If $f \circ g$ and $f$ are quasi-isomorphisms, is $g$ a quasi-isomorphism ?

# Composition of quasi-isomorphisms

homological-algebra

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# Composition of quasi-isomorphisms

homological-algebra

If $f \circ g$ and $f$ are quasi-isomorphisms, is $g$ a quasi-isomorphism ?

## Best Answer

Yes, because (co-)homolgy in degree $n\in\mathbb N$ is a functor and compatible with composition:

Let $g:C\Rightarrow D$ and $f:D\Rightarrow E$ be chain-maps such that $f\circ g$ and $f$ are quasiisomorphisms. By definition for any $n\in \mathbb N$ we have that $H_n(f\circ g)=H_n(f)\circ H_n(g)$ and $H_n(f)$ are isomorphisms. Then $H_n(g)=H_n(f)^{-1}\circ H_n(f\circ g)$ is an isomorphism as the composition of two isomorphisms. So for all $n\in \mathbb N$ we have that $H_n(g)$ is an isomorphism, that is $g$ as quasiisomorphism.