Editing Derived functor (section)

====[[Tor functor]]s====
The category of left <math>R</math>-modules also has enough projectives. If <math>A</math> is a fixed right <math>R</math>-module, then the [[tensor product]] with <math>A</math> gives a right exact covariant functor <math>A\otimes_R - : R\text{-Mod} \to Ab</math>; The category of modules has enough projectives so that left derived functors always exists. The left derived functors of the tensor functor are the [[Tor functor]]s <math>\operatorname{Tor}_i^R(A,-)</math>. Equivalently <math>\operatorname{Tor}_i^R(-,B)</math> can be defined symmetrically as the left derived functors of <math>-\otimes B</math>. In fact one can combine both definitions and define <math>\operatorname{Tor}_i^R(-,-)</math> as the left derived of <math>-\otimes-: \text{Mod-}R \times R\text{-Mod} \to Ab</math>.

This includes several notions of homology as special cases. This often mirrors the situation with Ext functors and cohomology.

* '''[[Group homology]]''' is the left derived functor of taking coinvariants <math>(-)_G: k[G]\text{-Mod}\to k\text{-Mod}</math> which is the same as <math>k\otimes_{k[G]}-</math>.
* '''[[Lie algebra homology]]''' is the left derived functor of taking coinvariants <math>\mathfrak{g}\text{-Mod}\to k\text{-Mod}, M\mapsto M/[\mathfrak{g},M]</math> which is the same as <math>k\otimes_{U(\mathfrak{g})}-</math>.
* '''[[Hochschild homology]]''' is the left derived functor of taking coinvariants <math>(A,A)\text{-Bimod}\to k\text{-Mod}, M\mapsto M/[A,M]</math> which is the same as <math>A \otimes_{A^e} -</math>.

Instead of taking individual left derived functors one can also take the total derived functor of the tensor functor. This gives rise to the [[derived tensor product]] <math>-\otimes^L-: D(\text{Mod-}R) \times D(R\text{-Mod}) \to D(Ab)</math> where <math>D</math> is the [[derived category]].