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| In mathematics, the '''base change map''' relates the [[Direct image functor|direct image]] and the [[Pullback (category theory)|pull-back]] of [[sheaf (mathematics)|sheaves]]. More precisely, it is the following [[natural transformation]] of sheaves:
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| :<math>g^*(R^r f_* \mathcal{F}) \to R^r f^'_*(g'^*\mathcal{F})</math>
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| where <math>f: X \to S, f':X' \to S', g':X' \to X, g:S' \to S</math> are continuous maps between topological spaces that form a [[Cartesian_square_(category_theory)|Cartesian square]]<!-- there really should be a diagram. --> and <math>\mathcal{F}</math> is a sheaf on ''X''.
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| 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]].
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| == General topology ==
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| If ''X'' is a Hausdorff topological space, ''S'' is a locally compact Hausdorff space and ''f'' is universally closed (i.e., <math>X \times_S T \to T</math> is closed for any continuous map <math>T \to S</math>), then
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| the base change map is an isomorphism.<ref>{{harvnb|Milne, Theorem 17.3}}</ref> Indeed, we have: for <math>s \in S</math>,
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| :<math>(R^r f_* \mathcal{F})_s = \varinjlim H^r(U, \mathcal{F}) = H^r(X_s, \mathcal{F}), \quad X_s = f^{-1}(s)</math> | |
| and so for <math>s = g(t)</math>
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| :<math>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.</math>
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| == Derivation ==
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| Since <math>g'^*</math> is left adjoint to <math>g'_*</math>, we have:
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| :<math>\operatorname{id} \to g'_* \circ g'^*</math>
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| and so | |
| :<math>R^r f_* \to R^r f_* \circ g'_* \circ g'^*.</math> | |
| The [[Grothendieck spectral sequence]] then gives the first map and the last map (they are edge maps) in:
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| :<math>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'^*.</math>
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| Combining this with the above we get
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| :<math>R^r f_* \to g_* \circ R^r f'_* \circ g'^*.</math> | |
| Again using the adjoint relation we get the desired map.
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| == See also ==
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| *[[Theorem on formal functions]]
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| == References ==
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| {{reflist}}
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| * [[James Milne (mathematician)|J. S. Milne]] (2012). "[http://www.jmilne.org/math/CourseNotes/LEC.pdf Lectures on Étale Cohomology]
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| {{geometry-stub}}
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| [[Category:Geometry]]
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