Varimax rotation: Difference between revisions

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{{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'')
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* [[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 &rarr; ''j''<sub>!</sub>''j''<sup>&lowast;</sup> ''F'' &rarr; ''F'' &rarr; ''i''<sub>&lowast;</sub>''i''<sup>&lowast;</sup> ''F'' &rarr; 0
Its Verdier dual reads
:''i''<sub>&lowast;</sub>''Ri''<sup>!</sup> ''F'' &rarr; ''F'' &rarr; ''Rj''<sub>&lowast;</sub>''j''<sup>&lowast;</sup> ''F'' &rarr; ''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]]

Latest revision as of 21:49, 23 April 2014


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