In this section, we always assume
to be a prime number.
be a connection with zero curvature. Then the
From the argument of the previous section we see that the p-curvature is
-linear.
A fairly good account on
-curvatures is given in [2].
Our treatment here is a bit different.
It is not so general, but is easy using only
arguments on rings and modules.
be a connection on
of parallel sections as a
is said to be locally generated by parallel sections
if there exists
an open covering
of
such that
is
generated by parallel sections.
be a connection on
Then the following conditions are equivalent.
Since the question is local, we may reduce the proposition to an
lemma which we describe later.
Before we do that, we need some preparation.
First we note that when
is smooth of relative dimension
,
we may locally choose a set of elements
(``coordinates'') such that
is freely generated by
over
.
Let
be an affine open subset of
on which such a local coordinate
system exists. Then there exists vector fields
on
Note also that from this observation we see
Next, let us set some more notation. We employ the graded lexicographic order on an index set
We define an order reversing map
Finally, for any
, we define
by
be a connection on
holds.
Since we know that curvature is
: obvious.
:
For any
and for any
, we put
Note that the condition (1) tells us that
Now we claim :
So it is enough to put
Assume now that the claim holds for all
.
Since
is well-ordered set, there exists an index
which is just before
. (That means,
is the largest index which is smaller than
.)
Let us put
Then for any
Thus we have
Then we put
We may easily see that
holds and thus the claim holds for
It is worthwhile to note that the coefficients
in the condition (3)
of the Lemma above is unique. Namely,
is bijective.
This contradicts the assumption.
Then every element
To obtain that, first we recall Taylor expansion
of a polynomial
By analogy we put
Then we see that
Indeed, we have the following lemma.
is parallel.
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as required.
Then:
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