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This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here by the full ring of differential operators I mean the same thing as the ring of divided-power differenial operators, which is the terminology used in the cited question.)

My question is:

Do people have experience using the full ring of differential operators successfully in characteristic $p$ (for localization, or other purposes)?

I always found this ring somewhat unpleasant (its sections over affines are not Noetherian, and, if I recall correctly a computation I made a long time ago, the structure sheaf ${\mathcal O}_X$ is not perfect over ${\mathcal D}_X$). Are there ways to get around these technical defects? (Or am I wrong in thinking of them as technical defects, or am I even just wrong about them full stop?)

EDIT: Let me add a little more motivation for my question, inspired in part by Hailong's answer and associated comments. A general feature of local cohomology in char. p is that you don't have the subtle theory of Bernstein polynomials that you have in char. 0. See e.g. the paper of Alvarez-Montaner, Blickle, and Lyubeznik cited by Hailong in his answer. What I don't understand is whether this means that (for example) localization with the full ring of differential ops is hopeless (because the answers would be too simple), or a wonderful prospect (because the answers would be so simple).

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You probably knew this already, but this conference sounds like a cool place to be for this question. –  Hailong Dao Feb 12 '10 at 20:20
Thanks for this; I hadn't heard about this conference. –  Emerton Feb 12 '10 at 20:37
Can you give a sense what kind of simplicity results you get for (presumably pushforwards of) D-modules from these results? –  David Ben-Zvi Feb 19 '10 at 15:00
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4 Answers

This might not be what you are looking for, as they use the actual full ring of differential operators (in Berthelot's theory, your "full ring" would be $D^{(0)}$, if I understand correctly), but the following papers are very beautiful in my opinion:

Gieseker, D. - Flat vector bundles and the fundamental group in non-zero characteristics.

dos Santos, João Pedro Pinto - Fundamental group schemes for stratified sheaves. J. Algebra 317 (2007), no. 2, 691--713.

Hélène Esnault, Vikram Mehta - Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles

You'll of course notice very quickly that in all cases, the D-Module flavor is lost, as a $O_X$-coherent D-module can be translated into the world of vector bundles thanks to Frobenius descent.

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By the full ring I mean Berthelot's ${\mathcal D}^{\infty}$, if I remember the notation correctly. Thank you for the references! –  Emerton Feb 12 '10 at 13:11
Oh, good, then we are talking about the same thing. I was confused because you wrote "divided power differential operators", which are D^(0) in Berthelots language (and PD-Diff in the language of his older writings on crystalline cohomology), if I understand correctly. The higher D^(n) are constructed via "partially-divided powers". –  Lars Feb 12 '10 at 14:27
Yes, sorry about this; I was copying the terminology of the earlier question that I linked to. I'm never sure what notation/terminology to use when discussing these various rings, since many non-arithmetic geometers don't know Berthelot's notation. –  Emerton Feb 12 '10 at 21:09
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Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module. It is not a direct translation of the characteristic zero definition (and this is certainly related to this inconvenience) but once given it seems to work as well as in characteristic zero:

MR1918185 (2003h:14030) Bögvad, Rikard(S-STOC) An analogue of holonomic D-modules on smooth varieties in positive characteristics. (English summary) The Roos Festschrift volume, 1. Homology Homotopy Appl. 4 (2002), no. 2, part 1, 83–116. 14F10 (16S32 32C38)

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Dear Matt: The people who are actively working with this whom I know are Genady Lyubeznik and Manuel Blickle. The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$-modules (here $F$ is the Frobenius). It has been applied to show that certain local cohomology modules over regular local rings in positive characteristic have finitely many associated primes. Examples are in the following papers:

D-module structure of R[F]-modules and

Generators of D-modules in positive characteristic

and the references there in. There is also this new preprint which might be of interest.

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Thanks! I guess this is the aspect of the theory that I know best, but a lot of weight is carried by the unit Frobenius action. I wonder if this is expected to be true in other contexts, like localization? –  Emerton Feb 11 '10 at 0:42
Dear Matt: I don't know much of other contexts. Are there some specific properties of $D_R$ you want to be true? –  Hailong Dao Feb 11 '10 at 0:58
Dear Hailong, I'm mostly just curious to hear what people have to say. My impression is that there are at least three different groups of mathematicians thinking about $\mathcal D_R$ in some form: $p$-adic cohomology people (of which I am, or at least once was, one); commutative algebraists; and representation-theorists. There is some interaction between the first two groups, but I don't know of as much interaction between either of them and the third group. I'm hoping this question might draw input from all three groups. –  Emerton Feb 12 '10 at 19:15
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Smith and Van den Bergh worked with it, and got some lovely results, in this paper. For example, they show that direct summands of polynomial rings have simple rings of differential operators in positive characteristic. This is still open (as far as I know) in characteristic 0.

It's a particularly interesting paper for the connections it makes with representation types.

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