Explicit ring of differential operators for polynomial algebras over the integers?

Question

Does anyone know of a reference or have any idea for an explicit description of the ring of differential operators for polynomial algebras over the integers? I'm hoping there is something analogous to the case of a polynomial ring over a field $K$, where if the field has characteristic $0$ an explicit description is given by the Weyl algebra. In particular, for $K[x_1,\ldots,x_n]$ this is the noncommutative $K$-algebra spanned by the symbols $x_1,\ldots,x_n$ and $d_1,\ldots,d_n$ where the $x_i$'s commute, the $d_i$'s commute and one has the Leibniz rule: $d_i x_j - x_j d_i = \delta_{i,j}$ where $\delta_{i,j}$ is the Kronecker delta.

If the field has characteristic $p > 0$, then one also has divided powers $\frac{1}{p!}(\frac{d^p}{dx_i^p})$, (Hasse derivatives or hyperdifferential operators) which are not in the Weyl algebra but can be constructed using Frobenius. Of course, for $\mathbf{Z}[x_1,\ldots,x_n]$ the divided powers $\frac{1}{t!}(\frac{d^t}{dx_i^t})$ for integers $t > 1$ are not in the Weyl algebra. Is the full ring of differential operators spanned by the Weyl algebra and these divided powers? Is there a finite generating set?

Answer 1

The answer to your first question is "yes". You can find a calculation of the full ring of differential operators on a suitably nice scheme here : Theoreme 16.11.2 on page 54 of EGA 4 IV, PIHES 32 (1967). Generally speaking, EGA 4 IV.16 is the original and definitive reference for differential operators on arbitrary schemes.

It follows from this result that the answer to your second question is also "yes". If $A$ is the polynomial algebra $K[x_1,\ldots, x_n]$ for any commutative ring $K$ and $m \geq 0$ is an integer, then the $A$-module $\mathcal{Diff}^m_{A/K}$ of all $K$-linear differential operators of order at most $m$ is free with basis precisely the divided powers $\frac{1}{\alpha !} \frac{\partial^\alpha}{\partial x^\alpha}$ as $\alpha$ varies over all multi-indices in $\mathbb{N}^n$ with $|\alpha| = \alpha_1 + \cdots + \alpha_n \leq m$.

The answer to your third question is "no". If there was a finite generating set for the algebra of full differential operators on $\mathbb{Z}[x_1,\ldots, x_n]$, then by passing to a field $k$ of positive characteristic $p$, we would obtain a finite generating set for $\mathcal{Diff}_{k[x_1,\ldots,x_n]/k}$. But it is known that any differential operator on $A = k[x_1,\ldots, x_n]$ is $A_r = k[x_1^{p^r}, \ldots, x_n^{p^r}]$-linear for some $r$ --- see the reference below. So if there was a finite generating set, then the whole differential operator algebra would be contained in the matrix algebra $End_{A_r}(A)$ for some fixed $r$; this can be seen not to be the case by considering a divided power of order $\geq p^r$.

A good reference for the computation of differential operators on $k[x_1,\ldots, x_n]$ is the following paper by S.P. Smith: "The global homological dimension of the ring of differential operators on a nonsingular variety over a field of positive characteristic", J. Algebra 107(1) 1987, 98---105. It should be possible to access it through www.sciencedirect.com .

I should perhaps mention at this point that Berthelot has developed a theory of so-called "arithmetic differential operators", which can be viewed as a refinement of Grothendieck's classical theory. You can find an introduction here.

Answer 2

In positive characteristic, there is another analog of the Weyl algebra, a ring generated by certain difference operators introduced by Carlitz. These operators are connected with intrinsic structures of local fields with positive characteristic. For the details see A. N. Kochubei, "Analysis in Positive Characteristic", Cambridge University Press, 2009.