Base change

In mathematics, especially in algebraic geometry, base change refers to a number of similar theorems concerning the cohomology of sheaves on algebro-geometric objects such as varieties or schemes.

The situation of a base change theorem typically is as follows: given two maps of, say, schemes, $f:Y\rightarrow X$, $g:X'\rightarrow X$, let $f'$ and $g'$ be the projections from the fiber product $Y':=Y\times _{X}X'$ to $X'$ and $Y$, respectively. Moreover, let a sheaf ${\mathcal {F}}$ on X' be given. Then, there is a natural map (obtained by means of adjunction)

$f^{*}R^{i}g_{*}{\mathcal {F}}\rightarrow R^{i}g'_{*}f'^{*}{\mathcal {F}}.$

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 $R^{i}g_{*}{\mathcal {F}}$ denotes the higher direct image of ${\mathcal {F}}$ 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".

Image functors for sheaves
direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
$f^{*}\leftrightarrows f_{*}$
$(R)f_{!}\leftrightarrows (R)f^{!}$
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Flat base change for quasi-coherent sheaves

The base change holds for a quasi-coherent sheaf ${\mathcal {F}}$ (on $X'$), 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.

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 quasi-compact.

See also

• Grothendieck's relative point of view in algebraic geometry
• Change of base (disambiguation)
• Base change lifting of automorphic forms

References

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