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...
13 KB (2,030 words) - 16:56, 21 August 2024
Homological algebra (section Tor functor)
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...
27 KB (3,857 words) - 20:15, 11 February 2024
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...
17 KB (3,050 words) - 12:27, 14 April 2024
Tensor-hom adjunction (category Adjoint functors)
motivates the definition of the Ext functor and the Tor functor. Currying Eckmann–Hilton_duality Ext functor Tor functor Change of rings May, J.P.; Sigurdsson...
5 KB (912 words) - 14:09, 13 June 2024
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...
19 KB (3,248 words) - 17:48, 21 August 2024
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...
30 KB (4,590 words) - 03:05, 9 August 2024
Indonesia Tor (fish), a genus of fish commonly known as mahseers Target of rapamycin, a regulatory enzyme Tor functor, in mathematics Tor (network),...
4 KB (531 words) - 03:05, 31 July 2024
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...
5 KB (718 words) - 03:34, 19 August 2024
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 to be Z/2Z, so that coefficients...
8 KB (1,181 words) - 13:40, 29 June 2023
Singular homology (redirect from Homology functor)
{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. Of note, if R is torsion-free, then T...
19 KB (3,239 words) - 08:00, 2 July 2024
resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract algebra) Spectral sequence Abelian...
987 bytes (78 words) - 14:16, 5 April 2022
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...
2 KB (226 words) - 14:35, 19 January 2024
Hochschild homology (section Loday functor)
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)}...
20 KB (3,394 words) - 10:15, 8 November 2023
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...
36 KB (5,016 words) - 01:30, 27 August 2024
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...
10 KB (1,708 words) - 22:11, 8 April 2024
tgn – tangent function. (Also written as tan, tg.) Thm – theorem. Tor – Tor functor. Tr – field trace. tr – trace of a matrix or linear transformation...
20 KB (1,989 words) - 11:23, 31 July 2024
Intersection number (section Serre's Tor formula)
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...
18 KB (3,298 words) - 13:08, 13 June 2024
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...
4 KB (641 words) - 18:00, 16 June 2024
formula. In the usual formulation, the formula involves the Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber...
14 KB (1,810 words) - 16:44, 31 July 2024
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...
19 KB (2,837 words) - 22:51, 25 November 2023
Mathematical Society in its inaugural class. Coherent duality Ext functor Tor functor "Luchezar Avramov". directory.unl.edu. University of Nebraska. Retrieved...
3 KB (186 words) - 19:40, 7 June 2024
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...
50 KB (8,651 words) - 19:03, 15 August 2024
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}}...
37 KB (6,320 words) - 09:24, 30 August 2024
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...
22 KB (3,600 words) - 20:26, 3 March 2024
{\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...
12 KB (1,657 words) - 16:54, 17 May 2024
Homology (mathematics) (section Homology functors)
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...
54 KB (8,218 words) - 14:50, 22 August 2024
{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...
14 KB (2,259 words) - 18:04, 23 July 2024
necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology...
8 KB (1,031 words) - 17:57, 2 September 2022
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...
8 KB (1,028 words) - 17:44, 11 January 2023
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...
48 KB (8,467 words) - 22:58, 6 April 2024