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...

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

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...

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...

function Smooth morphism Étale morphisms – The algebraic analogue of local diffeomorphisms. Resolution of singularities contraction morphism Here is the...

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}}...

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

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

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...

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...

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...

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...

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

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

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...

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...

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...

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...

In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions...

{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...

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...

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...

(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...

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...

\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...