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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.
The existence theorem for the twisted inverse image is the name given to the proof of the existence for what would be the counit for the comonad of the sought-for adjunction, namely a natural transformation
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}}} .
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
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.
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)
This change can be computed by substituting the "old" coordinates for their expressions in terms of the "new" coordinates. More precisely, if f(x) is the expression of the function in terms of the old coordinates, and if x = Ay is the change-of-base formula, then f(Ay) is the expression of the same function in terms of the new coordinates.
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.