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Let $R$ be a Noetherian local ring and $\{ M_n \}_{n \ge 0}$ be a family of finitely generated $R$-modules. Further, let $\phi_n : M_n \to M_{n-1}$ be a family of $R$-linear maps. The kernel and cokernel of the map $$ \prod M_n \to \prod M_n, $$ where $(x_1,\cdots)$ is mapped as $x_i \mapsto x_i - \phi_{i+1}(x_{i+1})$ are the inverse limit $\varprojlim_n M_n$ and the (first) right derived functor of the inverse limit functor detnoed as $\varprojlim_n^1 M_n$.

It is well-known that if an inverse system satisfies the Mittag-Leffler condition, then $\varprojlim^1_n M_n = 0$. However, it is not true that $\varprojlim_n^1 M_n$ is zero even in the case where $\varprojlim_n M_n = 0$. One can see a couple of examples in this nice answer https://math.stackexchange.com/a/1153465/84157.

In the examples in the linked post, the maps $\phi_{n}$ are all injective. I would like to ask the following question.

Is there an example such that
1) $\varprojlim_n M_n = 0$,
2) $\varprojlim_n^1 M_n \neq 0$, and
3) $\phi_n$ are not injective?

Youngsu
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    If you take an inverse system such that for every $n$, one has $\phi_n=0$, then $\varprojlim M_n=\varprojlim^1 M_n=0$. You can now add this system to the linked example. Thus an example will be the following : take $M_n=\mathbb{Z}^2$ and $\phi_n=\begin{pmatrix} p&0\0&0 \end{pmatrix}$ – Roland Aug 04 '17 at 14:27
  • @Roland Thank you for the comment. I see that your example provides a family of examples. – Youngsu Aug 04 '17 at 18:11

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