# Isomorphism preserves exactness

Let $R$ be a commutative ring with unity. Let $A_i$ be an R-module for every $i$. Consider a sequence of modules $$\xrightarrow{\delta_{i-1}}A_{i-1}\xrightarrow{\delta_{i}} A_i\xrightarrow{\delta_{i+1}} A_{i+1}.$$

Let a morphism of a sequence of modules be defined as in this question, which does not use exactness. I am wondering whether an isomorphism of sequences preserves exactness at a module $A_i$ in the sequence.

I cannot find an counterexample, but exactness is a set-theoretical property (kernel of $\delta_i$ is equal to the image of $\delta_{i-1}$ as subsets of the module $A_{i-1}$) which does not have to be preserved by an isomorphism.

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More generally, if the sequence is a not necessarily exact complex (i.e. $\delta_i\circ \delta_{i-1}=0$ for all $i$), an siomorphism of complexes induces an isomorphism of homologies (i.e. of the quotients $\ker \delta_i/\operatorname{im}\delta_{i-1}$).