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# Tor

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

$Tor_{R}^{n}(M,A)=H_{n}(P_{*};A)$ |

Defines:

Tor, Ext

Keywords:

homology, homological algebra

Related:

HomologyChainComplex, CohomologyOfACochainComplex

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

18G15*no label found*16E30

*no label found*

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## Corrections

Pointer to definitions by AxelBoldt ✓

typo by pahio ✓

H_n(C*,A) by AxelBoldt ✘

misclassification by rspuzio ✘

formatting by yark ✓

typo by pahio ✓

H_n(C*,A) by AxelBoldt ✘

misclassification by rspuzio ✘

formatting by yark ✓