Example of nonvanishing higher inverse limits

Question

It is well known that in the category of abelian groups, the limit over a cofiltered inverse system $\mathcal I$ of cofinality $\omega_n$ has nonvanishing derived functors only in degree $\le n+1$, i.e. $R^k\lim\limits_{\mathcal I} A_i$ vanishes for any system $A_i$ and any integer $k > n+1$. It is also known that $n+1$ is the best possible value.

However, I cannot construct any example such that $R^{n+1}\lim$ or even $R^2\lim$ does not vanish. Is there any simple examples?

Answer 1

In

• Mardesic, Sibe. Strong shape and homology. Springer Science & Business Media, 2013.

you will find examples in section 13.3. In the bibliographic notes in the end of the section the author writes

It is difficult to find in the literature sufficiently simple, explicitly described, inverse systems $\mathbf X$ with $\lim^n \mathbf X \ne 0$, $n \ge 2$.

He also gives two references to two such papers.