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Cohomologies.
Yoshifumi Tsuchimoto
L
EMMA
02.1
Let
be a ring. Let
be a homomorphism of
-modules. Then for any
-module
we may define:
A homomorphism
defined by
.
A homomorphism
defined by
.
P
ROPOSITION
02.2
Let
be a ring. Let
be an exact sequence of
-modules. Then for any
-module
, we have:
is exact. The third arrow
need not be surjective.
is exact. The third arrow
need not be surjective.
E
XERCISE
02.1
We consider an exact sequence
where
is the inclusion map. Show that
is not surjective
E
XERCISE
02.2
Assume
is a field. Then show that the third arrow which appear in the sequence (1) in Proposition
2.2
is surjective.
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2010-04-15