$\DeclareMathOperator\Spec{Spec}$Let $p$ be a prime number, and $A$ a commutative ring. Recall that a $p$-derivation on $A$, or a $\delta$-ring structure on $A$ is a set map $\delta : A \to A$ such that $\phi(x) = x^p + p\delta(x)$ is a ring homomorphism; if $p$ is a zero-divisor in $A$, it is additionally required that "$\delta$ satisfies all equations which this description would entail had $A$ been $p$-torsionfree" (see the link for a precise definition). Thus $\delta$ encodes a sort of "rigid lift of Frobenius" from $A/p$.
There is an analogy lurking in the term "$p$-derivation": the analogy says that $\delta$ is something like a derivation. I would like to understand this analogy better.
Question 1: In what ways is it useful to think of a $p$-derivation as being like a derivation?
Question 2: Are there places where the analogy comes up short, and we should not really think that a $p$-derivation is like a derivation?
Notes:
There are at least two books, both by Buium, which take this analogy and run with it. So I should probably start by reading the introductions to these books in detail.
There's also a theorem, again of Buium which says that in some weak "up to isogeny" sense the derivations, $p$-derviations, difference operators, and $p$-difference operators are the "only" plethories satisfying some conditions. But the result is not quite as crisp as one might like. See here for some discussion.
Note that all the references so far are due to a single author (Buium). I'd appreciate getting a perspective which puts Buium's work into context with number theory and algebraic geometry in general.
(Question 3:) One thing I'd like to understand in these terms is the universal property of the $p$-typical Witt vectors. I guess $\Spec W(A)$ is $\Spec A$ with a "$p$-tangent-direction freely added to it" or something? What is the analogous object when $p$-derivations are replaced with ordinary derivations?
