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Construction in homological algebra
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological
Ext_functor
Homological construction in category theory
right derived functors are the Ext functors Ext R i ( A , − ) {\displaystyle \operatorname {Ext} _{R}^{i}(A,-)} . Alternatively Ext R i ( − , B )
Derived_functor
Topics referred to by the same term
in telephony, commonly written as "ext." Ext functor, used in the mathematical field of homological algebra Ext (JavaScript library), a programming library
Ext
Branch of mathematics
independent subject with the study of objects such as the ext functor and the tor functor, among others. The notion of chain complex is central in homological
Homological_algebra
Functor mapping hom objects to an underlying category
to the tensor product functor – ⊗ {\displaystyle \otimes } R M: Ab → Mod-R. Ext functor Functor category Representable functor Also commonly denoted Cop
Hom_functor
Construction in homological algebra
mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central
Tor_functor
Category in mathematics
generalizing the Ext functor in an abelian category. In this notation, the first exact sequence above would be written: ⋯ → Ext i ( B , X ) → Ext i ( B ,
Triangulated_category
pairing between Ext groups of modules: Ext n ( M , N ) ⊗ Ext m ( L , M ) → Ext n + m ( L , N ) {\displaystyle \operatorname {Ext} ^{n}(M,N)\otimes
Yoneda_product
Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Tor functor and the Ext functor. A
Lift_(mathematics)
Algebraic structure
homological methods, such as the Ext functor. This functor is the derived functor of the functor HomR(M, −). The latter functor is exact if M is projective
Commutative_ring
Theory for associative algebras over rings
cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by H H n ( A , M ) = Tor n A e ( A , M ) {\displaystyle HH_{n}(A
Hochschild_homology
Transforming a function in such a way that it only takes a single argument
Hom functor and the tensor product functor might not lift to an exact sequence; this leads to the definition of the Ext functor and the Tor functor. In
Currying
Establish relationships between homology and cohomology theories
theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence 0 → Ext R 1 ( H i − 1 ( X ; R ) , G ) → H
Universal_coefficient_theorem
Tools for studying groups based on techniques from algebraic topology
identity) to M. Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism H n ( G , M ) = Ext Z [ G ] n ( Z , M ) . {\displaystyle
Group_cohomology
exp1m – exponential minus 1 function. (Also written as expm1.) Ext – Ext functor. ext – exterior. extr – a set of extreme points of a set. FFT – fast
List of mathematical abbreviations
List_of_mathematical_abbreviations
Spectral sequence
computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\displaystyle
Grothendieck spectral sequence
Grothendieck_spectral_sequence
Cohomology theory for Lie algebras
{g}};M):=\mathrm {Ext} _{U{\mathfrak {g}}}^{n}(R,M)} (see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left
Lie_algebra_cohomology
Generalisations of Serre duality in mathematics
still wider than Serre duality, Hartshorne (Algebraic Geometry) uses the Ext functor of sheaves; this is a kind of stepping stone to the derived category
Coherent_duality
Category with direct sums and certain types of kernels and cokernels
category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These
Abelian_category
Possible axiom for set theory in mathematics
{\displaystyle \mathrm {Ext} ^{1}(A,\mathbb {Z} )=0} is a free abelian group, where E x t {\displaystyle \mathrm {Ext} } is the Ext functor. The existence of
Axiom_of_constructibility
Topic in abstract algebra
It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis
Tilting_theory
Mathematical object in abstract algebra
modules. Injective resolutions can be used to define derived functors such as the Ext functor. The length of a finite injective resolution is the first index
Injective_module
submodule of M intersects non-trivially. exact exact sequence Ext functor Ext functor extension Extension of scalars uses a ring homomorphism from R
Glossary_of_module_theory
Bulgarian-American mathematician
Mathematical Society in its inaugural class. Coherent duality Ext functor Tor functor "Luchezar Avramov". University of Nebraska–Lincoln. Retrieved May
Luchezar_Avramov
Sheaf consisting of modules on a ringed space; generalizing vector bundles
{H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),} since both the sides are the right derived functors of the same functor Γ ( X , − ) = Hom O ( O , −
Sheaf_of_modules
Type of module over a ring
together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers
Simple_module
Group for which a given group is a normal subgroup
which is isomorphic to Ext Z 1 ( Q , N ) ; {\displaystyle \operatorname {Ext} _{\mathbb {Z} }^{1}(Q,N);} cf. the Ext functor. Several other general
Group_extension
Study of dimension in algebraic geometry
Ext functor satisfies depth M = sup { n ∣ Ext R i ( k , M ) = 0 , i < n } {\displaystyle \operatorname {depth} M=\sup\{n\mid \operatorname {Ext}
Dimension_theory_(algebra)
German mathematician (1902–1979)
group Baer ring Baer–Suzuki theorem Baer–Specker group Injective module Ext functor Scientific career Fields Mathematics Institutions University of Illinois
Reinhold_Baer
Concept in mathematics
preserve short exact sequences motivates the definition of the Ext functor and the Tor functor. We can illustrate the tensor-hom adjunction in the category
Tensor–hom_adjunction
Directed graph which is also a multigraph
forgetful functor from Cat (the category of small categories) to Quiv (the category of multidigraphs). Its left adjoint is a free functor which, from
Quiver_(mathematics)
Operation in algebra
f_{*}N=N_{R}} , formed by restriction of scalars. They are related as adjoint functors: f ∗ : Mod R ⇆ Mod S : f ∗ {\displaystyle f^{*}:{\text{Mod}}_{R}\leftrightarrows
Change_of_rings
Homological construction
'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational
Derived_category
Mathematical object
an algebraic family of curves Gauss–Manin connection Derived category Ext functor Kodaira (2005). Complex Manifolds and Deformation of Complex Structures
Kodaira–Spencer_map
In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are
Exalcomm
Spectral sequence
spaces over Fp. But we can now consider the derived functors of Hom in the category of A-modules, ExtAr(H*(Y), H*(X)). These acquire a second grading from
Adams_spectral_sequence
Construct in algebraic geometry
the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this definition to
Cotangent_complex
Homological algebra is the study of homological functors
resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract algebra) Spectral sequence Abelian
List of homological algebra topics
List_of_homological_algebra_topics
_{\mathbb {Z} }^{1}(H_{n-1}(Y,\mathbb {Z} ),\pi _{n}(Y))\cong 1} for the Ext functor. The Universal coefficient theorem then simplifies and claims: H n (
Hopf–Whitney_theorem
Mathematical object in sheaf cohomology
resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves:
Injective_sheaf
Algebraic structure used in topology
variable, the right derived functors of the Hom functor HomR(M,N). Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheaf E
Cohomology
History of maths
Lefschetz Singular homology of topological spaces. 1934 Reinhold Baer Ext groups, Ext functor (for abelian groups and with different notation). 1935 Witold Hurewicz
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
points of the spectrum form a closed subset. Ext The Ext functors, the derived functors of the Hom functor. extension 1. An extension of an ideal is the
Glossary of commutative algebra
Glossary_of_commutative_algebra
Soviet and Russian mathematician (1938–2024)
setting, since the basis of the spectral sequence is the initial data of Ext functors taken with respect to a ring of such operations, generalising the Steenrod
Sergei Novikov (mathematician)
Sergei_Novikov_(mathematician)
Topics referred to by the same term
of a module over a supplemented associative algebra Cohomology Ext functor Tor functor This disambiguation page lists mathematics articles associated
Cohomology_of_algebras
Mathematical object in category theory
(not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic
Injective_object
Mathematical relation in abstract algrebra
R-module, then: Ext R n ( N , R M ) ≅ Ext S n ( S ⊗ R N , M ) {\displaystyle \operatorname {Ext} _{R}^{n}(N,{}_{R}M)\cong \operatorname {Ext} _{S}^{n}(S\otimes
Shapiro's_lemma
admissible if the inclusion functor i : A → T {\displaystyle i\colon {\mathcal {A}}\to {\mathcal {T}}} has a left adjoint functor, written i ∗ {\displaystyle
Semiorthogonal_decomposition
Structure in algebraic geometry
equivalence, i.e. the above standard conjecture D, the functor H is an exact faithful tensor-functor. Applying the Tannakian formalism, one concludes that
Motive_(algebraic_geometry)
that the functor Ω−1 is actually an inverse to Ω. One important property of the stable module category is it allows defining the Ω functor for general
Stable_module_category
Theorem in algebraic geometry
that there is a natural isomorphism: Ext X i ( E , ω X ) ≅ H n − i ( X , E ) ∗ {\displaystyle \operatorname {Ext} _{X}^{i}(E,\omega _{X})\cong H^{n-i}(X
Serre_duality
Mathematical category with weak equivalences, fibrations and cofibrations
are maps that are monomorphisms in each nonzero degree. This explains why Ext-groups of R-modules can be computed by either resolving the source projectively
Model_category
Concept in algebraic geometry
duality functor D ( − ) {\displaystyle D(-)} , the local duality theorem may be expressed as the following isomorphism. H m n ( M ) ≅ D ( Ext R d − n
Local_cohomology
Type of vector space
Hecke algebra is a further generalization of Hecke algebras to derived functors. It was introduced by Peter Schneider in 2015 who, together with Rachel
Hecke_algebra
the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the
Cohomology_operation
Existence and uniqueness theorem for certain partial differential equations
homogeneous parts of each equation and the vanishing of a derived functor E x t 1 {\displaystyle Ext^{1}} . Let n ≤ m {\displaystyle n\leq m} . Set Y = { x 1 =
Cauchy–Kovalevskaya_theorem
Study of categorified structures
consider quantum double groupoids to be fundamental groupoids defined via a 2-functor, which allows one to think about the physically interesting case of quantum
Higher-dimensional_algebra
Various mathematical dualites
thought of as an A-module: ( A ! ) opp = Ext A ∗ ( k , k ) . {\displaystyle (A^{!})^{\text{opp}}=\operatorname {Ext} _{A}^{*}(k,k).} If an algebra A {\displaystyle
Koszul_duality
Invariant of algebraic varieties and of more general schemes
_{j=0}^{n}R(j)[2j],} where M ↦ M[1] denotes the shift or "translation functor" in the triangulated category DM(k; R). In these terms, motivic cohomology
Motivic_cohomology
Branch of mathematics
of large matrices. K-theory involves the construction of families of K-functors that map from topological spaces or schemes, or to be even more general:
K-theory
Fiber bundle
. In detail, define Ext ( C ) := C × H G = ( C × G ) / ∼ , ( p , g ) ∼ ( p ⋅ h , h − 1 g ) . {\displaystyle \operatorname {Ext} (C):=C\times _{H}G=(C\times
Associated_bundle
Concept from algebraic geometry
{\displaystyle \omega _{X}} is an object representing the contravariant functor F ↦ H n ( X , F ) ∗ {\displaystyle F\mapsto \operatorname {H} ^{n}(X
Dualizing_sheaf
Connects homology and cohomology groups for oriented closed manifolds
S\longmapsto DS} . Note that H k {\displaystyle H^{k}} is a contravariant functor while H n − k {\displaystyle H_{n-k}} is covariant. The family of isomorphisms
Poincaré_duality
Tool in homological algebra
Čech-to-derived functor spectral sequence from Čech cohomology to sheaf cohomology. Change of rings spectral sequences for calculating Tor and Ext groups of
Spectral_sequence
Specific algebraic group
quasi-inverse of the weights functor is given by a dualization functor from free abelian groups to tori, defined by its functor of points as: D ( M ) S (
Algebraic_torus
Ring whose ideals are projective
dimension is at most 1. Hence the usual derived functors such as E x t R i {\displaystyle \mathrm {Ext} _{R}^{i}} and T o r i R {\displaystyle \mathrm
Hereditary_ring
Homology theory for locally compact spaces
theorem: 0 → Ext Z 1 ( H c i + 1 ( X , Z ) , Z ) → H i B M ( X , Z ) → Hom ( H c i ( X , Z ) , Z ) → 0. {\displaystyle 0\to {\text{Ext}}_{\mathbb {Z}
Borel–Moore_homology
Abelian group equipped with compatible ring action on both sides
T-R-bimodule in a natural fashion. These statements extend to the derived functors Ext and Tor. Profunctors can be seen as a categorical generalization of bimodules
Bimodule
How spheres of various dimensions can wrap around each other
groups. The classical Adams spectral sequence has E2 term given by the Ext groups Ext∗,∗ A(p)(Zp, Zp) over the mod p Steenrod algebra A(p), and converges
Homotopy_groups_of_spheres
Generalization of vector bundles
codimension 2 subvarieties Y {\displaystyle Y} using a certain Ext 1 {\displaystyle {\text{Ext}}^{1}} -group calculated on X {\displaystyle X} . This is given
Coherent_sheaf
Algebra in algebraic topology
A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R {\displaystyle
Steenrod_algebra
Type of cellular automaton
org/3697/ Quantum Nano-Automata.: http://doc.cern.ch/archive/electronic/other/ext/ext-2004-125/Quantumnanoautomata.doc Categories of Quantum Automata.: [1] [2]
Quantum dot cellular automaton
Quantum_dot_cellular_automaton
have to provide the functor Map.Make with a module which defines the key type and the comparison function. The third-party library ExtLib provides a polymorphic
Comparison of programming languages (associative array)
Comparison_of_programming_languages_(associative_array)
EXT FUNCTOR
EXT FUNCTOR
Boy/Male
Indian, Telugu
Lord Muruga; Text
Boy/Male
Muslim
Acceptor. Next. Able.
Boy/Male
Arabic
Acceptor; Next; Able
Girl/Female
Hindu, Indian
To Eat Something
Boy/Male
Hindu, Indian, Kannada, Telugu
Sacret Text
Boy/Male
Arabic
Something to Eat
Boy/Male
Tamil
A vedic text
Boy/Male
Muslim
Following, Next
Boy/Male
Hindu, Indian
Invitng to Eat
Boy/Male
Tamil
Vedic text
Boy/Male
Hindu
Vedic text
Girl/Female
Egyptian
Eat.
Boy/Male
Arabic
Coming; Next
Boy/Male
Hindu
A vedic text
Boy/Male
Muslim
Coming. Next.
Boy/Male
Sikh
Next to God
Surname or Lastname
English (Devon)
English (Devon) : nickname from Middle English hext ‘tallest’, ‘highest’ (Old English hēhst, superlative of hēah ‘high’).
Boy/Male
Indian
Following, Next
Boy/Male
Arabic, Muslim
Coming; Next; Following
Boy/Male
Muslim/Islamic
Following next
EXT FUNCTOR
EXT FUNCTOR
Boy/Male
Tamil
Rising Sun, Born of the Sun
Female
Ukrainian
, defender of man.
Girl/Female
British, English
Noble Friend
Girl/Female
Muslim
A bond
Girl/Female
Arabic, Hindu, Indian
Sweet
Boy/Male
Indian
God's Chosen
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Fast; Progressive; Lord Shiva
Girl/Female
Hindu
Goddess Parvati, The first sound of universe aum called as Pranavi
Boy/Male
Hindu, Indian
One of the Childhood Name of Lord Krishna
Boy/Male
Biblical
Hearing, obedient.
EXT FUNCTOR
EXT FUNCTOR
EXT FUNCTOR
EXT FUNCTOR
EXT FUNCTOR
n.
A style of writing in large characters; text-hand also, a kind of type used in printing; as, German text.
superl.
Nearest in degree, quality, rank, right, or relation; as, the next heir was an infant.
v. t.
To eat or prey upon, as a moth eats a garment.
v. t.
To write in large characters, as in text hand.
pl.
of Ex-voto
n.
Any departure; the act of quitting the stage of action or of life; death; as, to make one's exit.
n.
A large hand in writing; -- so called because it was the practice to write the text of a book in a large hand and the notes in a smaller hand.
superl.
Nearest in time; as, the next day or hour.
pl.
of Ex officio
a.
Next; nearest.
n.
The common newt; -- called also asker, eft, evat, and ewt.
obs. imp.
of Eat.
v. t.
To chew and swallow as food; to devour; -- said especially of food not liquid; as, to eat bread.
adv.
In the time, place, or order nearest or immediately suceeding; as, this man follows next.