calculation of reduced trace

Lemma 12   The trace of a differential operator $\xi^k (d/d\xi)^l$ on a vector space $k[\xi]/(\xi^p)$ is non zero if and only if $k=l=p-1$.

[proof] If $k\neq l$ then $\xi^i(d/d\xi)^j$ is represented by a strictly triangular matrix and the trace is 0. If $k=l$, we have

$$(\xi^l(d/d\xi)^l).\xi^s=s(s-1)\dots(s-k+1) \xi^s.$$

Summing up this we obtain the result.

$\qedsymbol$

Corollary 6  

$$\operatorname{tr}(({\mu}+T)^k({\nu}+U)^l)= \begin{cases} -T^... ...+(p-1), l=l_0 p+(p-1)$} \\ 0 & \text{otherwise}\\ \end{cases}$$

Lemma 13  

1. If $0\leq i,j \leq p-1$, then $\operatorname{trd}(\xi^i \eta^j)$ is an element of $k$.

2. $$\operatorname{trd}(\xi^i\eta^j)= \begin{cases} -1 &\text{(if $i=j=p-1$)}\\ 0 & \text{otherwise} \end{cases}$$

A formula for the reduced trace when $n>1$ is easily obtained by noting that $A_n(k)$ is isomorphic to a tensor product $A_1(k)\otimes_kA_1(k)\otimes_k \dots\otimes_k A_1(k)$ of $A_1(k)$ 's.

Lemma 14   Let $k$ be a field of characteristic $p$. Let $\phi$ be a $k$-endomorphism of $A_n(k)$. Let $I,J$ be an element of $\{0,1,2,\dots,p-1\}^n$ (multi-index). Then for any $f \in Z(A_n(k))$, we have

$$\operatorname{trd}( \xi^I \eta^J f)= \begin{cases} -f & \tex... ... } I=J=(p-1,p-1,\dots,p-1) \\ 0 & \text{otherwise} \end{cases}$$

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