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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 vs. ! does hold for non-proper maps f.
For example, the product of affine spaces A m and A n over a field k is the affine space A m+n over k. For a scheme X over a field k and any field extension E of k, the base change X E means the fiber product X × Spec(k) Spec(E). Here X E is a scheme over E. For example, if X is the curve in the projective plane P 2
Barwise compactness theorem (mathematical logic) Base change theorems (algebraic geometry) Basel problem (mathematical analysis) Bass's theorem (group theory) Basu's theorem ; Bauer–Fike theorem (spectral theory) Bayes' theorem (probability) Beatty's theorem (Diophantine approximation) Beauville–Laszlo theorem (vector bundles)
More strongly, properness is local on the base in the fpqc topology. For example, if X is a scheme over a field k and E is a field extension of k, then X is proper over k if and only if the base change X E is proper over E. [3] Closed immersions are proper. More generally, finite morphisms are proper. This is a consequence of the going up theorem.
Let S be a scheme and denote the image of the structure map . The smooth base change theorem states the following: let f : X → S {\displaystyle f:X\to S} be a quasi-compact morphism , g : S ′ → S {\displaystyle g:S'\to S} a smooth morphism and F {\displaystyle {\mathcal {F}}} a torsion sheaf on X et {\displaystyle X_{\text{et}}} .
In mathematics, base change may mean: Base change map in algebraic geometry; Fiber product of schemes in algebraic geometry; Change of base (disambiguation) in linear algebra or numeral systems; Base change lifting of automorphic forms
Finite morphisms are closed, hence (because of their stability under base change) proper. [5] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra. Finite morphisms have finite fibers (that is, they are quasi-finite). [6] This follows from the fact that for a field k, every finite k-algebra is an Artinian ring.
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