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searching for Morphism of schemes 45 found (70 total)

Six operations (1,038 words) [view diff] exact match in snippet view article find links to article

sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y. The basic insight was that many of the elementary facts

Radicial morphism (208 words) [view diff] exact match in snippet view article find links to article

In algebraic geometry, a morphism of schemes f: X → Y is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is

Generic flatness (393 words) [view diff] exact match in snippet view article find links to article

an integral locally noetherian scheme, u : X → Y is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset

Étale morphism (2,496 words) [view diff] exact match in snippet view article find links to article

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

Quasi-separated morphism (415 words) [view diff] exact match in snippet view article find links to article

In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X × YX is quasi-compact (meaning that

Flat morphism (3,528 words) [view diff] exact match in snippet view article find links to article

called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two

Cotangent sheaf (891 words) [view diff] exact match in snippet view article find links to article

\Theta _{X}} . There are two important exact sequences: If S →T is a morphism of schemes, then f ∗ Ω S / T → Ω X / T → Ω X / S → 0. {\displaystyle f^{*}\Omega

Fpqc morphism (368 words) [view diff] exact match in snippet view article find links to article

fpqc morphism is fpqc. If f : X → Y {\displaystyle f:X\to Y} is a morphism of schemes and if there is an open covering V i {\displaystyle V_{i}} of Y such

Finite morphism (861 words) [view diff] exact match in snippet view article find links to article

finite morphism f: X → Y is finite. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is finite. This corresponds

Smooth morphism (1,564 words) [view diff] exact match in snippet view article find links to article

base space (by Ehresmann's theorem). Let f {\displaystyle f} be the morphism of schemes Spec C ( C [ x , y ] ( f = y 2 − x 3 − x − 1 ) ) → Spec ( C ) {\displaystyle

P-curvature (239 words) [view diff] exact match in snippet view article find links to article

but only exists in finite characteristic. Suppose X/S is a smooth morphism of schemes of finite characteristic p > 0, E a vector bundle on X, and ∇ {\displaystyle

Étale topos (321 words) [view diff] exact match in snippet view article find links to article

X\}_{i\in I}} , where each φ i {\displaystyle \varphi _{i}} is an étale morphism of schemes, such that the family is jointly surjective that is X = ⋃ i ∈ I φ

Smooth scheme (1,047 words) [view diff] exact match in snippet view article find links to article

locally of finite type. There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a

Étale algebra (353 words) [view diff] exact match in snippet view article find links to article

finite-dimensional over K, and the trace form Tr(xy) is nondegenerate. The morphism of schemes Spec ⁡ L → Spec ⁡ K {\displaystyle \operatorname {Spec} L\to \operatorname

Formally étale morphism (537 words) [view diff] exact match in snippet view article find links to article

to formally étale for the discrete topology for rings. That is, a morphism of schemes f : X → Y is formally étale if for every affine Y-scheme Z, every

Valuative criterion (277 words) [view diff] exact match in snippet view article find links to article

point of Spec A. Let X and Y be schemes, and let f : X → Y be a morphism of schemes. Then the following are equivalent: f is separated (resp. universally

Schlessinger's theorem (608 words) [view diff] exact match in snippet view article find links to article

definition is closely related to the notion of a formally smooth morphism of schemes. If in addition the map between the tangent spaces of F and G is

Exceptional inverse image functor (707 words) [view diff] exact match in snippet view article find links to article

form. Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor Rf!: D(Y) → D(X)

Ramification (mathematics) (1,116 words) [view diff] exact match in snippet view article

define étale morphisms. Let f : X → Y {\displaystyle f:X\to Y} be a morphism of schemes. The support of the quasicoherent sheaf Ω X / Y {\displaystyle \Omega

Picard group (1,140 words) [view diff] exact match in snippet view article find links to article

topological classification by intersection numbers. Let f: X →S be a morphism of schemes. The relative Picard functor (or relative Picard scheme if it is

Gauss–Manin connection (1,042 words) [view diff] exact match in snippet view article find links to article

connection is to be inferred from the flat sections. Consider a smooth morphism of schemes X→B{\displaystyle X\to B} over characteristic 0. If we consider these

Flat module (4,590 words) [view diff] exact match in snippet view article find links to article

saying that flatness is a local property. The definition of a flat morphism of schemes results immediately from the local property of flatness. A morphism

Regular embedding (1,182 words) [view diff] exact match in snippet view article find links to article

{\displaystyle f\colon X\rightarrow Y} is a any local complete intersection morphism of schemes, its cotangent complex L X / Y {\displaystyle L_{X/Y}} is perfect

Degeneration (algebraic geometry) (596 words) [view diff] exact match in snippet view article

is I. In general, given a pointed scheme (S, 0) and a scheme X, a morphism of schemes π: X' → S is called the deformation of a scheme X if it is flat and

Fundamental group scheme (1,710 words) [view diff] exact match in snippet view article find links to article

X → Spec ( k ) {\displaystyle X\to {\text{Spec}}(k)} is a proper morphism of schemes with X {\displaystyle X} reduced and connected scheme. Assuming the

Zariski topology (2,766 words) [view diff] exact match in snippet view article find links to article

by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper

Nisnevich topology (1,767 words) [view diff] exact match in snippet view article find links to article

by Yevsey Nisnevich, who was motivated by the theory of adeles. A morphism of schemes f : Y → X {\displaystyle f:Y\to X} is called a Nisnevich morphism

Dualizing sheaf (1,018 words) [view diff] exact match in snippet view article find links to article

holds for any locally free sheaf. Given a proper finitely presented morphism of schemes f : X → Y {\displaystyle f:X\to Y} , (Kleiman 1980) defines the relative

Ample line bundle (6,640 words) [view diff] exact match in snippet view article find links to article

Namely, let f : X → Y {\displaystyle f\colon X\to Y} be a proper morphism of schemes, and let L be a line bundle on X. Then the set of points y in Y such

Hilbert's Nullstellensatz (3,811 words) [view diff] exact match in snippet view article find links to article

In this vein, one has the following theorem: Any faithfully flat morphism of schemes f : Y → X {\textstyle f:Y\to X} locally of finite presentation admits

Spectrum of a ring (3,908 words) [view diff] exact match in snippet view article find links to article

{A}})),} where π : X → S {\displaystyle \pi \colon X\to S} is a morphism of schemes. The relative spec is the correct tool for parameterizing the family

Topos (4,265 words) [view diff] exact match in snippet view article find links to article

{\displaystyle {\text{Spec}}(k)\to {\text{Spec}}(\kappa (x))} is an etale morphism of schemes. More precisely, those are the global points. They are not adequate

Frobenius endomorphism (4,337 words) [view diff] exact match in snippet view article find links to article

Let φ : X → S be a morphism of schemes, and denote the absolute Frobenius morphisms of S and X by FS and FX, respectively. Define X(p) to be the base

Deformation (mathematics) (4,021 words) [view diff] exact match in snippet view article

(A')\end{matrix}}} the name smooth comes from the lifting criterion of a smooth morphism of schemes. Recall that the tangent space of a scheme X {\displaystyle X} can

Proj construction (3,567 words) [view diff] exact match in snippet view article find links to article

{A} _{\lambda }^{1}-\{0,1\}\end{matrix}}} which is also a smooth morphism of schemes (which can be checked using the Jacobian criterion). The projective

Étale cohomology (5,016 words) [view diff] exact match in snippet view article find links to article

functor coincides with the section functor Γ. More generally, a morphism of schemes f : X → Y induces a map f∗ from étale sheaves over X to étale sheaves

Algebraic variety (5,757 words) [view diff] exact match in snippet view article find links to article

subscheme defined by x = 0 (the origin). More generally, the fiber of a morphism of schemes X → Y at a point of Y may be non-reduced, even if X and Y are reduced

Main theorem of elimination theory (1,567 words) [view diff] exact match in snippet view article find links to article

{\displaystyle R[\mathbf {x} ]=R[x_{1},\ldots ,x_{n}]} defines a morphism of schemes (which are algebraic varieties if R is finitely generated over a

Blowing up (3,816 words) [view diff] exact match in snippet view article find links to article

{\displaystyle d^{2}} points by Bézout's theorem). The following projective morphism of schemes gives a model of blowing up P 2 {\displaystyle \mathbb {P} ^{2}}

Chow group (4,195 words) [view diff] exact match in snippet view article find links to article

Chow ring" and more generally a bivariant theory associated to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant functors

Kähler differential (4,378 words) [view diff] exact match in snippet view article find links to article

cotangent space for related notions). Moreover, it extends to a general morphism of schemes f : X → Y {\displaystyle f:X\to Y} by setting I {\displaystyle {\mathcal

Coherent sheaf cohomology (4,664 words) [view diff] exact match in snippet view article find links to article

theory includes generalizations to any coherent sheaf and any proper morphism of schemes, although the statements become less elementary. For example, for

Weil conjectures (6,101 words) [view diff] exact match in snippet view article find links to article

sheaves with weights ≤ β. Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif! takes mixed sheaves

Sheaf (mathematics) (10,835 words) [view diff] exact match in snippet view article

{\displaystyle E} into a scheme and π {\displaystyle \pi } into a morphism of schemes in such a way that π {\displaystyle \pi } retains the same universal

Cotangent complex (4,731 words) [view diff] exact match in snippet view article find links to article

infinitesimal thickening S → S ′ {\displaystyle S\to S'} , that is a morphism of schemes where the kernel I = ker { O S ′ → O S } {\displaystyle {\mathcal