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.
Specifically, suppose R is a ring, and denoted 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) = A⊗RB. 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
i.e., we take a projective resolution
then remove the A term and tensor the projective resolution with B to get the complex
(note that A⊗RB does not appear and the last arrow is just the zero map) and take the homology of this complex.
- For every n ≥ 1, TorR
n 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.
- As is true for every family of derived functors, every short exact sequence 0 → K → L → M → 0 induces a long exact sequence of the form
- If R is commutative and r in R is not a zero divisor then
from which the terminology Tor (that is, Torsion) comes: see torsion subgroup.
n(A,B) = 0 for all n ≥ 2. The reason: every abelian group A has a free resolution of length 1, since subgroups of free abelian groups are free abelian. So in this important special case, the higher Tor functors are invisible. In addition, TorZ
1(Z/kZ,A) = Ker(f) where f represents "multiplication by k".
- Furthermore, every free module has a free resolution of length zero, so by the argument above, if F is a free R-module, then TorR
n(F,B) = 0 for all n ≥ 1.
- The Tor functors preserve filtered colimits and arbitrary direct sums: there is a natural isomorphism
- From the classification of finitely generated abelian groups, we know that every finitely generated abelian group is the direct sum of copies of Z and Zk. This together with the previous three points allows us to compute TorZ
1(A, B) whenever A is finitely generated.
- A module M in Mod-R is flat if and only if TorR
1(M, -) = 0. In this case, we even have TorR
n(M, -) = 0 for all n ≥ 1 . In fact, to compute TorR
n(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).