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# Cohomologies.

Yoshifumi Tsuchimoto

DEFINITION 08.1   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 08.2   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 08.3   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 08.4   Let be an associative unital ring. Then:
1. .
2. For any , is a right exact functor.
3. For any right ideal of and for any -module , we have

In particular, if the ring is commutative, then for any ideals of , we have

DEFINITION 08.5   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 .

EXERCISE 08.1   Compute for .

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2010-06-16