Lemma 7
Let be a algebraically closed field with characteristic .
Let
be a -homomorphism,
its restriction to the center.
Then we have the
following.
extends uniquely to a homomorphism
extends uniquely to a homomorphism
Under the isomorphism of lemma 5,
may be identified with a map
where is an element of
and
is a polynomial map associated to the algebra homomorphism .
In other words, we have
(1): In any commutative integral domain of characteristic ,
-th root of an element is unique.
(2):
For each -valued point of
we obtain a homomorphism
. In other words, we obtain a morphism
But
since the sheaf cohomology
(in Zariski topology)
is trivial,
we have a lift of and the claim is proved.
Corollary 5 (invariance of trace under algebra endomorphism)
Lemma 8
The polynomial map of the lemma above is determined
by
uniquely up to
an addition of an element of
.
That means, if we have two 's, namely and ,
with the same , then we have