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Let $C_*$ be a chain complex of abelian groups.

Is it true that $H_i(C_*\otimes \mathbb{Z}/p)=0$ for all $i$ if and only if $H_i(C_*\otimes \mathbb{Z}_p)=0$ for all $i$, where $\mathbb{Z}_p$ is localization of $\mathbb{Z}$ away from $p$?

And I want to know that the precise definition of localization of $\mathbb{Z}$ away from $p$. Is it equal to $\mathbb{Z}[\frac{1}{p}]$ or $(\mathbb{Z}-p\mathbb{Z})^{-1}\mathbb{Z}$?

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No. Consider for instance the chain complex $0 \to \mathbb{Z}_p \stackrel{p}{\to} \mathbb{Z}_p$. This is acyclic but its reduction mod $p$ isn't. The problem is that $\mathbb{Z}/p$ is not a flat module over the localization $\mathbb{Z}_p$.

As for your second question, it's the latter one. You are localizing away from $p$, so you invert the elements not in $(p)$.