# Tor functor

In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

Specifically, suppose R is a ring, and denote by R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Pick a fixed module B in R-Mod. For A in Mod-R, set T(A) = ARB. Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors LnT are defined. We set

${\displaystyle \mathrm {Tor} _{n}^{R}(A,B)=(L_{n}T)(A)}$

i.e., we take a projective resolution

${\displaystyle \cdots \rightarrow P_{2}\rightarrow P_{1}\rightarrow P_{0}\rightarrow A\rightarrow 0}$

then remove the A term and tensor the projective resolution with B to get the complex

${\displaystyle \cdots \rightarrow P_{2}\otimes _{R}B\rightarrow P_{1}\otimes _{R}B\rightarrow P_{0}\otimes _{R}B\rightarrow 0}$

(note that ARB does not appear and the last arrow is just the zero map) and take the homology of this complex.

## Properties

• For every n ≥ 1, TorTemplate:Su is an additive functor from Mod-R × R-Mod to Ab. In the case when R is commutative, we have additive functors from Mod-R × Mod-R to Mod-R.
${\displaystyle \cdots \rightarrow \mathrm {Tor} _{2}^{R}(M,B)\rightarrow \mathrm {Tor} _{1}^{R}(K,B)\rightarrow \mathrm {Tor} _{1}^{R}(L,B)\rightarrow \mathrm {Tor} _{1}^{R}(M,B)\rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0}$.
${\displaystyle \mathrm {Tor} _{1}^{R}(R/(r),B)=\{b\in B:rb=0\},}$

from which the terminology Tor (that is, Torsion) comes: see torsion subgroup.

• Furthermore, every free module has a free resolution of length zero, so by the argument above, if F is a free R-module, then TorTemplate:Su(F,B) = 0 for all n ≥ 1.
${\displaystyle \mathrm {Tor} _{n}^{R}\left(\bigoplus _{i}A_{i},\bigoplus _{j}B_{j}\right)\simeq \bigoplus _{i}\bigoplus _{j}\mathrm {Tor} _{n}^{R}(A_{i},B_{j})}$
• A module M in Mod-R is flat if and only if TorTemplate:Su(M, -) = 0. In this case, we even have TorTemplate:Su(M, -) = 0 for all n ≥ 1 . In fact, to compute TorTemplate:Su(A,B), one may use a flat resolution of A or B, instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible).