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](http://www.jmilne.org/math/CourseNotes/ala.html); 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?



Infinite products of representations of the additive group
Question A: Demazure-Gabriel, Groupes Algébriques, II, §2, 2.6


Question B: It is easily checked that the maximal submodule of $\prod M\_i$ on which $\prod f\_i$ acts locally nilpotent is the categorical product. This may also be described as $\bigcup\_n \prod\_i \ker f\_i^n\subseteq\prod\_i M\_i$. Actually, since the set-valued functor $(M,f)\mapsto\ker f^n$ is represented by $(R[X]/(X^n),X)$, so commutes with products, and since the equality $M=\bigcup\_n \ker f^n$ is equivalent to $f$ acting locally nilpotent, it is clear that the underlying set of the categorical product must be $\bigcup\_n \prod\_i \ker f\_i^n$.


Question C: An example where the filtration $\ker f^n$ does not split is $R=k[[T]]$, $M=k((T))/k[[T]]$, $f=T$.


