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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 g ∗ {\displaystyle g_{*}} vs. R f ! {\displaystyle Rf_{!}} does hold for non-proper maps f .
In mathematics, change of base can mean any of several things: Changing numeral bases, such as converting from base 2 to base 10 . This is known as base conversion. The logarithmic change-of-base formula, one of the logarithmic identities used frequently in algebra and calculus.
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.
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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.
Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem ; Büchi-Elgot-Trakhtenbrot theorem (mathematical logic) Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory, Cantor's diagonal argument) Church–Rosser theorem (lambda calculus)
Finite morphisms are closed, hence (because of their stability under base change) proper. [4] This follows from the going up theorem of Cohen-Seidenberg in commutative algebra. Finite morphisms have finite fibers (that is, they are quasi-finite). [4] This follows from the fact that for a field k, every finite k-algebra is an Artinian ring.
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