differential operators in positive characteristics

We begin by noting the following easy fact.

LEMMA 7.1   Let $k$ be a field of characteristic $p\neq 0$ . Let $n$ be a positive integer. Then we have

$$(\frac{\partial}{\partial x_i})^p f(x)=0 \quad (i=1,2,\dots,n).$$

for any polynomial $f\in k[x_1,x_2,\dots,x_n]$

In short, we have $\partial_i^p=0$ . This explains why we defined $A_n(k)$ by a generator and relation rather than a ring of differential operators.

Of course, differential operations give important examples of representation of the Weyl algebras.

LEMMA 7.2   Let $k$ be a field of characteristic $p\neq 0$ . Let $n$ be a positive integer. There is a finite dimensional representation $\Phi$ of $A_n(k)$ on $k[x_1,x_2,\dots,x_n]/(x_1^p,x_2^p,\dots,x_n^p)$ defined as follows.

$$\Phi(\gamma_i) f=x_i f, \qquad \Phi(\gamma_{n+i}) f=\partial_i \cdot f \qquad (i=1,2,\dots,n)$$

LEMMA 7.3   Let $k$ be a field of characteristic $p$ . We have the following facts.

1. $\gamma_j^p$ belongs to the center of $A_n(k)$ for any $j\in\{1,2,3,\dots,2n\}$ .
2. More precisely, the center $Z_n(k)=Z(A_n(k))$ of the ring $A_n(k)$ is given by
   $$Z_n(k)=k[\gamma_1^p,\gamma_2^p,\gamma_3^p,\dots,\gamma_{2n}^p].$$

3. $A_n(k)$ is a free $Z_n(k)$ -module of rank $p^{2n}$ .
4. Let $\mathfrak{a}_0$ be an ideal of $A_n(k)$ defined as
   $$\mathfrak{a}_{0}=(\gamma_1^p,\gamma_2^p,\dots,\gamma_{2 n}^p).$$
   Then we have
   $$\mathfrak{a}_{0}= \sum_{j=0}^{2 n}A_n(k)\gamma_j^p$$

5. Any element of $A_n(k)/\mathfrak{a}_{0}$ is written uniquely as
   $$\sum_{j_1,j_2,j_3,\dots,j_n=0}^{p-1} a_{j_1 j_2 j_3\dots j_{2n}} ... ..._2^{j_2} \gamma_3^{j_3}\dots \gamma_{2 n-1}^{j_{2 n-1}} \gamma_{2 n}^{j_{2 n}}$$
   for some $a_\bullet\in k$ .

LEMMA 7.4   Let $\Phi$ be the representation given above. The kernel of $\Phi$ is equal to

$$\mathfrak{a}_{0}=(\gamma_1^p,\gamma_2^p,\dots,\gamma_n^p)$$

$\Phi$ gives rise to an algebra isomorphism

$$\overline{\Phi}:A_n(k)/\mathfrak{a}_{0}\cong M_{p^n}(k).$$

PROOF.. That $\mathfrak{a}_0$ is contained in $\operatorname{Ker}(\Phi)$ is obvious. Thus we obtain an well-defined algebra homomorphism $\overline{\Phi}$ .

To see the injectivity of $\overline{\Phi}$ , we employ a lexicographic order on multi index sets and see that

$$\partial^I x^J= \begin{cases} 0 &\text{ if } I>J\\ I! &\text{ if } I=J \end{cases}$$

holds for any multi-indices $I,J \subset \{0,1,2,3,\dots,p-1\}^n$ . Then by using the previous sublemma we see that $\overline{\Phi}$ is indeed injective.

The surjectivity of $\overline{\Phi}$ is verified by counting dimensions.

$\qedsymbol$