Coherent duality - Revision history

en>Yobot: /* Adjoint functor point of view */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T12:19:47Z

Adjoint functor point of view: WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 14:19, 5 May 2014
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	In mathematics, especially in [[algebraic geometry]], '''base change''' refers to a number of similar theorems concerning the [[sheaf cohomology\|cohomology]] of [[sheaf (mathematics)\|sheaves]] on algebro-geometric objects such as [[Algebraic variety\|varieties]] or [[scheme (mathematics)\|schemes]].

	The situation of a base change theorem typically is as follows: given two maps of, say, schemes, <math>f: Y \rightarrow X</math>, <math>g: X' \rightarrow X</math>, let <math>f'</math> and <math>g'</math> be the projections from the [[fiber product]] <math>Y' := Y \times_X X'</math> to <math>X'</math> and <math>Y</math>, respectively. Moreover, let a sheaf <math>\mathcal F</math> on X' be given. Then, there is a natural map (obtained by means of adjunction)
	~~:<math>f^* R^i g_* \mathcal F \rightarrow R^i g'_* f'^* \mathcal F.</math>~~
	Depending on the type of sheaf, and on the type of the morphisms ''g'' and ''f'', this map is an isomorphism (of sheaves on ''Y'') in some cases. Here <math>R^i g_* \mathcal F</math> denotes the [[higher direct image]] of <math>\mathcal F</math> under ''g''. As the [[stalk (mathematics)\|stalk]] of this sheaf at a point on ''Y'' is closely related to the cohomology of the fiber of the point under ''g'', this statement is paraphrased by saying that "cohomology commutes with base extension".<ref>{{harvtxt \| Hartshorne \| 1977 \| p=255 }}</ref>

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	~~==Flat base change for quasi~~-~~coherent sheaves==~~
	~~The base change holds for~~ a [[~~quasi~~-~~coherent sheaf~~]] <~~math~~>~~\mathcal F~~</~~math> (on~~ <~~math~~>X'<~~/math~~>~~), provided that~~ the ~~map ''f''~~ is [~~[flat morphism\|flat~~]~~] (together with~~ a ~~number~~ of ~~technical conditions~~: ~~''g'' needs~~ to be a ~~[[separated morphism\|separated~~]~~] [[morphism~~ of ~~finite type]],~~ the ~~schemes involved need~~ to be ~~Noetherian)~~.

	~~==Proper base change for etale sheaves==~~
	~~The base change holds for etale torsion sheaves~~, ~~provided~~ that ~~''g'' is~~ [~~[proper morphism\|proper]~~].<~~ref~~>~~{{harvtxt\|Milne\|1980\|loc=section VI~~.~~2}}~~<~~/ref~~>

	~~==Smooth base change for etale sheaves==~~
	~~The base change holds for etale torsion sheaves, whose torsion is prime~~ to ~~the residue characteristics of ''X'', provided ''f'~~' is [~~[smooth morphism\|smooth]~~] and ~~''g'' is quasi-compact~~.<~~ref~~>~~{{harvtxt\|Milne\|1980\|loc=section VI.4}}~~<~~/ref~~>

	~~==See also==~~
	*[[Grothendieck's relative point of ~~view]]~~ in ~~algebraic geometry~~
	*[[Change of base (disambiguation)]]
	*[[Base change lifting]] of automorphic forms

	~~==Notes==~~
	~~<references />~~

	~~==References==~~
	* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link=Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, ~~New York \| isbn=978-0-387-90244-9 \| oclc=13348052 \| mr=0463157 \| year=1977}}~~
	* {{Citation \| last1=Milne \| first1=James S. ~~\| title=Étale cohomology \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08238-7 \| year=1980}}~~

	~~[[Category:Sheaf theory]]~~

85.65.186.39: wikifying E. Matlis

2013-02-26T12:16:12Z

wikifying E. Matlis

← Older revision		Revision as of 14:16, 26 February 2013
Line 1:		Line 1:
	~~The person who wrote the post is called Marcelino~~ [~~http://Search.Huffingtonpost.com/search?q=Volkman&s_it=header_form_v1 Volkman~~]. Iowa is where my home [http://www.purevolume.com/melodymabrydn/posts/6899154/How+To+Find+Out+Everything+There+Is+To+Know+About+Dagens+Spilleautomater+M%C3%A5+Fjernes+Etter+At. spilleautomater villig nett] ~~is and he will by no means move. Playing basketball is a factor that I~~'~~m totally addicted~~ to~~. Software~~ [~~http://warrenmooretgqn.newsvine.com/_news/2014/08/07/25208810-what-everybody-else-does-when-it-comes-to-dagens-spilleautomater-m-fjernes-etter-at-tipping-and Vi presenterer spilleautomater på nettet~~] ~~developing has been his~~ [~~https://www.vocabulary.com/dictionary/day+occupation day occupation~~] ~~for a~~ [~~http://www.fizzlive.com/member/1212915/blog/view/3496255 spilleautomater~~] ~~while and he's~~ [~~http://bartiuoigoqfx.soup.io/?sessid=8796d4b334b0494b429b32caddfe5829 Hvis du liker å spille på spilleautomater online] doing fairly good monetarily. Check out the latest news~~ [~~http://warrenmooretgqn.newsvine.com/_news/2014/08/07/25211558-ten-reasons-p-spilleautomater-kan-du-spille-jackpot-6000-mega-joker-is-a-waste-of-time norskeautomater~~] on [http://rochellebaldessin.newsvine.com/_news/2014/08/08/25242759-how-to-make-de-fleste-spilleautomatene-som-er-tilgjengelig-p-mobilen norske spilleautomater er en morsom kilde til underholdning og her kan du prøve mange norske automater] ~~my web site: http://kellyeleyvazmcg.pen~~.~~io/<br><br>~~		In mathematics, especially in [[algebraic geometry]], '''base change''' refers to a number of similar theorems concerning the [[sheaf cohomology\|cohomology]] of [[sheaf (mathematics)\|sheaves]] on algebro-geometric objects such as [[Algebraic variety\|varieties]] or [[scheme (mathematics)\|schemes]].

	~~Here~~ is ~~my [http~~://~~www.purevolume.com~~/~~melodymabrydn~~/~~posts~~/~~6926265~~/~~Genghis+Khan%27s+Guide+To+De+Beste+Norske+Spilleautomater+Excellence on-line spille automater] web [http:~~//~~www~~.~~purevolume~~.~~com~~/~~shennaharley~~/~~posts/6897038/Beste+Spilleautomater+Explained spilleautomater~~] ~~blog [http:/~~/~~journals~~.~~fotki~~.~~com~~/~~chancegreenberg~~/~~chancegreenberg~~/~~entry/stsbtwstqkqgq/ På Spilleautomater kan du spille Jackpot 6000 Mega Joker~~] :: [~~http:~~/~~/kellyeleyvazmcg~~.~~pen~~.io/ ~~signifiant beste norske spilleautomater~~]		The situation of a base change theorem typically is as follows: given two maps of, say, schemes, <math>f: Y \rightarrow X</math>, <math>g: X' \rightarrow X</math>, let <math>f'</math> and <math>g'</math> be the projections from the [[fiber product]] <math>Y' := Y \times_X X'</math> to <math>X'</math> and <math>Y</math>, respectively. Moreover, let a sheaf <math>\mathcal F</math> on X' be given. Then, there is a natural map (obtained by means of adjunction)
			:<math>f^* R^i g_* \mathcal F \rightarrow R^i g'_* f'^* \mathcal F.</math>
			Depending on the type of sheaf, and on the type of the morphisms ''g'' and ''f'', this map is an isomorphism (of sheaves on ''Y'') in some cases. Here <math>R^i g_* \mathcal F</math> denotes the [[higher direct image]] of <math>\mathcal F</math> under ''g''. As the [[stalk (mathematics)\|stalk]] of this sheaf at a point on ''Y'' is closely related to the cohomology of the fiber of the point under ''g'', this statement is paraphrased by saying that "cohomology commutes with base extension".<ref>{{harvtxt \| Hartshorne \| 1977 \| p=255 }}</ref>

			{{Images of sheaves}}

			==Flat base change for quasi-coherent sheaves==
			The base change holds for a [[quasi-coherent sheaf]] <math>\mathcal F</math> (on <math>X'</math>), provided that the map ''f'' is [[flat morphism\|flat]] (together with a number of technical conditions: ''g'' needs to be a [[separated morphism\|separated]] [[morphism of finite type]], the schemes involved need to be Noetherian).

			==Proper base change for etale sheaves==
			The base change holds for etale torsion sheaves, provided that ''g'' is [[proper morphism\|proper]].<ref>{{harvtxt\|Milne\|1980\|loc=section VI.2}}</ref>

			==Smooth base change for etale sheaves==
			The base change holds for etale torsion sheaves, whose torsion is prime to the residue characteristics of ''X'', provided ''f'' is [[smooth morphism\|smooth]] and ''g'' is quasi-compact.<ref>{{harvtxt\|Milne\|1980\|loc=section VI.4}}</ref>

			==See also==
			*[[Grothendieck's relative point of view]] in algebraic geometry
			*[[Change of base (disambiguation)]]
			*[[Base change lifting]] of automorphic forms

			==Notes==
			<references />

			==References==
			* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link=Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90244-9 \| oclc=13348052 \| mr=0463157 \| year=1977}}
			* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08238-7 \| year=1980}}

			[[Category:Sheaf theory]]

en>Michael Hardy at 00:54, 26 August 2011

2011-08-26T00:54:58Z

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