TorTor2004-08-09 09:12:082006-10-10 19:04:14DefinitionTorExthomologyhomological algebra% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $R$ be a ring with multiplicative identity. Let $M$ be a (\PMlinkescapetext{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 (\PMlinkescapetext{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
$$
Tor_R^n(M,A)=H_n(P_*; A)
$$