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Assume $C_*$ is a chain complex of abelian groups and $R$ is a ring.

Assume $$ H_k(C_*;R)=0 $$ for some $k\geq 0$.

Is it true that $$ H_k(C_*;\mathbb{Z})=0? $$

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    $\begingroup$ Take $R=\mathbb{Z}_3$ and look a the homology of $\mathbb{R}P^2$. $\endgroup$
    – Tyrone
    Nov 17, 2017 at 11:42

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No, this can easily fail. For instance, if $R$ is the zero ring, then $H_k(C_*;R)=0$ no matter what $C_*$ is. A bit less trivially, if $R=\mathbb{Q}$, then $H_k(C_*;R)\cong H_k(C_*;\mathbb{Z})\otimes\mathbb{Q}$, so any chain complex with $H_k(C_*;\mathbb{Z})$ a nonzero torsion group will give a counterexample.

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