In algebraic geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation (ii)...

other of morphisms. There are two objects that are associated to every morphism, the source and the target. A morphism f from X to Y is a morphism with source...

digital animation for a Tide commercial with a Tide detergent bottle smoothly morphing into the shape of the United States. The effect was programmed by...

a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism. Contents:  !$@ A B C D E F G H I J K L M N O P Q R S T U V W XYZ...

which can also be read off the above computation. A morphism f of finite type is a smooth morphism if it is flat and if Ω X / Y {\displaystyle \Omega _{X/Y}}...

general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme X is smooth over a field k if and...

smooth. This also proves this morphism is not smooth from the equivalence between formally smooth morphisms locally of finite presentation and smooth...

In algebraic geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is...

Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. A morphism f : X → Y {\displaystyle f:X\to Y} of...

naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. If X...

this definition to the general situation of a morphism of ringed topoi, thereby incorporating morphisms of ringed spaces, schemes, and algebraic spaces...

H^{2\dim(X)-2d}(X,\mathbb {Q} ).} Now consider a proper morphism f : X → Y {\displaystyle f\colon X\to Y} between smooth quasi-projective schemes and a bounded complex...

mathematics, in particular in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat...

abelian; the kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.) A vector bundle morphism between vector bundles...

and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular...

quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism. § 8.6 of F...

the animator can then smoothly morph (or "blend") between the base shape and one or several morph targets. Typical examples of morph targets used in facial...

and the Hom functor on modules. Flat morphism, Smooth morphism, Proper morphism, Finite morphism, Étale morphism Stable curve Birational geometry Étale...

holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3. Take two smooth curves C and D in a smooth projective...

(which means a smooth rational curve of self-intersection number −1), then there exists a morphism from X {\displaystyle X} to another smooth projective surface...

sections of the cotangent bundle are called (differential) one-forms. A smooth morphism ϕ : M → N {\displaystyle \phi \colon M\to N} of manifolds induces a...

In algebraic geometry, given a morphism of schemes p : X → S {\displaystyle p:X\to S} , the diagonal morphism δ : X → X × S X {\displaystyle \delta :X\to...

as extension fields of k. A special case is a birational morphism f : X → Y, meaning a morphism which is birational. That is, f is defined everywhere, but...

through f have the same étale cohomology, locally. For example, a smooth morphism is universally locally acyclic. Milne, J. S. (1980), Étale cohomology...

&\operatorname {Spec} (A')\end{matrix}}} the name smooth comes from the lifting criterion of a smooth morphism of schemes. Recall that the tangent space of...

X {\displaystyle X} . A morphism φ : F → G {\displaystyle \varphi :{\mathcal {F}}\to {\mathcal {G}}} consists of a morphism φ U : F ( U ) → G ( U ) {\displaystyle...

{Spec} (A\otimes _{B}C).} The morphism X ×Y Z → Z is called the base change or pullback of the morphism X → Y via the morphism Z → Y. In some cases, the fiber...

morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism...

G-maps. The composition of two morphisms is again a morphism. If a morphism f is bijective, then its inverse is also a morphism. In this case f is called an...

the closed immersion determined by J, and every Y-morphism g : Z0 → X, there exists a unique Y-morphism s : Z → X such that g = si. It is equivalent to...