## Return to Question

2 added 636 characters in body

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).

1

# Have people successfully worked with the full ring of diferential operators in characteristic p?

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?)