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Concept in algebraic geometry
In algebraic geometry, a finite morphism between two affine varieties X , Y {\displaystyle X,Y} is a dense regular map which induces isomorphic inclusion
Finite_morphism
Type of morphism in algebraic geometry
algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent
Quasi-finite_morphism
Concept in mathematics
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
Morphism of algebraic varieties
Morphism_of_algebraic_varieties
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
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Term in algebraic geometry
compact. 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
Proper_morphism
analogous notion in terms of schemes is that a morphism f : X → Y {\displaystyle f:X\to Y} of schemes is of finite type if Y {\displaystyle Y} has a covering
Morphism_of_finite_type
Concept in algebraic geometry
an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation; the étale morphism is connected
Étale_morphism
schemes can be found in the article on finite morphisms. Finite morphism Finitely generated algebra Finitely generated module Atiyah & Macdonald (1969)
Finite_algebra
Theorem of algebraic geometry and commutative algebra
a proper birational morphism is connected. A generalization due to Grothendieck describes the structure of quasi-finite morphisms of schemes. Several
Zariski's_main_theorem
Topics referred to by the same term
Morphism of finite type, a morphism of schemes with underlying morphisms on affine opens given by algebras of finite type Scheme of finite type, a scheme
Finite_type
Scheme theory concept
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
Flat_morphism
_{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
Smooth_morphism
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 and (b) for
Unramified_morphism
Result of commutative algebra
map S ↪ A {\displaystyle S\hookrightarrow A} induces a surjective finite morphism of affine varieties X → A k d {\displaystyle X\to \mathbb {A} _{k}^{d}}
Noether_normalization_lemma
diagonal morphism is a closed immersion. Also, a morphism p : X → S {\displaystyle p:X\to S} locally of finite presentation is an unramified morphism if and
Diagonal morphism (algebraic geometry)
Diagonal_morphism_(algebraic_geometry)
Concept in algebraic geometry
morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism
Morphism_of_schemes
Concept in mathematics
a code. Every elementary morphism is a code. For L a subset of B∗, a finite subset T of L is a test set for L if morphisms f and g on B∗ agree on L if
Free_monoid
Stein factorization, any surjective projective morphism is a contraction morphism followed by a finite morphism. Examples include ruled surfaces and Mori fiber
Contraction_morphism
finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands for fidèlement
Flat_topology
spaces, states that a proper morphism of schemes can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly
Stein_factorization
Map raising elements to the pth power, in characteristic p
functor, because an S-morphism X → Y induces an S-morphism XF → YF. For example, consider a ring A of characteristic p > 0 and a finitely presented algebra
Frobenius_endomorphism
Type of category in mathematics
a geometric morphism X → Y is to give a functor u∗: Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi
Topos
Type of algebra
ISBN 9780201407518. Finitely generated module Finitely generated field extension Artin–Tate lemma Noether normalization lemma Finite algebra Morphism of finite type
Finitely_generated_algebra
Algebraic geometry
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
Formally_étale_morphism
Generalization of algebraic variety
and the Hom functor on modules. Flat morphism, Smooth morphism, Proper morphism, Finite morphism, Étale morphism Stable curve Birational geometry Étale
Scheme_(mathematics)
Collection of objects and morphisms
a morphism 1 x : x → x {\displaystyle 1_{x}:x\to x} (some authors write id x {\displaystyle \operatorname {id} _{x}} ) called the identity morphism for
Category_(mathematics)
General theory of mathematical structures
objects of the category, and the morphisms, which relate two objects called the source and the target of the morphism. A morphism is often represented by an
Category_theory
French mathematician (1928–2014)
Projective tensor product Proper morphism – Term in algebraic geometry Pursuing Stacks – Seminal math text Quasi-finite morphism Quot scheme Ramanujam–Samuel
Alexander_Grothendieck
Type of ringed space
{\displaystyle f^{-1}(U_{i})} are affine. For example, a finite morphism is affine. An affine morphism is quasi-compact and separated; in particular, the direct
Sheaf_of_algebras
Characterizing property of mathematical constructions
property of universal morphisms, given any morphism h : X 1 → X 2 {\displaystyle h:X_{1}\to X_{2}} there exists a unique morphism g : A 1 → A 2 {\displaystyle
Universal_property
Used to compare mixed characteristic situations with purely finite characteristic ones
a finite étale morphism of adic spaces over K and Y is perfectoid, then X also is perfectoid; A morphism X → Y of perfectoid spaces over K is finite étale
Perfectoid_space
Algebraic structure with an associative operation and an identity element
monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object.
Monoid
Projective variety that is also an algebraic group
abelian varieties carry the structure of a group. A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves
Abelian_variety
Tool to track locally defined data attached to the open sets of a topological space
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
Sheaf_(mathematics)
a universal homeomorphism is a morphism of schemes f : X → Y {\displaystyle f:X\to Y} such that, for each morphism Y ′ → Y {\displaystyle Y'\to Y}
Universal_homeomorphism
Algebraic variety in a projective space
^{r}} is a finite morphism. Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Start with
Projective_variety
A morphism from a quasi-compact scheme to an affine scheme is quasi-compact. Let f : X → Y {\displaystyle f:X\to Y} be a quasi-compact morphism between
Quasi-compact_morphism
Relation between algebraic varieties and polynomial ideals
locally of finite presentation admits a quasi-section, in the sense that there exists a faithfully flat and locally quasi-finite morphism g : X ′ → X
Hilbert's_Nullstellensatz
Transformations induced by a mathematical group
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
Group_action
Concept in algebraic geometry
morphism has the property that L {\displaystyle L} is the pullback f ∗ O ( 1 ) {\displaystyle f^{*}{\mathcal {O}}(1)} . Conversely, for any morphism f
Ample_line_bundle
Sheaf of rings in mathematics
{O}}_{X}} is a morphism from the structure sheaf of Y {\displaystyle Y} to the direct image of the structure sheaf of X. In other words, a morphism from ( X
Ringed_space
Faithfully flat morphism of schemes
flat morphism is fpqc. An fpqc morphism satisfies the following properties: The composite of fpqc morphisms is fpqc. A base change of an fpqc morphism is
Fpqc_morphism
Mathematical technique in algebraic geometry
the restriction of f to X′ such that X′ → T is a finite morphism and T → S is a smooth affine morphism with geometrically integral fibers of dimension
Dévissage
Type of map between algebraic groups
a finite kernel. In the case of abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres
Isogeny
Well-quasi-ordering of finite trees
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic
Kruskal's_tree_theorem
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 embedding. If Spec
Regular_embedding
Isomorphism of an object to itself
some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f : X → X {\displaystyle f:X\to X}
Automorphism
algébrique Fiber product of schemes Flat morphism Smooth scheme Finite morphism Quasi-finite morphism Proper morphism Semistable elliptic curve Grothendieck's
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Mathematical mapping between objects arising from their definitions
closely related notion is that of a structure map or structure morphism: the map or morphism that comes with the given structure on the object. These are
Canonical_map
Generalized alphabetical order
used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences
Lexicographic_order
Category-theoretic construction
canonical morphism X ⊕ Y → X × Y {\displaystyle X\oplus Y\rightarrow X\times Y} . This may be extended by induction to a canonical morphism from any finite coproduct
Coproduct
Object that is both a product and coproduct
A_{n}} (the embedding morphisms) satisfying p k ∘ i k = 1 A k {\textstyle p_{k}\circ i_{k}=1_{A_{k}}} , the identity morphism of A k , {\displaystyle
Biproduct
Category whose objects are finite sets and whose morphisms are functions
objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and
FinSet
affine variety X. Then the inclusion map S → A induces a surjective finite morphism of affine varieties X → A k d {\displaystyle X\to \mathbb {A} _{k}^{d}}
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Type of category in category theory
closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors.
Cartesian_closed_category
Mathematical structure
this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat, of finite presentation, and is quasi-finite. fpqc stands for
Grothendieck_topology
Type of commutative ring in mathematics
of finite type over a field K (for example, an affine variety). Let n be the dimension of X. By Noether normalization, there is a finite morphism f from
Cohen–Macaulay_ring
Type of mathematical object
to ensure representability of the resulting functor. When this morphism is along a finite extension of fields, it is known as Weil restriction. For any
Group_scheme
Algebraic structure
there is a surjective semigroup morphism from S to T. For example, (Z/2Z, +) is a quotient of (Z/4Z, +), using the morphism consisting of taking the remainder
Semigroup
mathematics, the image of a morphism is a generalization of the image of a function. Given a category C {\displaystyle C} and a morphism f : X → Y {\displaystyle
Image_(category_theory)
Mathematical concept
parallel pair of morphisms. Cokernels are coequalizers of a morphism and a parallel zero morphism. Pushouts are colimits of a pair of morphisms with common
Limit_(category_theory)
Category
are the same as finite coproducts, making them biproducts; given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists
Pre-abelian_category
Partially ordered set in which all subsets have both a supremum and infimum
pairs of elements need to have a supremum and an infimum. Every non-empty finite lattice is complete, but infinite lattices may be incomplete. Complete lattices
Complete_lattice
Topics referred to by the same term
mathematics, there are several finiteness theorems. Ahlfors finiteness theorem Finiteness theorem for a proper morphism Compactness theorem, in mathematical
Finiteness_theorem
Function, homomorphism, or morphism
for "morphism" or "arrow", which is a structure-respecting function and thus may imply more structure than "function" does. For example, a morphism f :
Map_(mathematics)
{\displaystyle X'\to X} is a proper morphism of finite presentation, Z → X {\displaystyle Z\to X} is a closed immersion of finite presentation, and X ′ → X {\displaystyle
H_topology
In mathematics, invertible homomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse
Isomorphism
Concept in algebraic geometry
finite birational morphism from any variety Y to X is an isomorphism. Normal varieties were introduced by Zariski. A morphism of varieties is finite if
Normal_scheme
Algebraic geometry analog of a principal bundle in algebraic topology
{\displaystyle a:Y\to T} is a X {\displaystyle X} -morphism and b : G → H {\displaystyle b:G\to H} is group-scheme morphism such that σ H ∘ ( a × b ) = a ∘ σ G {\displaystyle
Torsor_(algebraic_geometry)
Central object of study in category theory
, the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally
Natural_transformation
Special type of lattice
Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity (and thus be a morphism of distributive
Distributive_lattice
Generalized object in category theory
\mathbf {C} .} This universal morphism consists of an object X {\displaystyle X} of C {\displaystyle C} and a morphism ( X , X ) → ( X 1 , X 2 ) {\displaystyle
Product_(category_theory)
of finite type and quasi-separated) and for every valuation ring A, if Y' = Spec A and X' denotes the generic point of Y' , then for every morphism Y'
Valuative_criterion
Mathematical concept
resolution is a morphism that combines symplectic geometry and resolution of singularities. Let π : Y → X {\displaystyle \pi :Y\to X} be a morphism between complex
Symplectic_resolution
a finite product of finite separable field extensions of K. L is finite-dimensional over K, and the trace form Tr(xy) is nondegenerate. The morphism of
Étale_algebra
of maps (or "morphisms"). The key result is: Chevalley's theorem. If f : X → Y {\displaystyle f:X\to Y} is a finitely presented morphism of schemes and
Constructible_set_(topology)
and q be elements of the monoid M. Let φ be a morphism of monoids from M to N. It is said that the morphism φ separates p and q if ϕ ( p ) ≠ ϕ ( q ) {\displaystyle
Profinite_word
parallel morphisms x0 and x1 from X1 to X0 such that there exist a reflexivity morphism r from X0 to X1 such that x0r = x1r = 1X0; a symmetry morphism s from
Exact_completion
Category theory concept
π : A → X {\displaystyle \pi :A\to X} is a morphism in C {\displaystyle {\mathcal {C}}} . Then, a morphism between objects f : ( A , π ) → ( A ′ , π ′
Overcategory
Self-self morphism
object to itself. More generally in category theory, an endomorphism is a morphism from an object in some category to itself. An endomorphism that is also
Endomorphism
sends cartesian morphisms to cartesian morphisms. cartesian morphism 1. Given a functor π: C → D (e.g., a prestack over schemes), a morphism f: x → y in
Glossary_of_category_theory
Relationship between two functors abstracting many common constructions
every C-morphism f : FY → X, there is a unique D-morphism ΦY, X(f) = g : Y → GX, and for every D-morphism g : Y → GX, there is a unique C-morphism Φ−1Y,
Adjoint_functors
Branch of mathematics
f_{*}\mu _{f}} is equal to μ f {\displaystyle \mu _{f}} . Because f is a finite morphism, the pullback measure f ∗ μ f {\displaystyle f^{*}\mu _{f}} is also
Complex_dynamics
that if Y is 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
Generic_flatness
algebraic geometry, the projection formula states the following: For a morphism f : X → Y {\displaystyle f:X\to Y} of ringed spaces, an O X {\displaystyle
Projection_formula
Algebraic structure used in logic
from any Heyting algebra to itself is a morphism, and the composite g ∘ f of any two morphisms f and g is a morphism. Hence Heyting algebras form a category
Heyting_algebra
Relate the direct image and the pull-back of sheaves
with a number of technical conditions: f needs to be a separated morphism of finite type, the schemes involved need to be Noetherian). A far reaching
Base_change_theorems
Category in which all small limits exist
one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits
Complete_category
Surjective homomorphism
theory, an epimorphism is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: Y → Z, g 1 ∘ f =
Epimorphism
Branch of mathematics that studies the properties of groups
between 1960 and 2004, that culminated into a complete classification of finite simple groups. Group theory has three main historical sources: number theory
Group_theory
more generally, a similar statement holds for any quasi-finite morphism such as an étale morphism. If f is an open immersion, the exceptional inverse image
Exceptional inverse image functor
Exceptional_inverse_image_functor
Type of object in algebraic geometry
a Deligne–Mumford stack if the following conditions hold: The diagonal morphism Δ F : F → F × S F {\displaystyle \Delta _{F}\colon F\to F\times _{S}F}
Deligne–Mumford_stack
Completion of the usual space with "points at infinity"
projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an
Projective_space
Category with direct sums and certain types of kernels and cokernels
abelian. Specifically: AB1) Every morphism has a kernel and a cokernel. AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism
Abelian_category
Category of finite-dimension vector spaces
(or FdVect) is the category whose objects are all finite-dimensional vector spaces and whose morphisms are all linear maps between them. FinVect has two
FinVect
Mathematical object studied in the field of algebraic geometry
(irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type. An affine variety over an algebraically closed field
Algebraic_variety
Most general completion of a commutative square given two morphisms with same codomain
a pullback diagram, then the induced morphism ker(p2) → ker(f) is an isomorphism, and so is the induced morphism ker(p1) → ker(g). Every pullback diagram
Pullback_(category_theory)
Mathematics term
that is, a morphism that maps letter to letter. If a morphic word is constructed as the fixed point of a prolongable k-uniform morphism on A∗ then the
Morphic_word
Mathematical function such that every output has at least one input
above, on. Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of a morphism f is called a
Surjective_function
Formal semantics for non-classical logic systems
Kripke semantics are called p-morphisms (which is short for pseudo-epimorphism, but the latter term is rarely used). A p-morphism of Kripke frames ⟨ W , R
Kripke_semantics
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