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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
Diagonal morphism (algebraic geometry)
Diagonal_morphism_(algebraic_geometry)
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Generalization of algebraic spaces or schemes
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli
Algebraic_stack
Map (arrow) between two objects of a category
homological algebra and algebraic topology. They belong to the foundational tools of Grothendieck's scheme theory, a generalization of algebraic geometry that
Morphism
Concept in algebraic geometry
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by
Morphism_of_schemes
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Function, homomorphism, or morphism
operation is composition of permutations Regular map (algebraic geometry) – Morphism of algebraic varieties Weisstein, Eric W. "Map". mathworld.wolfram
Map_(mathematics)
Branch of mathematics
enormous role in algebraic topology. Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory
Homological_algebra
Generalization of a scheme
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory
Algebraic_space
Completion of the usual space with "points at infinity"
a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization
Projective_space
Structure in algebraic geometry
In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as
Motive_(algebraic_geometry)
Generalization of algebraic variety
commutative algebra can be viewed as an algebraic approach to affine algebraic varieties. However, many arguments in algebraic geometry work better for
Scheme_(mathematics)
Type of category in mathematics
a notion of localization. Grothendieck topoi find applications in algebraic geometry. They are generalized by elementary topoi, which are used in logic
Topos
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Generalisation of a sheaf; a fibered category that admits effective descent
topology on algebraic spaces. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the
Stack_(mathematics)
Algebra based on a vector space with a quadratic form
Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing μ2 for the algebraic group of square roots
Clifford_algebra
a {\displaystyle a} and every morphism in J {\displaystyle {\mathcal {J}}} to 1 a {\displaystyle 1_{a}} . The diagonal functor Δ : C → C J {\displaystyle
Diagonal_functor
Generalization of vector bundles
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric
Coherent_sheaf
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
Unramified_morphism
Construction in algebra
deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there
Hopf_algebra
Elements taken to zero by a homomorphism
a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group
Kernel_(algebra)
Algebraic geometry category satisfying lifting conditions
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying
Prestack
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
Mathematical category with weak equivalences, fibrations and cofibrations
language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results
Model_category
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
In algebraic geometry, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely
Ideal_sheaf
Most general completion of a commutative square given two morphisms with same codomain
is, the mediating morphism u : Q → P above is not required to be unique. Pullbacks in differential geometry Join (relational algebra) Mitchell, p. 9 Lee
Pullback_(category_theory)
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
Category with direct sums and certain types of kernels and cokernels
make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory
Abelian_category
Mapping between categories
in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects
Functor
Branch of mathematics
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local
Derived_algebraic_geometry
In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X ×Y X is quasi-compact (meaning that
Quasi-separated_morphism
Concept in category theory
{\displaystyle n:z\to y} is an f {\displaystyle f} -morphism, then there is precisely one T {\displaystyle T} -morphism a : z → x {\displaystyle a:z\to x} such that
Fibred_category
Study of categorified structures
"Non-commutative Geometry and Non-Abelian Algebraic Topology". PlanetPhysics. Archived from the original on 2009-08-14. Retrieved 2009-03-02. Non-Abelian Algebraic Topology
Higher-dimensional_algebra
Differential form in commutative algebra
standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers
Kähler_differential
Generalized notion of counting curve intersections
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves
Intersection_number
Embedding of categories into functor categories
{\mathcal {C}}} ) to the morphism f ∘ − {\displaystyle f\circ -} (composition with f {\displaystyle f} on the left) that sends a morphism g {\displaystyle g}
Yoneda_lemma
Type of object in algebraic geometry
In algebraic geometry, a Deligne–Mumford stack is a stack that behaves, in many respects, like an algebraic variety or an orbifold, while still allowing
Deligne–Mumford_stack
Moduli scheme of subschemes of a scheme, represents the flat-family-of-subschemes functor
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space
Hilbert_scheme
Mathematical parametrization of vector spaces by another space
Lie group), algebraic functions then the vector bundle is an algebraic vector bundle (this requires the matrix group to be an algebraic group). The C∞-vector
Vector_bundle
Indexed collection of objects and morphisms in a category
which sends every object of J to an object N of C and every morphism to the identity morphism on N. The limit of a diagram D is a universal cone to D. That
Diagram_(category_theory)
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
Mathematical construction used in homotopy theory
category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to
Simplicial_set
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules Ω
Cotangent_sheaf
Conjecture in algebraic geometry
specifically arithmetic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms
Tate_conjecture
Algebra in algebraic topology
In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {\displaystyle
Steenrod_algebra
In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which
Fourier–Mukai_transform
mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable
Diagonalizable_group
Group that is also a differentiable manifold with group operations that are smooth
algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and
Lie_group
Additional mathematical object
known as a morphism, and such maps are of special interest in many fields of mathematics. Examples include homomorphisms, which preserve algebraic structures;
Mathematical_structure
a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that f = g ∘ h (in
Lift_(mathematics)
Mathematical set of all subsets of a set
functor which sends a set S to P(S) and a morphism f: S → T (here, a function between sets) to the image morphism. That is, for A = {x1, x2, ...} ∈ P(S)
Power_set
Function that returns its argument unchanged
identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M {\displaystyle M} need
Identity_function
Mathematical concept
proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field
Elliptic_surface
Sequence of homomorphisms such that each kernel equals the preceding image
morphism t : B → A {\displaystyle t:B\to A} such that t ∘ f {\displaystyle t\circ f} is the identity on A {\displaystyle A} . There exists a morphism
Exact_sequence
Map in projective geometry
the set-theoretic sense: it is a closed immersion in the sense of algebraic geometry. That is, one can give a set of equations for the image. Except for
Segre_embedding
Special case of colimit in category theory
underlying sets equipped with a given algebraic structure, such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this
Direct_limit
Certain generalizations of groups
category of supermanifolds. An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more
Group_object
Mathematics study in geometry
mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories
Derived noncommutative algebraic geometry
Derived_noncommutative_algebraic_geometry
Scheme in algebraic geometry
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry
Normal cone (algebraic geometry)
Normal_cone_(algebraic_geometry)
Branch of mathematics
structures that are often specified via algebraic operations and defining identities are Heyting algebras and Boolean algebras, which both introduce a new operation
Order_theory
Concept in algebraic geometry
invariants to distinguish one algebraic variety from another. Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent
Coherent_sheaf_cohomology
Algebraic structure used in analysis
this notation, a Lie algebra can be defined as an object A {\displaystyle A} in the category of vector spaces together with a morphism [ ⋅ , ⋅ ] : A ⊗ A
Lie_algebra
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
Algebro-geometric stability condition
differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The
K-stability
, and for any morphism φ : X → B {\displaystyle \varphi :X\to B} from the class Φ {\displaystyle \Phi } there exists a unique morphism φ ′ : X ′ → B {\displaystyle
Envelope_(category_theory)
projective curves, using techniques from algebraic geometry. Han, Monsky, and Teixeira have treated diagonal hypersurfaces and various related hypersurfaces
Hilbert–Kunz_function
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
Abstract mathematics relationship
c} and all morphisms to 1 c {\displaystyle 1_{c}} . By contrast, the category C {\displaystyle C} with a single object and a single morphism is not equivalent
Equivalence_of_categories
Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension
Fulton–Hansen connectedness theorem
Fulton–Hansen_connectedness_theorem
Concept in algebraic topology
needed to define the concept of hypercovering in homotopical algebra and algebraic geometry. Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge
N-skeleton
Concept in mathematics
reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive
Reductive_group
{X}}(M)})} Any morphism ϕ : P → P ′ {\displaystyle \phi :P\to P'} of G {\displaystyle G} -principal bundles induces a Lie algebroid morphism d ϕ : T P /
Atiyah_algebroid
Mathematical group of the homotopy classes of loops in a topological space
(1998-07-22), An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1 Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston:
Fundamental_group
Vector bundle of cotangent spaces at every point in a manifold
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point
Cotangent_bundle
Category whose objects are rings and whose morphisms are ring homomorphisms
morphism is a monomorphism. This follows from the fact that the only ideals in a field F are the zero ideal and F itself. One can then view morphisms
Category_of_rings
Property of functions which is weaker than continuity
morphism of schemes of finite presentation, then n X / Y {\displaystyle n_{X/Y}} is lower semicontinuous. If f {\displaystyle f} is a proper morphism
Semi-continuity
History of maths
Dieudonné; The historical development of algebraic geometry Charles Weibel; History of homological algebra Peter Johnstone; The point of pointless topology
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Function that preserves distinctness
homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and
Injective_function
Abstract homotopical model for topological spaces
an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are
∞-groupoid
Array of numbers
matrix, or a matrix of dimension 2 × 3. In linear algebra, matrices are used as linear maps. In geometry, matrices are used for geometric transformations
Matrix_(mathematics)
Mathematical concept for comparing objects
symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is numerical
Equivalence_relation
Mathematics concept
the algebraic geometry of X (the derived category of coherent sheaves on X) and another triangulated category constructed from the symplectic geometry of
Homological_mirror_symmetry
Every set is smaller than its power set
{\displaystyle {\mathcal {C}}} such that a morphism f : T × T → Y {\displaystyle f:T\times T\to Y} can parameterize all morphisms T → Y {\displaystyle T\to Y} . In
Cantor's_theorem
Asymptotically stable in the sense of geometric invariant theory
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent
Stable_curve
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
One-to-one correspondence
(16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1
Bijection
Set with associative invertible operation
more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups
Group_(mathematics)
Operation in algebra and mathematics
is relevant in different fields such as topos theory and topics in algebraic geometry related to descent. A first example of a comonadic adjunction is the
Monad_(category_theory)
Concept in mathematical category theory
e., it also sends morphisms. If α : y ¯ 1 → y ¯ 2 {\displaystyle \alpha :{\overline {y}}_{1}\to {\overline {y}}_{2}} is a morphism in π − 1 ( y ) {\displaystyle
Category_of_elements
Tool in homological algebra
computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Motivated by problems in algebraic topology, Jean Leray introduced
Spectral_sequence
Type of group in group theory
follows: for any algebraic group G {\displaystyle \mathrm {G} } defined over Q {\displaystyle \mathbb {Q} } such that there is a morphism G ( R ) → G {\displaystyle
Arithmetic_group
Graphical representation of a morphism
a domain and codomain to each box, i.e. the input and output types. A morphism of monoidal signature F : Σ → Σ ′ {\displaystyle F:\Sigma \to \Sigma '}
String_diagram
Algebraic structure used in topology
mathematics, specifically in homology theory and algebraic topology, cohomology is a way of attaching algebraic invariants to a topological space or other mathematical
Cohomology
In algebraic geometry, a point with rational coordinates
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is
Rational_point
Continuous surjection satisfying a local triviality condition
triviality condition; cf. principal homogeneous space and torsor (algebraic geometry). In topology, a fibration is a mapping π : E → B {\displaystyle \pi
Fiber_bundle
Theory for associative algebras over rings
n_{+}\mapsto M\otimes A^{\otimes n}.} A morphism f : m + → n + {\displaystyle f:m_{+}\to n_{+}} is sent to the morphism f ∗ {\displaystyle f_{*}} given by
Hochschild_homology
Higher categorical generalization of a topos
Lurie 2009, Definition 6.1.0.4. Lurie 2009, Theorem 6.1.0.6. Spectral Algebraic Geometry - Charles Rezk (gives a down-enough-to-earth introduction) Lurie,
∞-topos
Mathematical term; concerning axioms used to derive theorems
projective algebraic variety found in a complex projective space. A generation later, with the publication of the textbook Algebraic Geometry by Robin Hartshorne
Axiomatic_system
Mapping theorem in topology
fixed-point index, provided that their number is finite. There is also algebraic geometry counterpart of this theorem called Lefschetz trace formula that allows
Lefschetz_fixed-point_theorem
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DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
DIAGONAL MORPHISM-ALGEBRAIC-GEOMETRY
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