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The proof that fiber products of schemes always do exist reduces the problem to the tensor product of commutative rings (cf. gluing schemes). In particular, when X, Y, and Z are all affine schemes, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A,B,C, the fiber product is the affine scheme
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the categorical fiber product exists in the category of schemes. If X and Z are schemes over a field k, their fiber product over Spec(k) may be called the product X × Z in the category of k-schemes.
In mathematics, the fiber or fibre (British English) of an element under a function is the ... in is the fiber product of schemes where () is ...
The fiber product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively. [ 9 ] Flatness and faithful flatness is preserved by base change: If f is flat or faithfully flat and g : Y ′ → Y {\displaystyle g\colon Y'\to Y} , then the fiber product f × g : X × Y Y ′ → Y ′ {\displaystyle f\times g ...
In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is , the fiber product of the closed immersions,.It is denoted by .. Locally, W is given as for some ring R and X, Y as (/), (/) for some ideals I, J.
The category of schemes admits finite pullbacks and in some cases finite pushouts; [4] they both are constructed by gluing affine schemes. For affine schemes, fiber products and pushouts correspond to tensor products and fiber squares of algebras.
In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions: [1] Every point x of X is isolated in its fiber f −1 (f(x)). In other words, every fiber is a discrete (hence finite) set.