PlanetMath: Tor

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Tor

(Definition)

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

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

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

For a (right) - module we may define

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

For a (left) - module we may define

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

"Tor" is owned by whm22.

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Also defines:

Tor, Ext

Keywords:

homology, homological algebra

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Cross-references: definitions, chain homotopy, sequence, map, cokernel, exact sequence, module, multiplicative identity, ring
There are 2 references to this entry.

This is version 9 of Tor, born on 2004-08-09, modified 2006-10-10.
Object id is 6090, canonical name is Tor.
Accessed 4616 times total.

Classification:

AMS MSC:	16E30 (Associative rings and algebras :: Homological methods :: Homological functors on modules )
	18G15 (Category theory; homological algebra :: Homological algebra :: Ext and Tor, generalizations, Künneth formula)

Pending Errata and Addenda

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