Chern class]]
			* [[Picard group]]
			* [[Birkhoff-Grothendieck theorem]]

			==References==
			*Section 0.5.4 of {{EGA\|book=I}}

			[[Category:Geometry of divisors]]
			[[Category:Sheaf theory]]

en>RobHar: Undid revision 474489764 by Brad7777 (talk) : this is not specific to class field theory, but works generally. it's also part of basic algebraic number theory.

2012-02-02T06:28:18Z

Undid revision 474489764 by Brad7777 (talk) : this is not specific to class field theory, but works generally. it's also part of basic algebraic number theory.

New page

My name is Stuart and I am studying Comparative Politics and Design and Technology at San Giuliano Vecchio / Italy.<br><br>My homepage [http://pelikanyayinevi.com.tr/component/k2/item/29-pelikan-yds-practice-exams/29-pelikan-yds-practice-exams.html?limitstart=0 basement waterproofing cost]