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# Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Tensor products and Tor.

DEFINITION 09.1   Let be an associative unital (but not necessarily commutative) ring. Let be a right -module. Let be a left -module. For any ( -)module , an map

is called an -balanced biadditive map if
1.      .
2.      .
3.      .

PROPOSITION 09.2   Let be an associative unital (but not necessarily commutative) ring. Then for any right -module and for any left -module , there exists a ( -)module together with a -balanced map

which is universal amoung -balanced maps.

DEFINITION 09.3   We employ the assumption of the proposition above. By a standard argument on universal objects, we see that such object is unique up to a unique isomorphism. We call it the tensor product of and and denote it by

LEMMA 09.4
1. .
2. For any , is a right exact functor.

DEFINITION 09.5   For any left -module , the left derived functor of is called the Tor functor and denoted by .

By definition, may be computed by using projective resolutions of .

DEFINITION 09.6   For any group , the derived functor of a functor

defined by

span

is called the homology of with coefficients in . We denote the homology group by .

LEMMA 09.7

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2009-07-10