# 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=
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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*) = *A*⊗_{R}*B*. 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 *L _{n}T* 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*⊗_{R}*B* 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*.

- 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.

- TorTemplate:Su(
*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, TorTemplate:Su(**Z**/*k***Z**,*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 TorTemplate:Su(*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**Z**_{k}. This together with the previous three points allows us to compute TorTemplate:Su(*A*,*B*) whenever*A*is finitely generated.

- 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).