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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
Homological construction in category theory
derived functors always exists. The left derived functors of the tensor functor are the Tor functors Tor i R ( A , − ) {\displaystyle \operatorname {Tor} _{i}^{R}(A
Derived_functor
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
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
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)
Establish relationships between homology and cohomology theories
result is that other coefficients A may be used, at the cost of using a Tor functor. For example, it is common to take A {\displaystyle A} to be Z / 2 Z
Universal_coefficient_theorem
Algebraic structure in ring theory
the Tor functors, the left derived functors of the tensor product. A left R {\displaystyle R} -module M {\displaystyle M} is flat if and only if Tor n R
Flat_module
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
Topics referred to by the same term
regulatory enzyme Tor functor, in mathematics Tor (network), an Internet communication method for enabling online anonymity The Tor Project, a software
Tor
Concept in algebraic topology
{Z} )\otimes R\to H_{n}(X;R)\to \mathrm {Tor} _{1}(H_{n-1}(X;\mathbb {Z} ),R)\to 0.} where Tor is the Tor functor. This sequence splits, though not naturally
Singular_homology
Topics referred to by the same term
theory Torsion group, in group theory and arithmetic geometry Tor functor, the derived functors of the tensor product of modules over a ring Torsion-free
Torsion
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
Theory for associative algebras over rings
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,M)=\operatorname {Tor} _{n}^{A^{e}}(A,M)}
Hochschild_homology
Relates the homology of two objects to the homology of their product
phenomena. This correction factor is expressed in terms of the Tor functor, the first derived functor of the tensor product. When R is a PID, then the correct
Künneth_theorem
Theorem in algebra mathematics
is the residue field of R {\displaystyle R} and Tor {\displaystyle {\text{Tor}}} is the tor functor. Nakayama's lemma is used to prove a version of the
Nakayama's_lemma
Topic in abstract algebra
is a finitely-generated left B-module. The tilting functors HomA(T,−), Ext1 A(T,−), −⊗BT and TorB 1(−,T) relate the category mod-A of finitely-generated
Tilting_theory
Algebraic structure associated with a topological space
uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain
Homology_(mathematics)
Mathematical operation on vector spaces
is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived
Tensor_product
tgn – tangent function. (Also written as tan, tg.) Thm – theorem. Tor – Tor functor. Tr – field trace. tr – trace of a matrix or linear transformation
List of mathematical abbreviations
List_of_mathematical_abbreviations
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
Cohomology theory for Lie algebras
{g}};M):=\mathrm {Tor} _{n}^{U{\mathfrak {g}}}(R,M)} (see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right
Lie_algebra_cohomology
Branch of mathematics
formula. In the usual formulation, the formula involves the Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber
Derived_algebraic_geometry
defined the intersection multiplicity of R/P and R/Q by means of their Tor functors. Below, ℓ R ( M ) {\displaystyle \ell _{R}(M)} denotes the length of
Serre's multiplicity conjectures
Serre's_multiplicity_conjectures
Construction in homological algebra
_{R}M)=\operatorname {Tor} _{i}^{S}(S/(y_{1},\dots ,y_{n}),M).} where Tor denotes the Tor functor and M is an S-module through S → R {\displaystyle S\to R} . Proof:
Koszul_complex
Generalized notion of counting curve intersections
A/J))} where length is the length of a module over a local ring, and Tor is the Tor functor. When V and W can be moved into a transverse position, this homological
Intersection_number
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
Submodule of a mathematical ring
of ideals is measured by the Tor functor: Tor 1 R ( R / a , R / b ) = ( a ∩ b ) / a b {\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}}
Ideal_(ring_theory)
Zero divisors in a module
{\displaystyle \operatorname {Tor} _{1}^{R}(M,R_{S}/R)} is the kernel of the localisation map of M. The symbol Tor denoting the functors reflects this relation
Torsion_(algebra)
graded algebra A over a commutative ring R, the derived tensor product functor is − ⊗ A L − : D ( M A ) × D ( A M ) → D ( R M ) {\displaystyle -\otimes
Derived_tensor_product
Tools for studying groups based on techniques from algebraic topology
homology in terms of the Tor functors, H n ( G , M ) = Tor n Z [ G ] ( Z , M ) {\displaystyle H_{n}(G,M)=\operatorname {Tor} _{n}^{\mathbb {Z} [G]}(\mathbb
Group_cohomology
Operation that pairs a left and a right R-module into an abelian group
D-modules; that is, tensor products over the sheaf of differential operators. Tor functor Tensor product of algebras Tensor product of fields Derived tensor product
Tensor_product_of_modules
Mathematical object in category theory
necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology
Injective_object
tensor Tensor product of modules topological A topological module Tor Tor functor torsion-free torsion-free module torsionless torsionless module uniform
Glossary_of_module_theory
In algebra, module with a finite generating set
to the finite case (e.g., the characterization of flatness with the Tor functor). An example of a link between finite generation and integral elements
Finitely_generated_module
Roughly, the number of k-dimensional holes on a topological surface
number, is given in detail by the universal coefficient theorem (based on Tor functors, but in a simple case). The Betti number sequence for a circle is 1,
Betti_number
Exact sequence used to describe the structure of an object
resolutions (and, more generally, flat resolutions) can be used to compute Tor functors. Projective resolution of a module M {\displaystyle M} is unique up to
Resolution_(algebra)
realisation of a Clifford algebra as a matrix algebra. Tor functors These are the derived functors of the tensor product, and feature strongly in homological
Glossary_of_tensor_theory
Topics referred to by the same term
a module over a supplemented associative algebra Cohomology Ext functor Tor functor This disambiguation page lists mathematics articles associated with
Cohomology_of_algebras
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
Relate the direct image and the pull-back of sheaves
{\mathcal {F}}} under f, i.e., the derived functor of the direct image (also known as pushforward) functor f ∗ {\displaystyle f_{*}} . This map exists
Base_change_theorems
Concept in category theory
E(FU_{*}M)} is the n-th left derived functor of E evaluated at M; i.e., Tor n R ( M , N ) {\displaystyle \operatorname {Tor} _{n}^{R}(M,N)} . Example (algebraic
Cotriple_homology
}(E))\Rightarrow H^{\ast }(E_{f}).} This is a generalization insofar as the zeroeth Tor functor is just the tensor product and in the above special case the cohomology
Eilenberg–Moore spectral sequence
Eilenberg–Moore_spectral_sequence
Homological construction
the '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
Pairing in algebra between ext groups of modules
application a la theorie des deformations" (PDF). p. 163. May, J. Peter. "Notes on Tor and Ext" (PDF). Universality of Ext functor using Yoneda extensions
Yoneda_product
Mathematical object in sheaf cohomology
derived functors of a right exact functor (such as Tor). This can sometimes be done by ad hoc means: for example, the left derived functors of Tor can be
Injective_sheaf
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
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
Algebraic structure used in topology
For example, for a ring R, the Tor groups ToriR(M,N) form a "homology theory" in each variable, the left derived functors of the tensor product M⊗RN of
Cohomology
Mathematical concept
objects from B {\displaystyle {\mathcal {B}}} . There is a canonical exact functor Q : A → A / B {\displaystyle Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal
Quotient of an abelian category
Quotient_of_an_abelian_category
Study of dimension in algebraic geometry
{\displaystyle M} be a finite R {\displaystyle R} -module. Then the Ext functor satisfies depth M = sup { n ∣ Ext R i ( k , M ) = 0 , i < n } {\displaystyle
Dimension_theory_(algebra)
Tool in homological algebra
derived functors of the tensor product are denoted Tor. Tor is defined using a projective resolution of its first argument. However, it turns out that Tor i
Spectral_sequence
Mathematical relation in abstract algrebra
Similarly, for NG the co-induced representation of N from H to G using the Hom functor, and for H ∗ {\displaystyle \ast } the group cohomology: H ∗ {\displaystyle
Shapiro's_lemma
Mathematical object
space can be considered as a cohomology theory. In fact, it defines a functor Σ ∞ : h CW → h Spectra {\displaystyle \Sigma ^{\infty }:h{\text{CW}}\to
Spectrum_(topology)
Ring whose ideals are projective
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 {Tor} _{i}^{R}} are trivial
Hereditary_ring
Scheme theory concept
to the vanishing of the Tor group Tor 1 C [ t ] ( C [ x , y , t ] x 2 + y 2 − t , C ) , {\displaystyle \operatorname {Tor} _{1}^{\mathbb {C} [t]}\left({\frac
Flat_morphism
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
Subject area in mathematics
hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K0 takes a ring A to the Grothendieck group of the set of isomorphism classes
Algebraic_K-theory
ideal generated by all qth powers of elements of I. Tor The Torsion functors, the derived functors of the tensor product. torsion 1. A torsion element
Glossary of commutative algebra
Glossary_of_commutative_algebra
defunction, defunctive, function, functional, functionality, functionary, functor, fungibility, fungible, malfunction, multifunctional, multifunctor, nonfunctional
List of Latin verbs with English derivatives
List_of_Latin_verbs_with_English_derivatives
TOR FUNCTOR
TOR FUNCTOR
Female
Scandinavian
 Feminine form of Scandinavian Tor, TORA means "Thor" or "thunder."
Surname or Lastname
Scandinavian (mainly Swedish)
Scandinavian (mainly Swedish) : from a personal name, a short form of any of the various Scandinavian personal names containing the first element Thor (Old Norse þórr), the name of the god of thunder in Scandinavian mythology.English : from the Anglo-Scandinavian name þÅr, þūr, probably short forms of Old Norse compound names in þór-, þúr- (see 1).German : habitational name for someone who lived by the gates of a town or a metonymic occupational name for someone responsible for guarding them, from Middle High German tor ‘gate’ (modern German Tor). Compare Portmann.German : nickname from Middle Low German dor, Middle High German tor ‘fool’; also ‘deaf person’.Southeast Asian : unexplained.
Male
Arthurian
, (Sir), 1st knight of the Round Table.
Boy/Male
English American
Fox. Tod is a Scottish nickname meaning a clever or wily person.
Boy/Male
African, Australian, British, Celtic, Danish, English, Irish, Norse, Scandinavian, Scottish, Swedish
King; Tower; Watchtower; God of Thunder; Victory; Castle
Surname or Lastname
English
English : topographic name for someone who lived by a tor or rocky hilltop (Old English torr, of Celtic origin), or a habitational name from any of the places named with this word, for example Torre or Torr in Devon, where the surname is frequent.English : nickname for someone thought to resemble a bull, Anglo-Norman French tor (Latin taurus).English : perhaps a habitational name from a minor place in Fife.
Male
Scandinavian
 Variant spelling of Scandinavian Tor, TORE means "Thor" or "thunder." Compare with another form of Tore.
Female
Hebrew
(תּï‹×¨Ö´×™) Hebrew name TORI means "my turtledove." Compare with another form of Tori.
Male
Hungarian
Hungarian form of Mongolian Baatar, BÃTOR means "warrior."
Female
Scandinavian
Short form of Scandinavian Tordis, TORD means "Thor's goddess" or "Thor's woman."
Male
English
Variant spelling of English Todd, TOD means "fox."
Male
Italian
 Italian short form of Latin Salvatore, TORE means "savior." Compare with another form of Tore.
Male
Japanese
(å¾¹)Â Japanese name TORU means "penetrating; wayfarer." Compare with another form of Toru.
Female
English
(Hebrew תּï‹×¨Ö´×™): English short form of Latin Victoria, TORI means "conqueror" or "victory." Compare with another form of Tori.
Surname or Lastname
English
English : nickname for a light-hearted or frivolous person, from Middle English toy ‘play’, ‘sport’ (of uncertain origin), or from an occasional medieval personal name, Toye.French : metonymic occupational name for a sheath maker, from Old French toie ‘sheath’ (Latin theca).
Boy/Male
Egyptian Norse Swedish Arthurian Legend Irish Scandinavian Scottish
King.
Male
English
Short form of English Thomas, TOM means "twin."
Female
Hebrew
(תּï‹×¨Ö¸×”) Hebrew name TORA means "bible, holy scripture." Compare with another form of Tora.
Male
Scandinavian
 Scandinavian form of Old Norse Þórr, TOR means "Thor" or "thunder." Compare with other forms of Tor.
Girl/Female
Australian, Scandinavian
Toy
TOR FUNCTOR
TOR FUNCTOR
Girl/Female
Indian, Tamil, Traditional
Daughter of Fire; Born from the Fire
Boy/Male
Arabic, Muslim
Gift of the Beneficent
Boy/Male
Tamil
Touchstone, Stone that turns iron to gold
Boy/Male
Australian, British, English
Homestead on the Boundary
Girl/Female
Indian
Famous, Eminent, Renowned
Biblical
friend
Boy/Male
Indian, Sanskrit
Surrounded by Fire
Male
English
Irish Anglicized form of Gaelic Fionnbarr, FINBAR means "fair-headed."
Girl/Female
Polish Czechoslovakian
A flower name.
Girl/Female
Latin American
Ardent. Eager. Industrious.
TOR FUNCTOR
TOR FUNCTOR
TOR FUNCTOR
TOR FUNCTOR
TOR FUNCTOR
adv.
Over; more than enough; -- noting excess; as, a thing is too long, too short, or too wide; too high; too many; too much.
v. t.
To rise to the top of; to go over the top of.
prep.
Indicating that in the character of or as being which anything is regarded or treated; to be, or as being.
v. t.
anything done successively, or by regular order; a turn; as, a tour of duty.
v. i.
To make a tourm; as, to tour throught a country.
v. t.
The act of towing, or the state of being towed; --chiefly used in the phrase, to take in tow, that is to tow.
n.
Top-boots.
v. t.
To take off the or upper part of; to crop.
conj.
A negative connective or particle, introducing the second member or clause of a negative proposition, following neither, or not, in the first member or clause (as or in affirmative propositions follows either). Nor is also used sometimes in the first member for neither, and sometimes the neither is omitted and implied by the use of nor.
n.
The highest rank; the most honorable position; the utmost attainable place; as, to be at the top of one's class, or at the top of the school.
v. i.
To hold or carry the toes (in a certain way).
n.
A tower; a turret.
n.
The prevailing fashion or mode; vogue; as, things of ton.
v. t.
To cover on the top; to tip; to cap; -- chiefly used in the past participle.
v. t.
To perform eminently, or better than before.
v. t.
To smear with tar, or as with tar; as, to tar ropes; to tar cloth.
n.
The highest part of anything; the upper end, edge, or extremity; the upper side or surface; summit; apex; vertex; cover; lid; as, the top of a spire; the top of a house; the top of a mountain; the top of the ground.
n.
Anything, or any part, corresponding to the toe of the foot; as, the toe of a boot; the toe of a skate.
v. t.
To touch or reach with the toes; to come fully up to; as, to toe the mark.
n.
High-pointed hill; a rocky pinnacle.