follows that Tor commutes with localization. That is, for a multiplicatively closed set S in R, For a commutative ring R and commutative R-algebras A and B, Tor(A,B) has the structure of a graded-commutative algebra over R. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree. Important special cases Group homology is defined by where G is a group, M is a representation of G over the integers, and is the group ring of G. For an algebra A over a field k and an A-bimodule M, Hochschild homology is defined by Lie algebra homology is defined by , where is a Lie algebra over a commutative ring R, M is a -module, and is the universal enveloping algebra. For a commutative ring R with a homomorphism onto a field k, is a graded-commutative Hopf algebra over k. (If R is a Noetherian local ring with residue field k, then the dual Hopf algebra to is Ext(k,k).) As an algebra, is the free graded-commutative divided power algebra on a graded vector space π*(R). When k has characteristic zero, π*(R) can be identified with the André-Quillen homology D*(k/R,k). See also Flat morphism Serre's intersection formula Derived tensor product Eilenberg–Moore spectral sequence Notes References External links Homological algebra Binary operations