Consider an abelian category $\mathcal{A}$ and two objects $A,B$ of $\mathcal{A}$. It is straightforward that an element $\eta \in \text{Ext}^{1}(A,B)$ of the form $0 \to B \xrightarrow{f} X \xrightarrow{g} A \to 0$ is zero in $\text{Ext}^{1}(A,B)$ if and only if $g$ has a right inverse, which is equivalent to say that $f$ has a left inverse.
Question: If we take $n > 1$ and consider an element $\eta \in \text{Ext}^{n}(A,B)$ of the form $0 \to B \xrightarrow{f} X_{n} \to \cdots \to X_{1} \xrightarrow{g} A \to 0$, is it true that $\eta$ is zero $\text{Ext}^{n}(A,B)$ if and only if $g$ has a right inverse (or $f$ has a left inverse)?
No. For example, consider the exact sequence $$0\to\mathbb{Z}/2\mathbb{Z}\xrightarrow{\times2}\mathbb{Z}/4\mathbb{Z}\xrightarrow{\times2}\mathbb{Z}/4\mathbb{Z}\xrightarrow{\times1}\mathbb{Z}/2\mathbb{Z}\to0$$ of abelian groups.