_{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...

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}}...

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...

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...

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...

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

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...

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

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...

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...

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...

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...

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...

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...

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...

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

field k ′ {\displaystyle k'} of k {\displaystyle k} . étale morphism formally smooth morphism Popescu's theorem Matsumura 1989, Theorem 25.3 Matsumura 1989...

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

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...

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...

The morphism from X′ to X does not depend on the embedding of X in W. Or in general, the sequence of blowings up is functorial with respect to smooth morphisms...

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

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...

{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...

\mathbb {R} } . The Legendre transformation of L {\textstyle L} is the smooth morphism F L : E → E ∗ {\displaystyle \mathbf {F} L:E\to E^{*}} defined by F...

_{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 ⋯ ×...

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...