In 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 concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.
In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950.[1] It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Algebra.[2]
mathematics, the Torfunctors 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 torfunctor, among others. The notion of chain complex is central in homological...
derived functors always exists. The left derived functors of the tensor functor are the TorfunctorsTor i R ( A , − ) {\displaystyle \operatorname {Tor} _{i}^{R}(A...
the Torfunctors, 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...
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Torfunctor, Ext is one of the core concepts of homological...
result is that other coefficients A may be used, at the cost of using a Torfunctor. For example it is common to take A to be Z/2Z, so that coefficients...
R\to H_{n}(X;R)\to Tor_{1}(H_{n-1}(X;\mathbb {Z} ),R)\to 0.} where Tor is the Torfunctor. Of note, if R is torsion-free, then Tor_1(G, R) = 0 for any...
theory Torsion group, in group theory and arithmetic geometry Torfunctor, the derived functors of the tensor product of modules over a ring Torsion-free...
in terms of the Torfunctor 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)}...
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 Torfunctor. In...
tgn – tangent function. (Also written as tan, tg.) Thm – theorem. Tor – Torfunctor. Tr – field trace. tr – trace of a matrix or linear transformation...
A/J))} where length is the length of a module over a local ring, and Tor is the Torfunctor. When V and W can be moved into a transverse position, this homological...
to the finite case (e.g., the characterization of flatness with the Torfunctor). An example of a link between finite generation and integral elements...
D-modules; that is, tensor products over the sheaf of differential operators. Torfunctor Tensor product of algebras Tensor product of fields Derived tensor product...
is not injective. Higher Torfunctors measure the defect of the tensor product being not left exact. All higher Torfunctors are assembled in the derived...
formula. In the usual formulation, the formula involves the Torfunctor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber...
{g}};M):=\mathrm {Tor} _{n}^{U{\mathfrak {g}}}(R,M)} (see Torfunctor for the definition of Tor), which is equivalent to the left derived functors of the right...
necessarily exact) sequence. This approach is used to define Ext, and Torfunctors and also the various cohomology theories in group theory, algebraic topology...
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...
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