Length function - Revision history

en>Yobot: /* Definition */WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T11:30:40Z

Definition: WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 13:30, 5 May 2014
Line 1:		Line 1:
	~~In [[mathematics]], in the field of [[sheaf theory]]~~ and ~~especially in [[algebraic geometry]], the '''direct image functor''' generalizes the notion of a [[section of a sheaf]] to~~ the ~~relative case~~.		Greetings! I am Marvella and I really feel comfy when individuals use the complete name. My day occupation is a meter reader. One of the very best issues in the globe for me is to do aerobics and now I'm attempting to make money with it. For years he's been living in North Dakota and his family members enjoys it.<br><br>Feel free to surf to my homepage - [http://nuvem.tk/altergalactica/AliceedMaurermy over the counter std test]

	~~==Definition==~~
	~~{{Images of sheaves}}~~
	~~Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[topological space]]s, and ''Sh''(–) the [[category (mathematics)\|category]] of sheaves of [[abelian group]]s on~~ a ~~topological space~~. ~~The '''direct image [[functor]]'''~~

	~~:<math>f_*: Sh(X) \to Sh(Y)</math>~~

	~~sends a sheaf ''F'' on ''X'' to its direct image presheaf~~

	~~:<math>f_*F : U \mapsto F(f^{-1}(U)),</math>~~

	~~which turns out be a sheaf on ''Y''. This assignment is functorial, i.e. a [[morphism of sheaves]] φ: ''F'' → ''G'' on ''X'' gives rise to a morphism~~ of ~~sheaves ''f''<sub>∗</sub>(φ): ''f''<sub>∗</sub>(''F'') → ''f''<sub>∗</sub>(''G'') on ''Y''.~~

	~~=== Example ===~~
	~~If ''Y'' is a point, then~~ the ~~direct image equals~~ the ~~[[global sections functor]].~~
	~~Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image~~ is ~~a functor~~
	~~f<sup>!</sup>: D(Y) → D(X).~~

	~~=== Variants ===~~
	~~A similar definition applies~~ to ~~sheaves on [[topos\|topoi]], such as [[etale\|etale sheaves]]. Instead of the above preimage ''f''<sup>−1</sup>(''U'') the [[fiber product]] of ''U''~~ and ''~~X'' over ''Y'' is used.~~

	~~== Higher direct images ==~~
	~~The direct image functor is left exact, but usually not right exact. Hence one can consider the right [[derived functor]]~~s ~~of the direct image. They are called '''higher direct images'''~~ and ~~denoted ''R<sup>q</sup> f''<sub>∗</sub>~~.

	~~One can show that there is a similar expression as above for higher direct images: for a sheaf ''F'' on ''X'', ''R~~<~~sup~~>q<~~/sup~~> ~~f''<sub>∗</sub>(''F'') is the sheaf associated~~ to ~~the presheaf~~
	~~:<math>U \mapsto H^q(f^{~~-~~1}(U), F)</math>~~

	~~== Properties ==~~
	* The direct image functor is [[adjoint functor\|right adjoint]] to the [[inverse image functor]], which means that for any continuous <math>f: X \to Y</math> and sheaves <math>\mathcal F, \mathcal G</math> respectively on ''X'', ''Y'', there is a natural isomorphism:
	:~~<math>\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)<~~/~~math>.~~
	* If ''f'' is the inclusion of a closed subspace ''X'' ⊂ ''Y'' then ''f''<sub>∗</sub> is exact. Actually, in this case ''f''<sub>∗</~~sub> is an [[equivalence of categories\|equivalence]] between sheaves on ''X'' and sheaves on ''Y'' supported on ''X''~~. ~~It follows from the fact that the stalk of <math>(f_* \mathcal F)_y</math> is <math>\mathcal F_y<~~/~~math> if <math>y \in X<~~/~~math> and zero otherwise (here~~ the ~~closeness of ''X'' in ''Y'' is used).~~

	~~== See also ==~~
	*[[Proper base change theorem]]

	~~==References==~~
	* {{Citation \| last1=Iversen \| first1=Birger \| title=Cohomology of sheaves \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Universitext \| isbn=978-3-540-16389-3 \| mr=842190 \| year=1986}}, esp. section II.4

	~~{{PlanetMath attribution\|id=1101\|title=Direct image (functor)}}~~

	~~{{DEFAULTSORT:Direct Image Functor}}~~
	~~[[Category:Sheaf theory]]~~
	~~[[Category:Continuous mappings]~~]

138.37.7.247: /* Properties */ consistency with above

2013-05-31T00:37:14Z

Properties: consistency with above

← Older revision		Revision as of 02:37, 31 May 2013
Line 1:		Line 1:
	The ~~author~~ is ~~called Irwin~~. ~~For many years I~~'~~ve been working as~~ a ~~payroll clerk~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my spouse reside~~. ~~What I adore performing~~ is ~~performing ceramics~~ but ~~I haven~~'~~t made~~ a ~~dime with it.~~<br><br>~~Review my blog; home std test~~ ([~~http~~://~~xrambo~~.~~com~~/~~blog~~/~~191590 simply click~~ the ~~next internet page~~])		In [[mathematics]], in the field of [[sheaf theory]] and especially in [[algebraic geometry]], the '''direct image functor''' generalizes the notion of a [[section of a sheaf]] to the relative case.

			==Definition==
			{{Images of sheaves}}
			Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[topological space]]s, and ''Sh''(–) the [[category (mathematics)\|category]] of sheaves of [[abelian group]]s on a topological space. The '''direct image [[functor]]'''

			:<math>f_*: Sh(X) \to Sh(Y)</math>

			sends a sheaf ''F'' on ''X'' to its direct image presheaf

			:<math>f_*F : U \mapsto F(f^{-1}(U)),</math>

			which turns out be a sheaf on ''Y''. This assignment is functorial, i.e. a [[morphism of sheaves]] φ: ''F'' → ''G'' on ''X'' gives rise to a morphism of sheaves ''f''<sub>∗</sub>(φ): ''f''<sub>∗</sub>(''F'') → ''f''<sub>∗</sub>(''G'') on ''Y''.

			=== Example ===
			If ''Y'' is a point, then the direct image equals the [[global sections functor]].
			Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor
			f<sup>!</sup>: D(Y) → D(X).

			=== Variants ===
			A similar definition applies to sheaves on [[topos\|topoi]], such as [[etale\|etale sheaves]]. Instead of the above preimage ''f''<sup>−1</sup>(''U'') the [[fiber product]] of ''U'' and ''X'' over ''Y'' is used.

			== Higher direct images ==
			The direct image functor is left exact, but usually not right exact. Hence one can consider the right [[derived functor]]s of the direct image. They are called '''higher direct images''' and denoted ''R<sup>q</sup> f''<sub>∗</sub>.

			One can show that there is a similar expression as above for higher direct images: for a sheaf ''F'' on ''X'', ''R<sup>q</sup> f''<sub>∗</sub>(''F'') is the sheaf associated to the presheaf
			:<math>U \mapsto H^q(f^{-1}(U), F)</math>

			== Properties ==
			* The direct image functor is [[adjoint functor\|right adjoint]] to the [[inverse image functor]], which means that for any continuous <math>f: X \to Y</math> and sheaves <math>\mathcal F, \mathcal G</math> respectively on ''X'', ''Y'', there is a natural isomorphism:
			:<math>\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)</math>.
			* If ''f'' is the inclusion of a closed subspace ''X'' ⊂ ''Y'' then ''f''<sub>∗</sub> is exact. Actually, in this case ''f''<sub>∗</sub> is an [[equivalence of categories\|equivalence]] between sheaves on ''X'' and sheaves on ''Y'' supported on ''X''. It follows from the fact that the stalk of <math>(f_* \mathcal F)_y</math> is <math>\mathcal F_y</math> if <math>y \in X</math> and zero otherwise (here the closeness of ''X'' in ''Y'' is used).

			== See also ==
			*[[Proper base change theorem]]

			==References==
			* {{Citation \| last1=Iversen \| first1=Birger \| title=Cohomology of sheaves \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Universitext \| isbn=978-3-540-16389-3 \| mr=842190 \| year=1986}}, esp. section II.4

			{{PlanetMath attribution\|id=1101\|title=Direct image (functor)}}

			{{DEFAULTSORT:Direct Image Functor}}
			[[Category:Sheaf theory]]
			[[Category:Continuous mappings]]

en>777sms at 03:35, 14 April 2012

2012-04-14T03:35:01Z

New page

The author is called Irwin. For many years I've been working as a payroll clerk. South Dakota is exactly where me and my spouse reside. What I adore performing is performing ceramics but I haven't made a dime with it.<br><br>Review my blog; home std test ([http://xrambo.com/blog/191590 simply click the next internet page])