_{S}^{n}\to S} where g is étale. A morphism of finite type is étale if and only if it is smooth and quasi-finite. A smooth morphism is stable under base change...

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...

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...

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

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...

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}}...

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...

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...

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

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...

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...

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...

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...

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...

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

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...

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...

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

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

_{i}N_{Y_{i}/X}|_{W}.} In particular, if X → S {\displaystyle X\to S} is a smooth morphism, then the normal bundle to the diagonal embedding Δ : X ↪ X × S ⋯ ×...

&\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...

scheme, in which case the pulled-back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also...

defines a morphism from the sheaf of smooth functions on N {\displaystyle N} to the direct image by ϕ {\displaystyle \phi } of the sheaf of smooth functions...

object x, there exists a morphism 1x : x → x (some authors write idx) called the identity morphism for x, such that every morphism f : a → x satisfies 1x...

type, then its canonical morphism p : P ( E ) → X {\displaystyle p:\mathbb {P} ({\mathcal {E}})\to X} is a projective morphism. For any x ∈ X {\displaystyle...

In algebraic geometry, an unramified morphism is a morphism f : X → Y {\displaystyle f:X\to Y} of schemes such that (a) it is locally of finite presentation...

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...

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...