Let $A$ be a ring (commutative and noetherian if it helps). Suppose we are given an inverse system $M_i$ of complexes of $A$-modules (where $i$ is a natural number), and integers $a<b$ such that for each $i$, the complex $M_i$ has non-zero cohomologies only in degrees $a<j<b$.

Consider the complex $\varprojlim M_i$. Does this complex have bounded cohomology?

Note that I am not assuming that this system satisfy a Mittag-Leffler condition.


No, even for $A=\mathbf{Z}$.

Take your favourite example of a complex of short exact sequences of abelian groups whose projective limit is not exact (see for example this math.stackexchange question).

Now just make a complex of long exact sequences by splicing together the short exact sequences a la

$$\to0\to A\to B\to C\to 0\to 0\to A\to B\to C\to 0\to\cdots$$

and you get failure of exactness of the projective limit all over the place, even though all the sequences in the complex are exact.


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