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image functors for sheaves : ウィキペディア英語版
image functors for sheaves

In mathematics, especially in 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 spaces, and the category ''Sh''(–) of sheaves of abelian groups on a topological space. The functors in question are
* direct image ''f'' : ''Sh''(''X'') → ''Sh''(''Y'')
* inverse image ''f'' : ''Sh''(''Y'') → ''Sh''(''X'')
* direct image with compact support ''f''! : ''Sh''(''X'') → ''Sh''(''Y'')
* exceptional inverse image ''Rf''! : ''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 categories only. Similar considerations apply to étale sheaves on schemes.
==Adjointness==
The functors are adjoint to each other as depicted at the right, where, as usual, F \leftrightarrows G means that ''F'' is left adjoint to ''G'' (equivalently ''G'' right adjoint to ''F''), i.e.
:''Hom''(''F''(''A''), ''B'') ≅ ''Hom''(''A'', ''G''(''B''))
for any two objects ''A'', ''B'' in the two categories being adjoint by ''F'' and ''G''.
For example, ''f'' is the left adjoint of ''f''
*
. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms \mathcal \rightarrow f_
*f^\mathcal and f^f_
*\mathcal \rightarrow \mathcal for \mathcal G on ''Y'' and \mathcal F on ''X'', respectively. However, these are ''almost never'' isomorphisms - see the localization example below.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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