Tor functor

The Tor functors are the derived functors of the tensor product functor in mathematics. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology.

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). For A in Mod-R, set T(A) = A⊗_RB, where B in R-Mod is fixed. Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in case R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors L_nT are defined. We set

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

i.e., we take a projective resolution

$P(A)\rightarrow A\rightarrow 0,$

compute

$P(A)\otimes _{R}B\rightarrow A\otimes _{R}B\rightarrow 0,$

and take the homology on the lefthand side.

Facts

• For every n ≥ 1, Tor_n^R is a functor from Mod-R × R-Mod to Ab. In case R is commutative, we have a functor from Mod-R × Mod-R to Mod-R.

• If R is commutative and r in R is not a zero divisor then

$\mathrm {Tor} _{1}^{R}(R/(r),B)=\{b\in B:rb=0\},$

from which the terminology Tor (that is, Torsion) comes.

• A module M in Mod-R is flat if and only if Tor₁^R(M, -) = 0. In fact, to compute Tor_n^R(A, B), one may use a flat resolution of A or B, instead of a projective resolution.