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Algebra endomorphism and splitting of Weyl algebra

Lemma 7   Let $ k$ be a algebraically closed field with characteristic $ p>0$. Let $ \phi:A_n(k)\to A_n(k)$ be a $ k$-homomorphism, $ \psi:Z_n(k)\to Z_n(k)$ its restriction to the center. Then we have the following.
  1. $ \psi$ extends uniquely to a homomorphism

    $\displaystyle \hat{\psi}: S_n(k) \to S_n(k)
$

  2. $ \phi$ extends uniquely to a homomorphism

    $\displaystyle \hat{\phi}:
A_n(k)\otimes_{Z_n(k)}S_n(k)
\to
A_n(k)\otimes_{Z_n(k)}S_n(k)
$

  3. Under the isomorphism $ \Phi$ of lemma 5, $ \hat{\phi}$ may be identified with a map

    $\displaystyle M_p(S_n(k)) \ni M(T,U) \mapsto G(T,U) M(f(T,U))G(T,U)^{-1}
$

    where $ G(T,U)$ is an element of $ \operatorname{GL}_p(S_n(k))$ and $ f={}^a\psi:\mathbb{A}^{2n} \to \mathbb{A}^{2n}$ is a polynomial map associated to the algebra homomorphism $ \hat \psi$. In other words, we have

    $\displaystyle \Phi(\phi(x))=G f^*(\Phi(x))G^{-1}
$

(1): In any commutative integral domain of characteristic $ p$, $ p$-th root of an element is unique.

(2): For each $ k$-valued point $ (t,u)$ of $ \mathbb{A}^{2n}$ we obtain a homomorphism $ \psi_{t,u}: M_{p^n}(k)\to M_{p^n}(k)$. In other words, we obtain a morphism

$\displaystyle \check G:\mathbb{A}^{2n}\to \operatorname{PGL}_p.
$

But since the sheaf cohomology $ H^1(\mathbb{A}^{2n},\mathbb{G}_m)$ (in Zariski topology) is trivial, we have a lift $ G$ of $ \check G$ and the claim is proved.

Corollary 5 (invariance of trace under algebra endomorphism)  

$\displaystyle \operatorname{trd}(\phi(a))=\phi(\operatorname{trd}(a))
$

Lemma 8   The polynomial map $ f$ of the lemma above is determined by $ G(T,U)$ uniquely up to an addition of an element of $ k[T^p,U^p]$. That means, if we have two $ f$'s, namely $ f_1$ and $ f_2$, with the same $ G(T,U)$, then we have

$\displaystyle f_1^{*}(T) -f_2^{*}(T)
\in k[T^p,U^p], \quad
f_1^{*}(U) -f_2^{*}(U)
\in k[T^p,U^p]
$

Put $ \alpha=\xi-T, \beta=\eta-U$. Then

  $\displaystyle \phi(\xi)-\phi(T)=\phi(\alpha)=G(T,U)\alpha G(T,U)^{-1},$ (3)
  $\displaystyle \phi(\eta)-\phi(U)=\phi(\beta)=G(T,U)\beta G(T,U)^{-1}.$ (4)

$\displaystyle \phi(\xi),\phi(\eta)\in k[\xi,\eta], \phi(T),\phi(U)\in k[T,U].
$


next up previous contents
Next: the geometric degree Up: First properties of Weyl Previous: definition of reduced trace   Contents
2003/3/3