Editing Homological algebra (section)

===Tor functor===
{{Main|Tor functor}}
Suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Fix a module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''&otimes;<sub>''R''</sub>''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''L<sub>n</sub>T'' are defined. We set

: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math>

i.e., we take a [[Projective module#Projective resolutions|projective resolution]]

: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math>

then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex

: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B  \rightarrow 0</math>

(note that ''A''&otimes;<sub>''R''</sub>''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex.