Base change
In mathematics, especially in algebraic geometry, base change refers to a number of similar theorems concerning the cohomology of sheaves on algebrogeometric objects such as varieties or schemes.
The situation of a base change theorem typically is as follows: given two maps of, say, schemes, , , let and be the projections from the fiber product to and , respectively. Moreover, let a sheaf on X' be given. Then, there is a natural map (obtained by means of adjunction)
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 denotes the higher direct image of under g. As the 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".^{[1]}
Image functors for sheaves 

direct image f_{∗} 
inverse image f^{∗} 
direct image with compact support f_{!} 
exceptional inverse image Rf^{!} 

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Flat base change for quasicoherent sheaves
The base change holds for a quasicoherent sheaf (on ), provided that the map f is flat (together with a number of technical conditions: g needs to be a 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.^{[2]}
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 and g is quasicompact.^{[3]}
See also
 Grothendieck's relative point of view in algebraic geometry
 Change of base (disambiguation)
 Base change lifting of automorphic forms
Notes
References
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