TITLE: Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
QUESTION [9 upvotes]: In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in terms of Grothendieck (or other) style geometric ideas such as thickenings, formal schemes, lambda rings, or crystalline cohomology?
A related and possibly equivalent question is whether there is any precise sense in which the "extra" material in divided power algebras (compared to polynomial or power-series rings) is dual to the "extra" differential operators in positive characteristic(s).

REPLY [5 votes]: In a nutshell, observe that $\partial_x^p(x^p)=0$ in characteristic $p$. This means that you can divide either $\partial_x^p$ or $x^p$ by $p!$ and get a sensible object, i.e., the second object will interact with the divided object and you get $=1$ in the formula. Dividing $\partial_x^p$ you get your missing differential operators. Dividing $x^p$ you lose it but discover "crystalline world", "sickening", etc.
To put it bluntly, No, in crystalline world, differential operators on a smooth scheme are generated in degree 1, but yes, your missing operators will live on a formal scheme...
What was your question exactly, Doc?