Grothendieck's relative point of view

Under a change of basis refers to 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 a pull -back.

Speaking of change of basis, so that the following situation is meant: One considers a morphism

As a family with base Y. Is now a morphism

Given, is "the by base change along g" resulting morphism is the canonical projection of the fiber product

The base Y was thus replaced by the base Y '. Then we also say briefly: "f 'is the base change of f under g "

The symmetry of the fiber product is completely ignored.

If g has additional properties such as Flatness, so it is also called " flat base change ", etc.

Special base change

Is a morphism, and the inclusion of a point with, then the base change along the formation of the fiber

Is a subset of, then the base change along the inclusion

The constraint on the portion of the family of the base.

" Stable under change of basis "

If P is a property of morphisms of a category exist in the fiber products, so called P stable under base change, if the validity of P f for a morphism: X → Y the validity of P for the morphism caused by a change of basis Y '→ Y

Implied.

Examples

• Monomorphisms
• Surjectivity in the categories of quantities or topological spaces, and in each category, the property of being a retraction
• Fibrations in model categories, in particular Serre- fibrations
• To be completed by the property of continuous maps of topological spaces, ie represent closed subsets to closed subsets, is not stable under base change: It is the image of the real line to a point; it is completed. Due to the change of basis is obtained, the canonical projection. It is not finished, for example, the sealed portion is not mapped to the closed set. Completed pullback - stable, however, are the completed pictures with compact fibers.
• Many of the properties of morphisms of schemes which are considered in algebraic geometry, are stable under base change. If this is not for a property P is the case, it is called the property of a morphism that any change of basis P met, "universal P": for example, a morphism f is then completed universal if every base change is complete of f.

• Algebraic Geometry
• Category theory