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

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?

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You may want to take a look at the paper of Kontsevich "Holonomic D-modules in positive characteristic" section 2. –  B. Bischof Feb 27 '11 at 23:53
Uma Iyer showed in her dissertation that considering the Weyl algebra in positive characteristic is related to the Azumaya algebra for the Heisenberg algebra. The relevant reference from Arxiv is arxiv.org/pdf/math/0002014v1. I am not sure if this is exactly what you are looking for, thus the comment. –  B. Bischof Feb 28 '11 at 4:44

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.

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Thank for the very explicit answer Konstantin. I guess I was a bit vague in the last question. Of course the ring of differential operators in characteristic p > 0 is not finitely generated, but there we can get the divided powers through action of Frobenius. I guess I was wondering if in a similar sense, the divided powers appearing where we replace the field k with the integers $\mathbf{Z}$ all also appear in some finite process. –  lmiller Feb 28 '11 at 15:32
I'm not aware of any such finite process. For example, writing $\partial^{(n)} = \frac{1}{n!} \frac{d^n}{dx^n}$, how might you "generate" the divided powers $\partial^{(2)}, \partial^{(3)}, \partial^{(5)}, \partial^{(7)}, \cdots$ from the integral Weyl algebra $\mathbb{Z}[x;\partial]$ in some "finite process"? Grothendieck's definition of $\mathcal{Diff}_X$ as the inductive limit of a sequence of $\mathcal{O}_X$-duals of larger and larger infinitesimal neighbourhoods of the diagonal in $X \times X$ is a "uniform" way of getting all of the divided powers in one go. –  Konstantin Ardakov Feb 28 '11 at 15:52

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.

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