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| :''There is also a proper base change theorem in topology. For that, see [[base change map]].''
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| In [[algebraic geometry]], there are at least two versions of proper base change theorems: one for ordinary cohomology and the other for étale cohomology.
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| ==In ordinary cohomology==
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| The '''proper base change theorem''' states the following: let <math>f: X \to S</math> be a [[proper morphism]] between [[noetherian scheme]]s, and <math>\mathcal{F}</math> ''S''-[[flat morphism|flat]] [[coherent sheaf]] on <math>X</math>. If <math>S = \operatorname{Spec} A</math>, then there is a finite complex <math>0 \to K^0 \to K^1 \to \cdots \to K^n \to 0</math> of [[finitely generated projective module|finitely generated projective ''A''-modules]] and a natural isomorphism of functors
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| :<math>H^p(X \times_S \operatorname{Spec} -, \mathcal{F} \otimes_A -) \to H^p(K^\bullet \otimes_A -), p \ge 0</math>
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| on the category of <math>A</math>-algebras.
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| There are several corollaries to the theorem, some of which are also referred to as proper base change theorems: (the [[higher direct image]] <math>R^p f_* \mathcal{F}</math> is coherent since ''f'' is [[proper morphism|proper]].)
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| '''Corollary 1''' (semicontinuity theorem): Let ''f'' and <math>\mathcal{F}</math> as in the theorem (but ''S'' may not be affine). Then we have:
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| *(i) For each <math>p \ge 0</math>, the function <math>s \mapsto \dim_{k(s)} H^p (X_s, \mathcal{F}_s): S \to \mathbb{Z}</math> is upper [[semicontinuous]].
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| *(ii) The function <math>s \mapsto \chi(\mathcal{F}_s)</math> is locally constant, where <math>\chi(\mathcal{F})</math> denotes the [[Euler characteristic]].
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| '''Corollary 2''': Assume ''S'' is reduced and connected. Then for each <math>p \ge 0</math> the following are equivalent
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| *(i) <math>s \mapsto \dim_{k(s)} H^p (X_s, \mathcal{F}_s)</math> is constant.
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| *(ii) <math>R^p f_* \mathcal{F}</math> is locally free and the natural map
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| ::<math>R^p f_* \mathcal{F} \otimes_{\mathcal{O}_S} k(s) \to H^p(X_s, \mathcal{F}_s)</math>
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| :is an isomorphism for all <math>s \in S</math>.
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| :Furthermore, if these conditions hold, then the natural map
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| ::<math>R^{p-1} f_* \mathcal{F} \otimes_{\mathcal{O}_S} k(s) \to H^{p-1}(X_s, \mathcal{F}_s)</math>
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| :is an isomorphism for all <math>s \in S</math>. | |
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| '''Corollary 3''': Assume that for some ''p'' <math>H^p(X_s, \mathcal{F}_s) = 0</math> for all <math>s \in S</math>. Then
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| the natural map
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| ::<math>R^{p-1} f_* \mathcal{F} \otimes_{\mathcal{O}_S} k(s) \to H^{p-1}(X_s, \mathcal{F}_s)</math>
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| :is an isomorphism for all <math>s \in S</math>. | |
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| ==In étale cohomology==
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| In nutshell, the proper base change theorem states that the higher direct image <math>R^i f_* \mathcal{F}</math> of a [[torsion sheaf]] <math>\mathcal{F}</math> along a proper morphism ''f'' commutes with base change. A closely related, the finiteness theorem states that the étale cohomology groups of a [[constructible sheaf]] on a complete variety are finite. Two theorems are usually proved simultaneously.
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| '''Theorem''' (finiteness): Let ''X'' be a variety over a [[separably closed field]] and <math>\mathcal{F}</math> a constructible sheaf on <math>X_\text{et}</math>. Then <math>H^r(X, \mathcal{F})</math> are finite in each of the following cases: (i) ''X'' is complete, or (ii) <math>\mathcal{F}</math> has no ''p''-torsion, where ''p'' is the characteristic of ''k''.
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| <!--
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| ==See also==
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| *[[Base change morphism]]
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| -->
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| ==References==
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| * [[Robin Hartshorne]], ''[[Algebraic Geometry (book)|Algebraic Geometry]]''.
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| * [[David Mumford]], ''Abelian Varieties''.
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| * Vakil's notes
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| * SGA 4
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| * Milne, ''Étale cohomology''
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| * Gabber, "[http://www.math.polytechnique.fr/~laszlo/gdtgabber/abelien.pdf Finiteness theorems for étale cohomology of excellent schemes]"
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| [[Category:Theorems in algebraic geometry]]
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