Lemma 59.63.2. Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $V$ be a finite dimensional $k$-vector space. Let $F : V \to V$ be a frobenius linear map, i.e., an additive map such that $F(\lambda v) = \lambda ^ p F(v)$ for all $\lambda \in k$ and $v \in V$. Then $F - 1 : V \to V$ is surjective with kernel a finite dimensional $\mathbf{F}_ p$-vector space of dimension $\leq \dim _ k(V)$.
Proof. If $F = 0$, then the statement holds. If we have a filtration of $V$ by $F$-stable subvector spaces such that the statement holds for each graded piece, then it holds for $(V, F)$. Combining these two remarks we may assume the kernel of $F$ is zero.
Choose a basis $v_1, \ldots , v_ n$ of $V$ and write $F(v_ i) = \sum a_{ij} v_ j$. Observe that $v = \sum \lambda _ i v_ i$ is in the kernel if and only if $\sum \lambda _ i^ p a_{ij} v_ j = 0$. Since $k$ is algebraically closed this implies the matrix $(a_{ij})$ is invertible. Let $(b_{ij})$ be its inverse. Then to see that $F - 1$ is surjective we pick $w = \sum \mu _ i v_ i \in V$ and we try to solve
\[ (F - 1)(\sum \lambda _ iv_ i) = \sum \lambda _ i^ p a_{ij} v_ j - \sum \lambda _ j v_ j = \sum \mu _ j v_ j \]
This is equivalent to
\[ \sum \lambda _ j^ p v_ j - \sum b_{ij} \lambda _ i v_ j = \sum b_{ij} \mu _ i v_ j \]
in other words
\[ \lambda _ j^ p - \sum b_{ij} \lambda _ i = \sum b_{ij} \mu _ i, \quad j = 1, \ldots , \dim (V). \]
The algebra
\[ A = k[x_1, \ldots , x_ n]/ (x_ j^ p - \sum b_{ij} x_ i - \sum b_{ij} \mu _ i) \]
is standard smooth over $k$ (Algebra, Definition 10.137.6) because the matrix $(b_{ij})$ is invertible and the partial derivatives of $x_ j^ p$ are zero. A basis of $A$ over $k$ is the set of monomials $x_1^{e_1} \ldots x_ n^{e_ n}$ with $e_ i < p$, hence $\dim _ k(A) = p^ n$. Since $k$ is algebraically closed we see that $\mathop{\mathrm{Spec}}(A)$ has exactly $p^ n$ points. It follows that $F - 1$ is surjective and every fibre has $p^ n$ points, i.e., the kernel of $F - 1$ is a group with $p^ n$ elements. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: