= Prime coefficient detection of quasi-isomorphisms
{title2=$f\otimes\mathbb F_p\text{ a quasi-isomorphism for all }p\Rightarrow f\text{ a quasi-isomorphism}$}
Let $f$ be a <chain map> between degreewise finitely generated <free abelian groups>. If its reduction modulo every <prime number> is a <quasi-isomorphism>, the <mapping cone acyclicity criterion> makes $M(f)\otimes\mathbb F_p$ acyclic. The <short exact sequence of chain complexes> $0\to M\xrightarrow{p}M\to M\otimes\mathbb F_p\to0$ implies that multiplication by $p$ is an <isomorphism> on every $H_i(M)$. These are <finitely generated abelian groups>. Their free summands would make multiplication by $p$ fail to be surjective, and any nonzero finite summand would make it fail to be injective for a prime dividing its order. Thus $H_i(M)=0$ and $f$ is a <quasi-isomorphism>. Degreewise finite generation is essential to this argument.
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