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Relations of derivations.
curvature revisited
In this subsection we prove an important proposition on curvature. But before we do that, we do two things. Firstly, we prove the following lemma.
L
EMMA
5.7
Let
be a commutative ring. Let
be a commutative
-algebra. Then for any
and for any
, we have
P
ROOF.
.
Secondly, we add a notation.
D
EFINITION
5.8
Let
be a commutative ring. Let
be a commutative
-algebra. Let
be a
-module with a connection
Then for any
,
is an
-linear map. We shall denote it by
.
P
ROPOSITION
5.9
Let
be a commutative ring. Let
be a commutative
-algebra. Let
be a
-module with a connection
Let
be the connection
-form of
. Then for any
, we have
P
ROOF.
. Let
be an element of
. Since
is an element of
, we may write it as :
for some
. Then we compute
On the other hand, we have
So we may proceed
Thus, together with the Lemma above, we see
C
OROLLARY
5.10
If the curvature
of
is equal to zero, then
is a Lie algebra homomorphism.
Next:
a formula for -powers
Up:
some linear algebra
Previous:
Relations of derivations.
2007-12-26