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".^[1]

Image functors for sheaves
direct image f_∗
inverse image f^∗
direct image with compact support f_!
exceptional inverse image Rf^!
$f^* \leftrightarrows f_*$
$(R)f_! \leftrightarrows (R)f^!$

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.^[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 quasi-compact.^[3]

Notes

↑ Hartshorne (1977, p. 255)
↑ Milne (1980, section VI.2)
↑ Milne (1980, section VI.4)

References