Cohomologies.

Yoshifumi Tsuchimoto

LEMMA 02.1   Let be a ring. Let be a homomorphism of -modules. Then for any -module we may define:
1. A homomorphism defined by .
2. A homomorphism defined by .

PROPOSITION 02.2   Let be a ring. Let

be an exact sequence of -modules. Then for any -module , we have:
1. is exact. The third arrow need not be surjective.
2. is exact. The third arrow need not be surjective.

EXERCISE 02.1   We consider an exact sequence

where is the inclusion map. Show that

is not surjective

EXERCISE 02.2   Assume is a field. Then show that the third arrow which appear in the sequence (1) in Proposition 2.2 is surjective.