Varimax rotation - Revision history

en>Jayprich: tried to clarify that the coordinates are not being rotated .. it is the basis that is re-expressed

2014-04-23T20:49:41Z

tried to clarify that the coordinates are not being rotated .. it is the basis that is re-expressed

← Older revision		Revision as of 22:49, 23 April 2014
Line 1:		Line 1:
	~~{{Images of sheaves}}~~
	In [[mathematics]], especially in [[sheaf (mathematics)\|sheaf theory]], a domain applied in areas such as [[topology]], [[logic]] and [[algebraic geometry]], there are four '''image functors for sheaves''' which belong together in various senses.

	Given a [[continuous mapping]] ''f'': ''X'' → ''Y'' of [[topological space]]s, and the [[category (mathematics)\|category]] ''Sh''(–) of sheaves of [[abelian group]]s on a topological space. The functors in question are

	* [[direct image functor\|direct image]] ''f''<sub>∗</sub> : ''Sh''(''X'') → ''Sh''(''Y'')		Surely the second option would be more beneficial for any website. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Setting Up Your Business Online Using Free Wordpress Websites. User friendly features and flexibility that Word - Press has to offer is second to none. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>If you loved this article and also you would like to collect more info concerning [http://carspelletjes.nl/profile/rowainewri.html backup plugin] kindly visit the webpage. Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Internet is not only the source for information, it is also one of the source for passive income. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. The 2010 voting took place from July 7 through August 31, 2010.
	* [[inverse image functor\|inverse image]] ''f''<sup>∗</sup> : ''Sh''(''Y'') → ''Sh''(''X'')
	* [[direct image with ~~compact support]] ''f''~~<~~sub~~>!<~~/sub~~> ~~: ''Sh''(''X'') → ''Sh''(''Y'')~~
	* [[exceptional inverse image functor\|exceptional inverse image]] ''Rf''<sup>!</sup> : ''D''(''Sh''(''Y'')) → ''D''(''Sh''(''X'')).

	~~The [[exclamation mark]] is often pronounced "[[shriek]]" (slang for exclamation mark)~~, and the ~~maps called "''f'' shriek"~~ or "'~~'f'' lower shriek"~~ and ~~"''f'' upper shriek" – see also [[shriek map]]~~.

	~~The exceptional inverse image is in general defined~~ on the ~~level of [[derived category\|derived categories]] only~~. ~~Similar considerations apply to [[étale\|étale sheaves]] on [[scheme (mathematics)\|schemes]].~~

	~~==Adjointness==~~
	~~The functors~~ are ~~[[adjoint functor\|adjoint]]~~ to ~~each other as depicted at the right~~, ~~where~~, ~~as usual~~, <~~math~~>~~F \leftrightarrows G~~<~~/math~~> ~~means~~ that ~~''F''~~ is ~~left adjoint~~ to ~~''G'' (equivalently ''G'' right adjoint~~ to ~~''F''), i~~.e.
	~~:''[[morphism\|Hom]]''(''F''(''A'')~~, ~~''B'') &cong; ''Hom''(''A''~~, ~~''G''(''B''))~~
	for ~~any two objects ''A'', ''B''~~ in ~~the two categories being adjoint by ''F''~~ and ~~''G''.~~

	~~For example, ''f''<sup>∗</sup>~~ is ~~the left adjoint of ''f''<sub></sub>~~. ~~By the standard reasoning with adjointness relations~~, ~~there~~ are ~~natural unit~~ and ~~counit morphisms~~ <~~math~~>~~\mathcal{G} \rightarrow f_f^{}\mathcal{G}~~<~~/math~~> and ~~<math>f^{}f_*\mathcal{F} \rightarrow \mathcal{F}<~~/~~math> for <math>\mathcal G<~~/~~math> on ''Y'' and <math>\mathcal F<~~/~~math> on ''X'', respectively. However, these are ''almost never'' isomorphisms - see the localization example below~~.

	~~==Verdier duality==~~
	~~[[Verdier duality~~]~~] gives another link between them: morally speaking, it exchanges "&lowast;" and "!", i.e. in the synopsis above it exchanges functors along~~ the ~~diagonals~~. ~~For example the direct image~~ is ~~dual to~~ the ~~direct image with compact support. This phenomenon is studied and used in~~ the ~~theory~~ of ~~[[perverse sheaf\|perverse sheaves]]~~.

	~~==Base Change==~~
	~~Another useful property of the image functors is~~ [~~[Base change map\|base change~~]~~]. Given continuous maps <math>f:X \rightarrow Z</math>~~ and ~~<math>g:Y \rightarrow Z</math>, which induce morphisms <math>\bar f:X\times_Z Y \rightarrow Y</math>~~ and ~~<math>\bar g:X\times_Z Y \rightarrow X</math>~~.
	~~Then there exists~~ a ~~canonical isomorphism <math>R \bar f_* R\bar g^! \cong Rf^! Rg_*</math>.~~

	~~==Localization==~~
	In the ~~particular situation~~ of a ~~[[closed subset\|closed subspace]] ''i'': ''Z'' ⊂ ''X''~~ and the ~~[[complement (set theory)\|complementary]] [[open subset]] ''j'': ''U'' ⊂ ''X''~~, ~~the situation simplifies insofar that for ''j''<sup>∗</sup>=''j''<sup>!</sup>~~ and '~~'i''<sub>!</sub>=''i''<sub>∗</sub> and for any sheaf ''F'' on ''X'', one gets [[exact sequence]]~~s
	~~:0 → ''j''<sub>!</sub>''j''<sup>&lowast;</sup> ''F'' → ''F'' → ''i''<sub>&lowast;</sub>''i''<sup>&lowast;</sup> ''F'' → 0~~
	~~Its Verdier dual reads~~
	~~:''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F'' → ''F'' → ''Rj''<sub>&lowast;</sub>''j''<sup>&lowast;</sup> ''F'' → ''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F''[1],~~
	~~a [[distinguished triangle]]~~ in the ~~derived category of sheaves on ''X''~~.

	~~The adjointness relations read in this case~~
	:<~~math~~>~~i^* \leftrightarrows i_*=i_! \leftrightarrows i^!~~<~~/math~~>
	~~and~~
	~~:<math>j_! \leftrightarrows j^!=j^* \leftrightarrows j_*</math>.~~

	~~==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 \| id={{MathSciNet \| id = 842190}} \| year=1986}} treats~~ the ~~topological setting~~
	*{{cite book
	~~\| first = Michael~~
	~~\| last = Artin~~
	~~\| authorlink = Michael Artin~~
	~~\| coauthors = [[Alexandre Grothendieck]], [[Jean-Louis Verdier]]~~, ~~eds.~~
	~~\| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64~~ - ~~Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3~~
	~~\|series=Lecture notes~~ in ~~mathematics \|volume=305~~
	~~\| year = 1972~~
	~~\| publisher = [[Springer Science+Business Media\|Springer-Verlag]]~~
	~~\| location = Berlin; New York~~
	~~\| language = French~~
	~~\| pages = vi+640~~
	~~\|doi=10~~.~~1007/BFb0070714~~
	~~\|isbn= 978-3-540-06118-2~~
	~~}} treats~~ the ~~case~~ of ~~étale sheaves on schemes~~. ~~See Exposé XVIII, section 3~~.
	* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08238-7 ~~\| year=1980}} is another reference for the étale case~~.

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

175.144.213.146 at 02:50, 3 February 2014

2014-02-03T02:50:57Z

← Older revision		Revision as of 04:50, 3 February 2014
Line 1:		Line 1:
	~~Got nothing~~ to ~~write about me~~ at ~~all~~.<br>~~Hurrey Im here and a part~~ of ~~this site~~.<br>~~I just hope I~~'~~m useful at all~~<br><br>~~Also visit my blog~~ ... [~~http~~://~~carspelletjes~~.nl/~~profile~~/~~rowainewri~~.~~html backup plugin~~]		{{Images of sheaves}}
			In [[mathematics]], especially in [[sheaf (mathematics)\|sheaf theory]], a domain applied in areas such as [[topology]], [[logic]] and [[algebraic geometry]], there are four '''image functors for sheaves''' which belong together in various senses.

			Given a [[continuous mapping]] ''f'': ''X'' → ''Y'' of [[topological space]]s, and the [[category (mathematics)\|category]] ''Sh''(–) of sheaves of [[abelian group]]s on a topological space. The functors in question are

			* [[direct image functor\|direct image]] ''f''<sub>∗</sub> : ''Sh''(''X'') → ''Sh''(''Y'')
			* [[inverse image functor\|inverse image]] ''f''<sup>∗</sup> : ''Sh''(''Y'') → ''Sh''(''X'')
			* [[direct image with compact support]] ''f''<sub>!</sub> : ''Sh''(''X'') → ''Sh''(''Y'')
			* [[exceptional inverse image functor\|exceptional inverse image]] ''Rf''<sup>!</sup> : ''D''(''Sh''(''Y'')) → ''D''(''Sh''(''X'')).

			The [[exclamation mark]] is often pronounced "[[shriek]]" (slang for exclamation mark), and the maps called "''f'' shriek" or "''f'' lower shriek" and "''f'' upper shriek" – see also [[shriek map]].

			The exceptional inverse image is in general defined on the level of [[derived category\|derived categories]] only. Similar considerations apply to [[étale\|étale sheaves]] on [[scheme (mathematics)\|schemes]].

			==Adjointness==
			The functors are [[adjoint functor\|adjoint]] to each other as depicted at the right, where, as usual, <math>F \leftrightarrows G</math> means that ''F'' is left adjoint to ''G'' (equivalently ''G'' right adjoint to ''F''), i.e.
			:''[[morphism\|Hom]]''(''F''(''A''), ''B'') &cong; ''Hom''(''A'', ''G''(''B''))
			for any two objects ''A'', ''B'' in the two categories being adjoint by ''F'' and ''G''.

			For example, ''f''<sup>∗</sup> is the left adjoint of ''f''<sub></sub>. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms <math>\mathcal{G} \rightarrow f_f^{}\mathcal{G}</math> and <math>f^{}f_*\mathcal{F} \rightarrow \mathcal{F}</math> for <math>\mathcal G</math> on ''Y'' and <math>\mathcal F</math> on ''X'', respectively. However, these are ''almost never'' isomorphisms - see the localization example below.

			==Verdier duality==
			[[Verdier duality]] gives another link between them: morally speaking, it exchanges "&lowast;" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of [[perverse sheaf\|perverse sheaves]].

			==Base Change==
			Another useful property of the image functors is [[Base change map\|base change]]. Given continuous maps <math>f:X \rightarrow Z</math> and <math>g:Y \rightarrow Z</math>, which induce morphisms <math>\bar f:X\times_Z Y \rightarrow Y</math> and <math>\bar g:X\times_Z Y \rightarrow X</math>.
			Then there exists a canonical isomorphism <math>R \bar f_* R\bar g^! \cong Rf^! Rg_*</math>.

			==Localization==
			In the particular situation of a [[closed subset\|closed subspace]] ''i'': ''Z'' ⊂ ''X'' and the [[complement (set theory)\|complementary]] [[open subset]] ''j'': ''U'' ⊂ ''X'', the situation simplifies insofar that for ''j''<sup>∗</sup>=''j''<sup>!</sup> and ''i''<sub>!</sub>=''i''<sub>∗</sub> and for any sheaf ''F'' on ''X'', one gets [[exact sequence]]s
			:0 → ''j''<sub>!</sub>''j''<sup>&lowast;</sup> ''F'' → ''F'' → ''i''<sub>&lowast;</sub>''i''<sup>&lowast;</sup> ''F'' → 0
			Its Verdier dual reads
			:''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F'' → ''F'' → ''Rj''<sub>&lowast;</sub>''j''<sup>&lowast;</sup> ''F'' → ''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F''[1],
			a [[distinguished triangle]] in the derived category of sheaves on ''X''.

			The adjointness relations read in this case
			:<math>i^* \leftrightarrows i_*=i_! \leftrightarrows i^!</math>
			and
			:<math>j_! \leftrightarrows j^!=j^* \leftrightarrows j_*</math>.

			==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 \| id={{MathSciNet \| id = 842190}} \| year=1986}} treats the topological setting
			*{{cite book
			\| first = Michael
			\| last = Artin
			\| authorlink = Michael Artin
			\| coauthors = [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds.
			\| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3
			\|series=Lecture notes in mathematics \|volume=305
			\| year = 1972
			\| publisher = [[Springer Science+Business Media\|Springer-Verlag]]
			\| location = Berlin; New York
			\| language = French
			\| pages = vi+640
			\|doi=10.1007/BFb0070714
			\|isbn= 978-3-540-06118-2
			}} treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.
			* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08238-7 \| year=1980}} is another reference for the étale case.

			[[Category:Sheaf theory]]

en>Melcombe: add cat

2012-07-14T11:55:09Z

add cat

New page

Got nothing to write about me at all.<br>Hurrey Im here and a part of this site.<br>I just hope I'm useful at all<br><br>Also visit my blog ... [http://carspelletjes.nl/profile/rowainewri.html backup plugin]

← Older revision		Revision as of 22:49, 23 April 2014
Line 1:		Line 1:
	~~{{Images of sheaves}}~~
	In [[mathematics]], especially in [[sheaf (mathematics)\|sheaf theory]], a domain applied in areas such as [[topology]], [[logic]] and [[algebraic geometry]], there are four '''image functors for sheaves''' which belong together in various senses.

	Given a [[continuous mapping]] ''f'': ''X'' → ''Y'' of [[topological space]]s, and the [[category (mathematics)\|category]] ''Sh''(–) of sheaves of [[abelian group]]s on a topological space. The functors in question are

	* [[direct image functor\|direct image]] ''f''<sub>∗</sub> : ''Sh''(''X'') → ''Sh''(''Y'')		Surely the second option would be more beneficial for any website. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. This plugin allows a blogger get more Facebook fans on the related fan page. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Setting Up Your Business Online Using Free Wordpress Websites. User friendly features and flexibility that Word - Press has to offer is second to none. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. <br><br>If you loved this article and also you would like to collect more info concerning [http://carspelletjes.nl/profile/rowainewri.html backup plugin] kindly visit the webpage. Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. The most important plugins you will need are All-in-One SEO Pack, some social bookmarking plugin, a Feedburner plugin and an RSS sign up button. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Internet is not only the source for information, it is also one of the source for passive income. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. The 2010 voting took place from July 7 through August 31, 2010.
	* [[inverse image functor\|inverse image]] ''f''<sup>∗</sup> : ''Sh''(''Y'') → ''Sh''(''X'')
	* [[direct image with ~~compact support]] ''f''~~<~~sub~~>!<~~/sub~~> ~~: ''Sh''(''X'') → ''Sh''(''Y'')~~
	* [[exceptional inverse image functor\|exceptional inverse image]] ''Rf''<sup>!</sup> : ''D''(''Sh''(''Y'')) → ''D''(''Sh''(''X'')).

	~~The [[exclamation mark]] is often pronounced "[[shriek]]" (slang for exclamation mark)~~, and the ~~maps called "''f'' shriek"~~ or "'~~'f'' lower shriek"~~ and ~~"''f'' upper shriek" – see also [[shriek map]]~~.

	~~The exceptional inverse image is in general defined~~ on the ~~level of [[derived category\|derived categories]] only~~. ~~Similar considerations apply to [[étale\|étale sheaves]] on [[scheme (mathematics)\|schemes]].~~

	~~==Adjointness==~~
	~~The functors~~ are ~~[[adjoint functor\|adjoint]]~~ to ~~each other as depicted at the right~~, ~~where~~, ~~as usual~~, <~~math~~>~~F \leftrightarrows G~~<~~/math~~> ~~means~~ that ~~''F''~~ is ~~left adjoint~~ to ~~''G'' (equivalently ''G'' right adjoint~~ to ~~''F''), i~~.e.
	~~:''[[morphism\|Hom]]''(''F''(''A'')~~, ~~''B'') &cong; ''Hom''(''A''~~, ~~''G''(''B''))~~
	for ~~any two objects ''A'', ''B''~~ in ~~the two categories being adjoint by ''F''~~ and ~~''G''.~~

	~~For example, ''f''<sup>∗</sup>~~ is ~~the left adjoint of ''f''<sub></sub>~~. ~~By the standard reasoning with adjointness relations~~, ~~there~~ are ~~natural unit~~ and ~~counit morphisms~~ <~~math~~>~~\mathcal{G} \rightarrow f_f^{}\mathcal{G}~~<~~/math~~> and ~~<math>f^{}f_*\mathcal{F} \rightarrow \mathcal{F}<~~/~~math> for <math>\mathcal G<~~/~~math> on ''Y'' and <math>\mathcal F<~~/~~math> on ''X'', respectively. However, these are ''almost never'' isomorphisms - see the localization example below~~.

	~~==Verdier duality==~~
	~~[[Verdier duality~~]~~] gives another link between them: morally speaking, it exchanges "&lowast;" and "!", i.e. in the synopsis above it exchanges functors along~~ the ~~diagonals~~. ~~For example the direct image~~ is ~~dual to~~ the ~~direct image with compact support. This phenomenon is studied and used in~~ the ~~theory~~ of ~~[[perverse sheaf\|perverse sheaves]]~~.

	~~==Base Change==~~
	~~Another useful property of the image functors is~~ [~~[Base change map\|base change~~]~~]. Given continuous maps <math>f:X \rightarrow Z</math>~~ and ~~<math>g:Y \rightarrow Z</math>, which induce morphisms <math>\bar f:X\times_Z Y \rightarrow Y</math>~~ and ~~<math>\bar g:X\times_Z Y \rightarrow X</math>~~.
	~~Then there exists~~ a ~~canonical isomorphism <math>R \bar f_* R\bar g^! \cong Rf^! Rg_*</math>.~~

	~~==Localization==~~
	In the ~~particular situation~~ of a ~~[[closed subset\|closed subspace]] ''i'': ''Z'' ⊂ ''X''~~ and the ~~[[complement (set theory)\|complementary]] [[open subset]] ''j'': ''U'' ⊂ ''X''~~, ~~the situation simplifies insofar that for ''j''<sup>∗</sup>=''j''<sup>!</sup>~~ and '~~'i''<sub>!</sub>=''i''<sub>∗</sub> and for any sheaf ''F'' on ''X'', one gets [[exact sequence]]~~s
	~~:0 → ''j''<sub>!</sub>''j''<sup>&lowast;</sup> ''F'' → ''F'' → ''i''<sub>&lowast;</sub>''i''<sup>&lowast;</sup> ''F'' → 0~~
	~~Its Verdier dual reads~~
	~~:''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F'' → ''F'' → ''Rj''<sub>&lowast;</sub>''j''<sup>&lowast;</sup> ''F'' → ''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F''[1],~~
	~~a [[distinguished triangle]]~~ in the ~~derived category of sheaves on ''X''~~.

	~~The adjointness relations read in this case~~
	:<~~math~~>~~i^* \leftrightarrows i_*=i_! \leftrightarrows i^!~~<~~/math~~>
	~~and~~
	~~:<math>j_! \leftrightarrows j^!=j^* \leftrightarrows j_*</math>.~~

	~~==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 \| id={{MathSciNet \| id = 842190}} \| year=1986}} treats~~ the ~~topological setting~~
	*{{cite book
	~~\| first = Michael~~
	~~\| last = Artin~~
	~~\| authorlink = Michael Artin~~
	~~\| coauthors = [[Alexandre Grothendieck]], [[Jean-Louis Verdier]]~~, ~~eds.~~
	~~\| title = Séminaire de Géométrie Algébrique du Bois Marie - 1963-64~~ - ~~Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3~~
	~~\|series=Lecture notes~~ in ~~mathematics \|volume=305~~
	~~\| year = 1972~~
	~~\| publisher = [[Springer Science+Business Media\|Springer-Verlag]]~~
	~~\| location = Berlin; New York~~
	~~\| language = French~~
	~~\| pages = vi+640~~
	~~\|doi=10~~.~~1007/BFb0070714~~
	~~\|isbn= 978-3-540-06118-2~~
	~~}} treats~~ the ~~case~~ of ~~étale sheaves on schemes~~. ~~See Exposé XVIII, section 3~~.
	* {{Citation \| last1=Milne \| first1=James S. \| title=Étale cohomology \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08238-7 ~~\| year=1980}} is another reference for the étale case~~.

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