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, ${\displaystyle f:Y\rightarrow X}$, ${\displaystyle g:X'\rightarrow X}$, let ${\displaystyle f'}$ and ${\displaystyle g'}$ be the projections from the fiber product ${\displaystyle Y':=Y\times _{X}X'}$ to ${\displaystyle X'}$ and ${\displaystyle Y}$, respectively. Moreover, let a sheaf ${\displaystyle {\mathcal {F}}}$ on X' be given. Then, there is a natural map (obtained by means of adjunction)

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

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

Flat base change for quasi-coherent sheaves

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