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A function between topological spaces is called monotone if every fiber is a ... is a morphism of schemes, the fiber of a point in is the fiber product ...
Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X × Y Spec(k(y)); this is a scheme over the field k(y). [3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
A cartesian morphism: is called an inverse image of its projection = (); the object is called an inverse image of by . The cartesian morphisms of a fibre category F S {\displaystyle F_{S}} are precisely the isomorphisms of F S {\displaystyle F_{S}} .
A morphism is quasi-finite if it is of finite type and has finite fibers. quasi-projective A quasi-projective variety is a locally closed subvariety of a projective space. quasi-separated A morphism f : Y → X is called quasi-separated or (Y is quasi-separated over X) if the diagonal morphism Y → Y × X Y is quasi-compact.
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.
For every morphism :, there is an object Z (called a cone or cofiber of the morphism u) fitting into an exact triangle [] The ... is called a fiber of the morphism ...
Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety. A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth.
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.