Tor functor

In higher mathematics, the Tor functors of homological algebra 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 case "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

: $mathrm\{Tor\}\_n^R(A,B)=(L\_nT)(A)$ i.e., we take a projective resolution

: $cdots\; ightarrow\; P\_3\; ightarrow\; P\_2\; ightarrow\; P\_1\; ightarrow\; A\; ightarrow\; 0$

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

: $cdots\; ightarrow\; P\_3otimes\; B\; ightarrow\; P\_2otimes\; B\; ightarrow\; P\_1otimes\; B\; ightarrow\; 0$

and take the homology of this complex.

Properties

* 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:$0\; ightarrow\; K\; ightarrow\; L\; ightarrow\; M\; ightarrow\; 0$

induces a long exact sequence of the form:$cdots\; ightarrowmathrm\{Tor\}\_2^R(M,B)\; ightarrowmathrm\{Tor\}\_1^R(K,B)\; ightarrowmathrm\{Tor\}\_1^R(L,B)\; ightarrowmathrm\{Tor\}\_1^R(M,B)\; ightarrow\; Kotimes\; B\; ightarrow\; Lotimes\; B\; ightarrow\; Motimes\; B\; ightarrow\; 0$.

* If "R" is commutative and "r" in "R" is not a zero divisor then:$mathrm\{Tor\}\_1^R(R/(r),B)=\{bin\; B:rb=0\},$

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).