Proof
We consider the short exact sheaf sequences
-
on
coming from the resolution for
(
is just the structure sheaf).
Because all these sheaves are locally free, taking the absolute Frobenius
(and all its iterations)
is exact, therefore we get short exact sequences
-
and cohomology pull-backs
.
Note also that for
and
we get
-
so the image of this map inside
is exactly
. By the universal property of the absolute Frobenius and of the connecting homomorphisms in cohomology we have
-
![{\displaystyle {}F^{e*}(c_{j})=F^{e*}(\delta ^{j}(f))=\delta ^{j}(F^{e*}f)=\delta ^{j}(f^{q})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363a949c73bc00a74b99cd04c95e3aa16fa35538)
and also
-
![{\displaystyle {}zF^{e*}(c_{j})=\delta ^{j}(zf^{q})\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a34aaf42ad68413fae0356b58cf8f6032b3f9875)
Because of the injectivity of
in the given range we have that
belongs to the ideal
if and only if
if and only if
.