Let $R$ be a ring with multiplicative identity. Let $M$ be a (right) module over $R$ . We may assume there exists an exact sequence $P_*$ :

$$ \dots\dots\rightarrow P_2\rightarrow P_1\rightarrow P_0 $$

with the $P_n$ projective and the cokernel of the last map $M$ . Given $M$ , this sequence is unique up to chain homotopy. Hence we may make the following definitions.

For a (right) $R$ - module $A$ we may define

$$ Ext_R^n(M,A)=H^n(P_*; A) $$

For a (left) $R$ - module $A$ we may define

$$ Tor_R^n(M,A)=H_n(P_*; A) $$