I apologize for the lengthy introduction. It is mainly for context and to introduce a certain phenomenon.
$\newcommand{\Z}{\mathbb{Z}}$ Consider the groups $\Z[[x]]$ of formal power series and $\Z[x]$ of polynomials with integer coefficients. I am using $\Z$ here for simplicity, but it may be replaced with any abelian group, as we will see.
I am interested in the quotient group $X_{\Z}=\Z[[x]]/\Z[x]$. For some reason it is difficult to find anything about this group in the literature, but some properties can be deduced. Let us call it the group of 'asymptotes' of $\Z$.
The asymptotes of $\Z$, i.e. elements of $X_{\Z}$ are represented countable sequences of integers $a=(a_i)_{i\in\Z}$, which represent the same element if and only if they differ in at most finitely many degrees. In particular, two sequences represent the same element if and only if they eventually agree. This justifies the intuition of $X_{\Z}$ as modeling the asymptotic behaviour of power series.
There is an exact sequence of abelian groups $$0\longrightarrow \Z[x]\longrightarrow \Z[[x]] \longrightarrow X_{\Z} \longrightarrow 0$$ which has the property that given any abelian group $A$, tensoring with $A$ everywhere preserves exactness (i forget what this property of a short exact sequence is called): We have $\Z[[x]]\otimes A \simeq A[[x]]$ and $\Z[x]\otimes A\simeq A[x]$, and the inclusion $A[x]\rightarrow A[[x]]$ remains injective, so the following sequence is exact. $$0\rightarrow A[x]\rightarrow A[[x]]\rightarrow X_{\Z}\otimes A\rightarrow 0$$ In particular, $X_A\simeq X_{\Z}\otimes A$ and the asymptotes of $A$ are generated by $X_{\Z}$.
In the analogous situation with $n$ variables, the inclusion $\Z[x_1,...,x_n]\rightarrow\Z[[x_1,...,x_n]]$ preserves the $\mathbb{N}^n$-grading, so the same result holds for $n$-dimensional asymptotes.
Let $E$ be an abelian group and $E=F_0\supseteq F_1 \supseteq ...$ be a decending filtration of $E$. Let $\hat{E}$ be the completion of $E$ with respect to this filtration. Unless I am grossly mistaken, a necessary and sufficient criterion for the canonical homomorphism $E\rightarrow \hat{E}$ to be a monomorphism is that for every nonzero $e\in E$, there exists some stage $F_i$ of the filtration such that $e\not\in F_i$. Assuming this constraint, we obtain a short exact sequence $$0\longrightarrow E\longrightarrow \hat{E} \longrightarrow X \longrightarrow 0$$ Does this sequence have the property that tensoring with any abelian group preserves exactness?
Also, if anyone is aware of any references where this is treated, I would also accept that as an answer.
I would also appreciate other interesting properties $X_{\Z}$ possesses.