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...

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...

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...

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...

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...

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...

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...

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...

{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...

regulatory enzyme Tor functor, in mathematics Tor (network), an Internet communication method for enabling online anonymity The Tor Project, a software...

resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract algebra) Spectral sequence Abelian...

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...

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...

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...

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...

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...

tgn – tangent function. (Also written as tan, tg.) Thm – theorem. Tor – Tor functor. Tr – field trace. tr – trace of a matrix or linear transformation...

necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology...

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)}...

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...

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...

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...

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...

{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...

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...

formula. In the usual formulation, the formula involves the Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber...

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}}...

_{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:...

{\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...

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...