Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ *locally nilpotent* if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. Equivalently, for every finitely generated submodule $N \subseteq M$ there is some $n \in \mathbb{N}$ with $f^n |_N = 0$. In particular, for a finitely generated $M$ we don't get anything new.

Now consider the following category $C$: Objects are pairs $(M,f)$, consisting of an $R$-module and a locally nilpotent endomorphism $f$ of $M$. A morphism is just a commutative diagram.

**Question A.** Is there a reference in the literature for the following observation: $C$ is isomorphic to the category of representations of the additive group scheme $\mathbb{G}_{a}$ over $R$.

This is not hard to prove. In fact, one can use the same method as in Example I.8.1 in Milne's script on algebraic groups; but there $R$ is a field and $M$ is supposed to be finite dimensional, so that locally nilpotent = nilpotent. But the same works in general, the $R[T]$-comodule structure on $M$ corresponding to $(M,f)$ is given by $M \to M[T]$, $m \mapsto \sum\limits_{n \geq 0} \frac{f^n(m) T^n}{n!}$.

**Question B.** How do infinite products look like in $C$?

Note that the description of $C$ above shows that $C$ is cocomplete as as well as complete. The forgetful functor $C \to \mathrm{Mod}(R)$ creates colimits, so they are easy to describe. The same is true for *finite* limits, so the only limits missing are infinite products. The forgetful functor doesn't create them: For a familiy of objects $(M_i,f_i)$, the endomorphism $\prod_i f_i$ of $\prod_i M_i$ does not have to be locally nilpotent. Thus they will be more complicated.

**Question C.** What are interesting examples of locally nilpotent endomorphisms which are not nilpotent?

Of course there are many examples: If $M$ is an $\mathbb{N}$-graded module and $f$ is of negative degree, then $f$ is locally nilpotent, but usually $f$ is not nilpotent. A specific example is the derivative $\partial : R[X] \to R[X]$. Are there more interesting examples which don't arise from gradings?