Change of base (fiber product)

A change of base is understood to mean a special view of the formation of a fiber product in relative situations, especially in algebraic geometry . In this context, the fiber product is often referred to as pull-back .

When one speaks of a change of base, the following situation is meant: One is considering a morphism

${\displaystyle f\colon X\to Y}$

as a family with base Y . Is now a morphism

${\displaystyle g\colon Y'\to Y}$

given, then “the morphism resulting from the change of base along g ” is the canonical projection of the fiber product

${\displaystyle f'\colon X':=X\times _{Y}Y'\to Y'.}$

The base Y has thus been replaced by the base Y ′. One then also says briefly: " f ′ is the change in base of f under g ."

The symmetry of the fiber product is completely ignored.

Has g additional properties such. B. Flatness, one also speaks of "flat base change" etc.

Special base changes

If there is a morphism and the inclusion of a point with , then the base change along is the formation of the fiber ${\displaystyle f\colon X\to Y}$ ${\displaystyle i\colon {*}\to Y}$ ${\displaystyle i({*})=y}$ ${\displaystyle i}$

${\displaystyle f^{-1}(y)=X\times _{Y,i}{*}\to {*}.}$

If a subset of , the base change is along the inclusion ${\displaystyle U\subseteq Y}$ ${\displaystyle Y}$

${\displaystyle X\times _{Y}U\to U}$

the restriction of the family to the part of the base. ${\displaystyle X}$ ${\displaystyle U}$

"Stable under change of base"

If P is a property of morphisms of a category in which fiber products exist, then P is said to be stable under change of base if the validity of P for a morphism f :  X  →  Y is the validity of P for the morphism resulting from a change of base Y ′ →  Y

${\displaystyle f_{Y'}\colon X\times _{Y}Y'\longrightarrow Y'}$

implies.

Examples

• Monomorphisms
• Surjectivity in the categories of amounts or topological spaces , and in each category the property, a retraction to be
• Fibers in model categories , especially Serre fibers
• The property of continuous mappings of topological spaces to be closed , i.e. H. Mapping closed subsets to closed subsets is not stable under a change of basis: Let it be the mapping of the real straight line to a point; it is complete. By changing the base you get the canonical projection. It is not completed, for example the completed subset is mapped to the uncompleted set . In contrast, the closed images with compact fibers are closed in a pullback-stable manner. ${\displaystyle f\colon \mathbb {R} \to *}$ ${\displaystyle \mathbb {R} \to *}$ ${\displaystyle f'\colon \mathbb {R} ^{2}\to \mathbb {R} }$ ${\displaystyle \{(x,y)\mid xy=1\}}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$
• Many of the properties of morphisms of schemes considered in algebraic geometry are stable under change of base. If this is not the case for a property P , the property of a morphism that satisfies every change of base P is called "universal P ": for example, a morphism f is universally closed if every change of base of f is closed.