Editing Tor functor (section)

==Definition==
Let ''R'' be a [[ring (mathematics)|ring]]. Write ''R''-Mod for the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and Mod-''R'' for the category of right ''R''-modules. (If ''R'' is [[commutative ring|commutative]], the two categories can be identified.) For a fixed left ''R''-module ''B'', let <math>T(A) = A\otimes_R B</math> for ''A'' in Mod-''R''. This is a [[right exact functor]] from Mod-''R'' to the [[category of abelian groups]] Ab, and so it has left [[derived functor]]s <math>L_i T</math>. The Tor groups are the abelian groups defined by
<math display="block">\operatorname{Tor}_i^R(A,B) = (L_iT)(A),</math>
for an [[integer]] ''i''. By definition, this means: take any [[projective module#Projective resolutions|projective resolution]]
<math display="block">\cdots\to P_2 \to P_1 \to P_0 \to A\to 0,</math>
and remove ''A'', and form the [[chain complex]]:
<math display="block">\cdots \to P_2\otimes_R B \to P_1\otimes_R B \to P_0\otimes_R B \to 0</math>

For each integer ''i'', the group <math>\operatorname{Tor}_i^R(A,B)</math> is the [[chain complex|homology]] of this complex at position ''i''. It is zero for ''i'' negative. Moreover, <math>\operatorname{Tor}_0^R(A,B)</math> is the [[cokernel]] of the map <math>P_1\otimes_R B \to P_0\otimes_R B</math>, which is [[isomorphic]] to <math>A \otimes_R B</math>.

Alternatively, one can define Tor by fixing ''A'' and taking the left derived functors of the right exact functor <math>G(B)=A\otimes_RB</math>. That is, tensor ''A'' with a projective resolution of ''B'' and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups.<ref>Weibel (1994), section 2.4 and Theorem 2.7.2.</ref> Moreover, for a fixed ring ''R'', Tor is a functor in each variable (from ''R''-modules to abelian groups).

For a commutative ring ''R'' and ''R''-modules ''A'' and ''B'', <math>\operatorname{Tor}^R_i(A,B)</math> is an ''R''-module (using that <math>A\otimes_RB</math> is an ''R''-module in this case). For a non-commutative ring ''R'', <math>\operatorname{Tor}^R_i(A,B)</math> is only an abelian group, in general. If ''R'' is an [[algebra over a ring]] ''S'' (which means in particular that ''S'' is commutative), then <math>\operatorname{Tor}^R_i(A,B)</math> is at least an ''S''-module.