physics - Tor functor

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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). 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 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

$\mathrm{Tor}_n^R(A,B)=(L_nT)(A),$

i.e., we take a projective resolution

$\cdots\rightarrow P_3 \rightarrow P_2 \rightarrow P_1 \rightarrow A\rightarrow 0,$

then chop off the last term A and tensor it with B to get the complex

$\cdots \rightarrow P_3\otimes B \rightarrow P_2\otimes B \rightarrow P_1\otimes B \rightarrow 0,$

and take the homology of this complex.

Facts

For every n ≥ 1, Tor_n^R is an additive functor from Mod-R × R-Mod to Ab. In case 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

$0\rightarrow K\rightarrow L\rightarrow M\rightarrow 0$

induces a long exact sequence of the form

$\cdots\rightarrow\mathrm{Tor}_2^R(M,B)\rightarrow\mathrm{Tor}_1^R(K,B)\rightarrow\mathrm{Tor}_1^R(L,B)\rightarrow\mathrm{Tor}_1^R(M,B)\rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0$

is induced.

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: see torsion subgroup.

In the case of abelian groups (i.e. if R is the ring of integers Z), then Tor_n^Z(A,B) = 0 for all n ≥ 2. The reason: every abelian group A has a free resolution of length 2, since subgroups of free abelian groups are free abelian. So in this important special case, the higher Tor functors are invisible.

The Tor functors commute with arbitrary direct sums: there is a natural isomorphism

$\mathrm{Tor}_n^R(\oplus_i A_i, \oplus_j B_j) \simeq \oplus_i \oplus_j \mathrm{Tor}_n^R(A_i,B_j)$

A module M in Mod-R is flat if and only if Tor₁^R(M, -) = 0. In this case, we even have Tor_n^R(M, -) = 0 for all n. In fact, to compute Tor_n^R(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).

Categories: Homological algebra