1
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.