base change theorems

A simple base change phenomenon arises in commutative algebra when A is a commutative ring and B and A' are two A-algebras. Let $B\text{'}\; =\; B\; \backslash otimes\_A\; A\text{'}$. In this situation, given a B-module M, there is an isomorphism (of A'-modules):

:$(M\; \backslash otimes\_B\; B\text{'})\_\{A\text{'}\}\; \backslash cong\; (M\_A)\; \backslash otimes\_A\; A\text{'}.$

Here the subscript indicates the forgetful functor, i.e., $M\_A$ is M, but regarded as an A-module.

Indeed, such an isomorphism is obtained by observing

:$M\; \backslash otimes\_B\; B\text{'}\; =\; M\; \backslash otimes\_B\; B\; \backslash otimes\_A\; A\text{'}\; \backslash cong\; M\; \backslash otimes\_A\; A\text{'}.$

Thus, the two operations, namely forgetful functors and tensor products, commute in the sense of the above isomorphism.

The base change theorems discussed below are statements of a similar kind.

{{Images of sheaves}}

The base change theorems presented below all assert that (for different types of sheaves, and under various assumptions on the maps involved), that the following base change map

:$g^*(R^r\; f\_*\; \backslash mathcal\{F\})\; \backslash to\; R^r\; f\text{'}\_*(g\text{'}^*\backslash mathcal\{F\})$

is an isomorphism, where

:

f' \downarrow & & \downarrow f \\

S' & \stackrel g \to & S\\ \end{array}

are continuous maps between topological spaces that form a Cartesian square and $\backslash mathcal\{F\}$ is a sheaf on X.

The roles of $X$ and $S\text{'}$ are symmetric, and in some contexts (especially smooth base change) the more familiar formulation is the other one (dealing instead with the map $f^*\; R^i\; g\_*\; \backslash mathcal\; G\; \backslash rightarrow\; R^i\; g\text{'}\_*\; f\text{'}^*\; \backslash mathcal\; G$ for $\backslash mathcal\; G$ a sheaf on $S\text{'}$). For consistency, the results in this article below are all stated for the same situation, namely the map $g^*\; R^i\; f\_*\; \backslash mathcal\; F\; \backslash rightarrow\; R^i\; f\text{'}\_*\; g\text{'}^*\; \backslash mathcal\; F$; but readers should be sure to check this against their expectations. Here $R^i\; f\_*\; \backslash mathcal\; F$ denotes the higher direct image of $\backslash mathcal\; F$ under f, i.e., the derived functor of the direct image (also known as pushforward) functor $f\_*$.

This map exists without any assumptions on the maps f and g. It is constructed as follows: since $g\text{'}^*$ is left adjoint to $g\text{'}\_*$, there is a natural map (called unit map)

:$\backslash operatorname\{id\}\; \backslash to\; g\text{'}\_*\; \backslash circ\; g\text{'}^*$

and so

:$R^r\; f\_*\; \backslash to\; R^r\; f\_*\; \backslash circ\; g\text{'}\_*\; \backslash circ\; g\text{'}^*.$

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

:$R^r\; f\_*\; \backslash circ\; g\text{'}\_*\; \backslash circ\; g\text{'}^*\; \backslash to\; R^r(f\; \backslash circ\; g\text{'})\_*\; \backslash circ\; g\text{'}^*\; =\; R^r(g\; \backslash circ\; f\text{'})\_*\; \backslash circ\; g\text{'}^*\; \backslash to\; g\_*\; \backslash circ\; R^r\; f\text{'}\_*\; \backslash circ\; g\text{'}^*.$

Combining this with the above yields

:$R^r\; f\_*\; \backslash to\; g\_*\; \backslash circ\; R^r\; f\text{'}\_*\; \backslash circ\; g\text{'}^*.$

Using the adjointness of $g^*$ and $g\_*$ finally yields the desired map.

The above-mentioned introductory example is a special case of this, namely for the affine schemes $X\; =\; \backslash operatorname\{Spec\}\; (B),\; S\; =\; \backslash operatorname\{Spec\}\; (A),\; S\text{'}\; =\; \backslash operatorname\{Spec\}\; (A\text{'})$ and, consequently, $X\text{'}\; =\; \backslash operatorname\{Spec\}\; (B\text{'})$, and the quasi-coherent sheaf $\backslash mathcal\; F\; :=\; \backslash tilde\; M$ associated to the B-module M.

It is conceptually convenient to organize the above base change maps, which only involve only a single higher direct image functor, into one which encodes all $R^r\; f\_*$ at a time. In fact, similar arguments as above yield a map in the derived category of sheaves on S':

:$g^*\; Rf\_*\; (\backslash mathcal\{F\})\; \backslash to\; Rf\text{'}\_*(g\text{'}^*\backslash mathcal\{F\})$

where $Rf\_*$ denotes the (total) derived functor of $f\_*$.

If X is a Hausdorff topological space, S is a locally compact Hausdorff space and f is universally closed (i.e., $X\; \backslash times\_S\; T\; \backslash to\; T$ is a closed map for any continuous map $T\; \backslash to\; S$), then

the base change map

:$g^*\; R^r\; f\_*\; \backslash mathcal\; F\; \backslash to\; R^r\; f\text{'}\_*\; g\text{'}^*\; \backslash mathcal\; F$

is an isomorphism.{{harvtxt|Milne|2012|loc=Theorem 17.3}} Indeed, we have: for $s\; \backslash in\; S$,

:$(R^r\; f\_*\; \backslash mathcal\{F\})\_s\; =\; \backslash varinjlim\; H^r(U,\; \backslash mathcal\{F\})\; =\; H^r(X\_s,\; \backslash mathcal\{F\}),\; \backslash quad\; X\_s\; =\; f^\{-1\}(s)$

and so for $s\; =\; g(t)$

:$g^*\; (R^r\; f\_*\; \backslash mathcal\{F\})\_t\; =\; H^r(X\_s,\; \backslash mathcal\{F\})\; =\; H^r(X\text{'}\_t,\; g\text{'}^*\; \backslash mathcal\{F\})\; =\; R^r\; f\text{'}\_*\; (g\text{'}^*\; \backslash mathcal\{F\})\_t.$

To encode all individual higher derived functors of $f\_*$ into one entity, the above statement may equivalently be rephrased by saying that the base change map

:$g^*\; Rf\_*\; \backslash mathcal\; F\; \backslash to\; Rf\text{'}\_*\; g\text{'}^*\; \backslash mathcal\; F$

is a quasi-isomorphism.

The assumptions that the involved spaces be Hausdorff have been weakened by {{harvtxt|Schnürer|Soergel|2016}}.

{{harvtxt|Lurie|2009}} has extended the above theorem to non-abelian sheaf cohomology, i.e., sheaves taking values in simplicial sets (as opposed to abelian groups).{{harvtxt|Lurie|2009|loc=Theorem 7.3.1.16}}

If the map f is not closed, the base change map need not be an isomorphism, as the following example shows (the maps are the standard inclusions) :

:$\backslash begin\{array\}\{rcl\}$

\emptyset & \stackrel {g'} \to & \mathbb C \setminus \{0\} \\

f' \downarrow & & \downarrow f \\

\{0\} & \stackrel g \to & \mathbb C

\end{array}

One the one hand $f\text{'}\_*\; g\text{'}^*\; \backslash mathcal\; F$ is always zero, but if $\backslash mathcal\; F$ is a local system on $\backslash mathbb\; C\; \backslash setminus\; \backslash \{0\backslash \}$ corresponding to a representation of the fundamental group $\backslash pi\_1(X)$ (which is isomorphic to Z), then $g^*\; f\_*\; \backslash mathcal\; F$ can be computed as the invariants of the monodromy action of $\backslash pi\_1(X,\; x)$ on the stalk $\backslash mathcal\; F\_x$ (for any $x\; \backslash ne\; 0$), which need not vanish.

To obtain a base-change result, the functor $f\_*$ (or its derived functor) has to be replaced by the direct image with compact support $Rf\_!$. For example, if $f:\; X\; \backslash to\; S$ is the inclusion of an open subset, such as in the above example, $Rf\_!\; \backslash mathcal\; F$ is the extension by zero, i.e., its stalks are given by

:

In general, there is a map $Rf\_!\; \backslash mathcal\; F\; \backslash to\; Rf\_*\; \backslash mathcal\; F$, which is a quasi-isomorphism if f is proper, but not in general. The proper base change theorem mentioned above has the following generalization: there is a quasi-isomorphism{{harvtxt|Iversen|1986}}, the four spaces are assumed to be locally compact and of finite dimension.

:$g^*\; Rf\_!\; \backslash mathcal\; F\; \backslash to\; Rf\text{'}\_!\; g\text{'}^*\; \backslash mathcal\; F.$

Proper base change theorems for quasi-coherent sheaves apply in the following situation: $f:\; X\; \backslash to\; S$ is a proper morphism between noetherian schemes, and $\backslash mathcal\{F\}$ is a coherent sheaf which is flat over S (i.e., $\backslash mathcal\; F\_x$ is flat over $\backslash mathcal\; O\_\{S,\; f(x)\}$). In this situation, the following statements hold:{{harvtxt|Grothendieck|1963|loc=Section 7.7}}, {{harvtxt|Hartshorne|1977|loc=Theorem III.12.11}}, {{harvtxt|Vakil|2015|loc=Chapter 28 Cohomology and base change theorems}}

• "Semicontinuity theorem":
• For each $p\; \backslash ge\; 0$, the function $s\; \backslash mapsto\; \backslash dim\_\{k(s)\}\; H^p\; (X\_s,\; \backslash mathcal\{F\}\_s):\; S\; \backslash to\; \backslash mathbb\{Z\}$ is upper semicontinuous.
• The function $s\; \backslash mapsto\; \backslash chi(\backslash mathcal\{F\}\_s)$ is locally constant, where $\backslash chi(\backslash mathcal\{F\})$ denotes the Euler characteristic.
• "Grauert's theorem": if S is reduced and connected, then for each $p\; \backslash ge\; 0$ the following are equivalent
• $s\; \backslash mapsto\; \backslash dim\_\{k(s)\}\; H^p\; (X\_s,\; \backslash mathcal\{F\}\_s)$ is constant.
• $R^p\; f\_*\; \backslash mathcal\{F\}$ is locally free and the natural map

::$R^p\; f\_*\; \backslash mathcal\{F\}\; \backslash otimes\_\{\backslash mathcal\{O\}\_S\}\; k(s)\; \backslash to\; H^p(X\_s,\; \backslash mathcal\{F\}\_s)$

:is an isomorphism for all $s\; \backslash in\; S$.

:Furthermore, if these conditions hold, then the natural map

::$R^\{p-1\}\; f\_*\; \backslash mathcal\{F\}\; \backslash otimes\_\{\backslash mathcal\{O\}\_S\}\; k(s)\; \backslash to\; H^\{p-1\}(X\_s,\; \backslash mathcal\{F\}\_s)$

:is an isomorphism for all $s\; \backslash in\; S$.

• If, for some p, $H^p(X\_s,\; \backslash mathcal\{F\}\_s)\; =\; 0$ for all $s\; \backslash in\; S$, then the natural map

::$R^\{p-1\}\; f\_*\; \backslash mathcal\{F\}\; \backslash otimes\_\{\backslash mathcal\{O\}\_S\}\; k(s)\; \backslash to\; H^\{p-1\}(X\_s,\; \backslash mathcal\{F\}\_s)$

:is an isomorphism for all $s\; \backslash in\; S$.

As the stalk of the sheaf $R^p\; f\_*\; \backslash mathcal\; F$ is closely related to the cohomology of the fiber of the point under f, this statement is paraphrased by saying that "cohomology commutes with base extension".{{harvtxt | Hartshorne | 1977 | p=255 }}

These statements are proved using the following fact, where in addition to the above assumptions $S\; =\; \backslash operatorname\{Spec\}\; A$: there is a finite complex $0\; \backslash to\; K^0\; \backslash to\; K^1\; \backslash to\; \backslash cdots\; \backslash to\; K^n\; \backslash to\; 0$ of finitely generated projective A-modules and a natural isomorphism of functors

:$H^p(X\; \backslash times\_S\; \backslash operatorname\{Spec\}\; -,\; \backslash mathcal\{F\}\; \backslash otimes\_A\; -)\; \backslash to\; H^p(K^\backslash bullet\; \backslash otimes\_A\; -),\; p\; \backslash ge\; 0$

on the category of $A$-algebras.

The base change map

:$g^*(R^r\; f\_*\; \backslash mathcal\{F\})\; \backslash to\; R^r\; f\text{'}\_*(g\text{'}^*\backslash mathcal\{F\})$

is an isomorphism for a quasi-coherent sheaf $\backslash mathcal\; F$ (on $X$), provided that the map $g:\; S\text{'}\; \backslash rightarrow\; S$ is flat (together with a number of technical conditions: f needs to be a separated morphism of finite type, the schemes involved need to be Noetherian).{{harvtxt|Hartshorne|1977|loc=Proposition III.9.3}}

A far reaching extension of flat base change is possible when considering the base change map

:$Lg^*\; Rf\_*\; (\backslash mathcal\{F\})\; \backslash to\; Rf\text{'}\_*(Lg\text{'}^*\backslash mathcal\{F\})$

in the derived category of sheaves on S', similarly as mentioned above. Here $Lg^*$ is the (total) derived functor of the pullback of $\backslash mathcal\; O$-modules (because $g^*\; \backslash mathcal\; G\; =\; \backslash mathcal\; O\_X\; \backslash otimes\_\{g^\{-1\}\; \backslash mathcal\; O\_S\}\; g^\{-1\}\; \backslash mathcal\; G$ involves a tensor product, $g^*$ is not exact when {{math|g}} is not flat and therefore is not equal to its derived functor $Lg^*$).

This map is a quasi-isomorphism provided that the following conditions are satisfied:{{harvtxt|Berthelot|Grothendieck|Illusie|1971|loc=SGA 6 IV, Proposition 3.1.0}}

• $S$ is quasi-compact and $f$ is quasi-compact and quasi-separated,
• $\backslash mathcal\; F$ is an object in $D^b(\backslash mathcal\{O\}\_X\backslash text\{-mod\})$, the bounded derived category of $\backslash mathcal\{O\}\_X$-modules, and its cohomology sheaves are quasi-coherent (for example, $\backslash mathcal\; F$ could be a bounded complex of quasi-coherent sheaves)
• $X$ and $S\text{'}$ are Tor-independent over $S$, meaning that if $x\; \backslash in\; X$ and $s\text{'}\; \backslash in\; S\text{'}$ satisfy $f(x)\; =\; s\; =\; g(s\text{'})$, then for all integers $p\; \backslash ge\; 1$,

:$\backslash operatorname\{Tor\}\_p^\{\backslash mathcal\{O\}\_\{S,s\}\}(\backslash mathcal\{O\}\_\{X,x\},\; \backslash mathcal\{O\}\_\{S\text{'},s\text{'}\})\; =\; 0$.

• One of the following conditions is satisfied:
• $\backslash mathcal\; F$ has finite flat amplitude relative to $f$, meaning that it is quasi-isomorphic in $D^-(f^\{-1\}\backslash mathcal\; O\_S\backslash text\{-mod\})$ to a complex $\backslash mathcal\; F\text{'}$ such that $(\backslash mathcal\; F\text{'})^i$ is $f^\{-1\}\backslash mathcal\; O\_S$-flat for all $i$ outside some bounded interval $[m,\; n]$; equivalently, there exists an interval $[m,\; n]$ such that for any complex $\backslash mathcal\; G$ in $D^-(f^\{-1\}\backslash mathcal\; O\_S\backslash text\{-mod\})$, one has $\backslash operatorname\{Tor\}\_i(\backslash mathcal\; F,\; \backslash mathcal\; G)\; =\; 0$ for all $i$ outside $[m,\; n]$; or
• $g$ has finite Tor-dimension, meaning that $\backslash mathcal\{O\}\_\{S\text{'}\}$ has finite flat amplitude relative to $g$.

One advantage of this formulation is that the flatness hypothesis has been weakened. However, making concrete computations of the cohomology of the left- and right-hand sides now requires the Grothendieck spectral sequence.

Derived algebraic geometry provides a means to drop the flatness assumption, provided that the pullback $X\text{'}$ is replaced by the homotopy pullback. In the easiest case when X, S, and $S\text{'}$ are affine (with the notation as above), the homotopy pullback is given by the derived tensor product

:$X\text{'}\; =\; \backslash operatorname\{Spec\}\; (B\text{'}\; \backslash otimes^L\_B\; A)$

Then, assuming that the schemes (or, more generally, derived schemes) involved are quasi-compact and quasi-separated, the natural transformation

:$L\; g^*\; R\; f\_*\; \backslash mathcal\{F\}\; \backslash to\; Rf\text{'}\_*\; Lg\text{'}^*\; \backslash mathcal\{F\}$

is a quasi-isomorphism for any quasi-coherent sheaf, or more generally a complex of quasi-coherent sheaves.{{harvtxt|Toën|2012|loc=Proposition 1.4}}

The afore-mentioned flat base change result is in fact a special case since for g flat the homotopy pullback (which is locally given by a derived tensor product) agrees with the ordinary pullback (locally given by the underived tensor product), and since the pullback along the flat maps g and g' are automatically derived (i.e., $Lg^*\; =\; g^*$). The auxiliary assumptions related to the Tor-independence or Tor-amplitude in the preceding base change theorem also become unnecessary.

In the above form, base change has been extended by {{harvtxt|Ben-Zvi|Francis|Nadler|2010}} to the situation where X, S, and S' are (possibly derived) stacks, provided that the map f is a perfect map (which includes the case that f is a quasi-compact, quasi-separated map of schemes, but also includes more general stacks, such as the classifying stack BG of an algebraic group in characteristic zero).

Proper base change also holds in the context of complex manifolds and complex analytic spaces.{{harvtxt|Grauert|1960}}

The theorem on formal functions is a variant of the proper base change, where the pullback is replaced by a completion operation.

The see-saw principle and the theorem of the cube, which are foundational facts in the theory of abelian varieties, are a consequence of proper base change.{{harvtxt|Mumford|2008}}

A base-change also holds for D-modules: if X, S, X', and S' are smooth varieties (but f and g need not be flat or proper etc.), there is a quasi-isomorphism

:$g^\backslash dagger\; \backslash int\_f\; \backslash mathcal\; F\; \backslash to\; \backslash int\_\{f\text{'}\}\; g\text{'}^\backslash dagger\; \backslash mathcal\; F,$

where $-^\backslash dagger$ and $\backslash int$ denote the inverse and direct image functors for D-modules.{{harvtxt|Hotta|Takeuchi|Tanisaki|2008|loc=Theorem 1.7.3}}

For étale torsion sheaves $\backslash mathcal\; F$, there are two base change results referred to as proper and smooth base change, respectively: base change holds if $f:\; X\; \backslash rightarrow\; S$ is proper.{{harvtxt|Artin|Grothendieck|Verdier|1972|loc=Exposé XII}}, {{harvtxt|Milne|1980|loc=section VI.2}} It also holds if g is smooth, provided that f is quasi-compact and provided that the torsion of $\backslash mathcal\; F$ is prime to the characteristic of the residue fields of X.{{harvtxt|Artin|Grothendieck|Verdier|1972|loc=Exposé XVI}}

Closely related to proper base change is the following fact (the two theorems are usually proved simultaneously): let X be a variety over a separably closed field and $\backslash mathcal\{F\}$ a constructible sheaf on $X\_\backslash text\{et\}$. Then $H^r(X,\; \backslash mathcal\{F\})$ are finite in each of the following cases:

• X is complete, or
• $\backslash mathcal\{F\}$ has no p-torsion, where p is the characteristic of k.

Under additional assumptions, {{harvtxt|Deninger|1988}} extended the proper base change theorem to non-torsion étale sheaves.

In close analogy to the topological situation mentioned above, the base change map for an open immersion f,

:$g^*\; f\_*\; \backslash mathcal\; F\; \backslash to\; f\text{'}\_*\; g\text{'}^*\; \backslash mathcal\; F$

is not usually an isomorphism.{{harvtxt|Milne|2012|loc=Example 8.5}} Instead the extension by zero functor $f\_!$ satisfies an isomorphism

:$g^*\; f\_!\; \backslash mathcal\; F\; \backslash to\; f\text{'}\_!\; g^*\; \backslash mathcal\; F.$

This fact and the proper base change suggest to define the direct image functor with compact support for a map f by

:$Rf\_!\; :=\; Rp\_*\; j\_!$

where $f\; =\; p\; \backslash circ\; j$ is a compactification of f, i.e., a factorization into an open immersion followed by a proper map.

The proper base change theorem is needed to show that this is well-defined, i.e., independent (up to isomorphism) of the choice of the compactification.

Moreover, again in analogy to the case of sheaves on a topological space, a base change formula for $g\_*$ vs. $Rf\_!$ does hold for non-proper maps f.

For the structural map $f:\; X\; \backslash to\; S\; =\; \backslash operatorname\{Spec\}\; k$ of a scheme over a field k, the individual cohomologies of $Rf\_!\; (\backslash mathcal\; F)$, denoted by $H^*\_c(X,\; \backslash mathcal\; F)$ referred to as cohomology with compact support. It is an important variant of usual étale cohomology.

Similar ideas are also used to construct an analogue of the functor $Rf\_!$ in A¹-homotopy theory.{{citation|author=Ayoub|first=Joseph|title=Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I.|

isbn=978-2-85629-244-0|year=2007|publisher=Société Mathématique de France|zbl=1146.14001}}{{citation|author=Cisinski|first1=Denis-Charles|last2=Déglise|first2=Frédéric|title=Triangulated Categories of Mixed Motives|chapter=|series=Springer Monographs in Mathematics|year=2019|doi=10.1007/978-3-030-33242-6|arxiv=0912.2110|bibcode=2009arXiv0912.2110C|isbn=978-3-030-33241-9|s2cid=115163824}}

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

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