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Version 6 |
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Let $R$ be a ring with multiplicative identity. Let $M$ be a (\PMlinkescapeword{right}) module over $R$. We may assume there exists an exact sequence $C_*$:
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Let $R$ be a ring with multiplicative identity. Let $M$ be a (right) module over R. We may assume there exists an exact sequence $C_*$:
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$$ |
| \dots\dots\rightarrow P_2\rightarrow P_1\rightarrow P_0 |
\dots\dots\rightarrow P_2\rightarrow P_1\rightarrow P_0 |
| $$ |
$$ |
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| with the $P_n$ projective and the cokernel of the last map $M$. Given $M$, this sequence is unique up to chain homotopy. Hence we may make the following definitions. |
with the $P_n$ projective and the cokernel of the last map $M$. Given $M$, this sequence is unique up to chain homotopy. Hence we may make the following definitions. |
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| For a (right) $R$- module $A$ we may define |
For a (right) $R$- module $A$ we may define |
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| $$ |
$$ |
| Ext_R^n(M,A)=H^n(C_*; A) |
Ext_R^n(M,A)=H^n(C_*; A) |
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$$ |
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| For a (left) $R$- module $A$ we may define |
For a (left) $R$- module $A$ we may define |
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| $$ |
$$ |
| Tor_R^n(M,A)=H_n(C_*; A) |
Tor_R^n(M,A)=H_n(C_*; A) |
| $$ |
$$ |
