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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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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 L_nT are defined. We set

${\mathrm{T}\mathrm{o}\mathrm{r}}_n^R(A,B)=(L_{n}T)(A)$

i.e., we take a projective resolution

$\cdots \to P_2\to P_1\to P_0\to A\to 0$

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

$\cdots \to P_2{\otimes }_{R}B\to P_1{\otimes }_{R}B\to P_0{\otimes }_{R}B\to 0$

(note that A⊗_RB 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

$\cdots \to {\mathrm{T}\mathrm{o}\mathrm{r}}_2^R(M,B)\to {\mathrm{T}\mathrm{o}\mathrm{r}}_1^R(K,B)\to {\mathrm{T}\mathrm{o}\mathrm{r}}_1^R(L,B)\to {\mathrm{T}\mathrm{o}\mathrm{r}}_1^R(M,B)\to K\otimes B\to L\otimes B\to M\otimes B\to 0$.

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

${\mathrm{T}\mathrm{o}\mathrm{r}}_1^R(R/(r),B)=\{b\in B:rb=0\},$

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/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 TorTemplate:Su(F,B) = 0 for all n ≥ 1.

• The Tor functors preserve filtered colimits and arbitrary direct sums: there is a natural isomorphism

${\mathrm{T}\mathrm{o}\mathrm{r}}_n^R(\underset{i}{⨁}{A}_i,\underset{j}{⨁}{B}_j)\simeq \underset{i}{⨁}\underset{j}{⨁}{\mathrm{T}\mathrm{o}\mathrm{r}}_n^R(A_i,B_j)$

• 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).

See also

• Ext functor

References

• Template:Weibel IHA