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A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is monotone in this topological sense if and only if it is non-increasing or non-decreasing , which is the usual meaning of " monotone function " in real analysis .
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 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.
(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). 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.
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A principal -bundle, where denotes any topological group, is a fiber bundle: together with a continuous right action such that preserves the fibers of (i.e. if then for all ) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each and , the map sending to is a homeomorphism.
Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X × Y Z → Z is proper. Properness is a local property on the base (in the Zariski topology). That is, if Y is covered by some open subschemes Y i and the restriction of f to all f −1 (Y i) is proper ...
The choice of a (normalised) cleavage for a fibred -category specifies, for each morphism : in , a functor:; on objects is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms.