Coherent duality: Difference between revisions

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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>


{{Images of sheaves}}
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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]]

Latest revision as of 14:19, 5 May 2014


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