Give examples of maps \(f, g\) in \(\mathbf{C h}(\mathcal{A})\) such that (1) \(f=\)
0 in \(\mathbf{D}(\mathcal{A})\), but \(f\) is not null homotopic, and (2) \(g\)
induces the zero map on cohomology, but \(g \neq 0\) in \(\mathbf{D}(\mathcal{A})
.\) Hint: For \((2)\) try \(X: 0 \rightarrow \mathbb{Z}
\stackrel{2}{\longrightarrow} \mathbb{Z} \rightarrow 0\), \(Y: 0 \rightarrow
\mathbb{Z} \stackrel{1}{\longrightarrow} \mathbb{Z} / 3 \rightarrow 0,
g=(1,2)\).
