Editing Tor functor (section)

==Important special cases==

*[[Group homology]] is defined by <math>H_*(G,M)=\operatorname{Tor}^{\Z[G]}_*(\Z, M),</math> where ''G'' is a group, ''M'' is a [[group representation|representation]] of ''G'' over the integers, and <math>\Z[G]</math> is the [[group ring]] of ''G''.
*For an [[algebra over a field|algebra]] ''A'' over a field ''k'' and an ''A''-[[bimodule]] ''M'', [[Hochschild homology]] is defined by <math display="block">HH_*(A,M)=\operatorname{Tor}_*^{A\otimes_k A^{\text{op}}}(A, M).</math>
*[[Lie algebra homology]] is defined by <math>H_*(\mathfrak g,M)=\operatorname{Tor}_*^{U\mathfrak g}(R,M)</math>, where <math>\mathfrak g</math> is a [[Lie algebra]] over a commutative ring ''R'', ''M'' is a <math>\mathfrak g</math>-module, and <math>U\mathfrak g</math> is the [[universal enveloping algebra]].
*For a commutative ring ''R'' with a homomorphism onto a field ''k'', <math>\operatorname{Tor}_*^R(k,k)</math> is a graded-commutative [[Hopf algebra]] over ''k''.<ref>Avramov & Halperin (1986), section 4.7.</ref> (If ''R'' is a [[Noetherian local ring]] with residue field ''k'', then the dual Hopf algebra to <math>\operatorname{Tor}_*^R(k,k)</math> is [[Ext functor#Important special cases|Ext]]{{supsub|*|''R''}}(''k'',''k'').) As an algebra, <math>\operatorname{Tor}_*^R(k,k)</math> is the free graded-commutative divided power algebra on a graded vector space π<sub>*</sub>(''R'').<ref>Gulliksen & Levin (1969), Theorem 2.3.5; Sjödin (1980), Theorem 1.</ref> When ''k'' has [[characteristic of a field|characteristic]] zero, π<sub>*</sub>(''R'') can be identified with the [[André-Quillen homology]] ''D''<sub>*</sub>(''k''/''R'',''k'').<ref>Quillen (1970), section 7.</ref>