Base change map

In mathematics, the base change map relates the direct image and the pull-back of sheaves. More precisely, it is the following natural transformation of sheaves:

g^{*}(R^{r}f_{*}{\mathcal {F}})\to R^{r}f'_{*}(g'^{*}{\mathcal {F}})

where $f:X\to S,f':X'\to S',g':X'\to X,g:S'\to S$ are continuous maps between topological spaces that form a Cartesian square and ${\mathcal {F}}$ is a sheaf on X.

In general topology, the map is an isomorphism under some mild technical conditions. An analogous result holds for étale cohomologies (with topological spaces replaced by sites), though more difficult. See proper base change theorem.

General topology

If X is a Hausdorff topological space, S is a locally compact Hausdorff space and f is universally closed (i.e., $X\times _{S}T\to T$ is closed for any continuous map $T\to S$ ), then the base change map is an isomorphism.^[1] Indeed, we have: for $s\in S$ ,

(R^{r}f_{*}{\mathcal {F}})_{s}=\varinjlim H^{r}(U,{\mathcal {F}})=H^{r}(X_{s},{\mathcal {F}}),\quad X_{s}=f^{{-1}}(s)

and so for $s=g(t)$

g^{*}(R^{r}f_{*}{\mathcal {F}})_{t}=H^{r}(X_{s},{\mathcal {F}})=H^{r}(X'_{t},g'^{*}{\mathcal {F}})=R^{r}f'_{*}(g'^{*}{\mathcal {F}})_{t}.

Derivation

Since $g'^{*}$ is left adjoint to $g'_{*}$ , we have:

\operatorname {id}\to g'_{*}\circ g'^{*}

and so

R^{r}f_{*}\to R^{r}f_{*}\circ g'_{*}\circ g'^{*}.

The Grothendieck spectral sequence then gives the first map and the last map (they are edge maps) in:

R^{r}f_{*}\circ g'_{*}\circ g'^{*}\to R^{r}(f\circ g')_{*}\circ g'^{*}=R^{r}(g\circ f')_{*}\circ g'^{*}\to g_{*}\circ R^{r}f'_{*}\circ g'^{*}.

Combining this with the above we get

R^{r}f_{*}\to g_{*}\circ R^{r}f'_{*}\circ g'^{*}.

Again using the adjoint relation we get the desired map.

Base change in algebraic geometry

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".^[2]

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

Main article: proper base change theorem

The base change holds for etale torsion sheaves, provided that g is proper.^[3]

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.^[4]

References

↑ Milne, Theorem 17.3
↑ Hartshorne (1977, p. 255)
↑ Milne (1980, section VI.2)
↑ Milne (1980, section VI.4)

Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052
Milne, James S. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7
J. S. Milne (2012). "Lectures on Étale Cohomology

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