# 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.

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, Tor*R**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.

- Tor
**Z***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 vanish. In addition, Tor**Z**

1(**Z**/*k***Z**,*A*) = ker(*f*) where*f: A → A*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 Tor*R**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**Z**_{k}. This together with the previous three points allows us to compute Tor**Z**

1(*A*,*B*) whenever*A*is finitely generated. - A module
*M*in**Mod**-*R*is flat if and only if Tor*R*

1(*M*, –) = 0. In this case, we even have Tor*R**n*(*M*, –) = 0 for all*n*≥ 1. In fact, to compute Tor*R**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). - Symmetry: if
*R*is commutative, then there is a natural isomorphism Tor*R**n*(*L*

1,*L*

2) ≅ Tor*R**n*(*L*

2,*L*

1). Here is the idea for abelian groups (i.e., the case*R*=**Z**and*n*= 1). Fix a free resolution of*L**i*as follows

- so that
*M**i*and*K**i*are free abelian groups. This gives rise to a double-complex with exact rows and columns

- Start with
*x*∈ Tor**Z**

1(*L*

1,*L*

2), so*β*_{03}(*x*) ∈ ker(*β*_{13}). Let*x*

12 ∈*M*

1 ⊗*K*

2 be such that*α*_{12}(*x*

12) =*β*_{03}(*x*). Then

- i.e.,
*β*_{12}(*x*

12) ∈ ker(*α*_{22}). By exactness of the second row, this means that*β*_{12}(*x*

12) =*α*_{21}(*x*

21) for some unique*x*

21 ∈*K*

1 ⊗*M*

2. Then

- i.e.,
*β*_{21}(*x*

21) ∈ ker(*α*_{31}). By exactness of the bottom row, this means that*β*_{21}(*x*

21) =*α*_{30}(*y*) for some unique*y*∈ Tor**Z**

1(*L*

2,*L*

1).

- Upon checking that
*y*is uniquely determined by*x*(not dependent on the choice of*x*

12), this defines a function Tor**Z***n*(*L*

1,*L*

2) → Tor**Z***n*(*L*

2,*L*

1), taking*x*to*y*, which is a group homomorphism. One may check that this map has an inverse, namely the function Tor**Z***n*(*L*

2,*L*

1) → Tor**Z***n*(*L*

1,*L*

2) defined in a similarly manner. One can also check that the function does not depend on the choice of free resolutions.

## See also

## References

- Weibel, Charles A. (1994).
*An introduction to homological algebra*. Cambridge Studies in Advanced Mathematics.**38**. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259. MR 1269324.