Lemma 86.3.4. Let $R$ be a ring. Let $M^\bullet $ be a complex of modules over $R$ with $M^ i = 0$ for $i > 0$ and $M^0$ a projective $R$-module. Let $K^\bullet $ be a second complex.

If $K^ i = 0$ for $i \leq -2$, then $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(M^\bullet , K^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(R)}(M^\bullet , K^\bullet )$,

If $K^ i = 0$ for $i \leq -3$ and $\alpha \in \mathop{\mathrm{Hom}}\nolimits _{D(R)}(M^\bullet , K^\bullet )$ composed with $K^\bullet \to K^{-2}[2]$ comes from an $R$-module map $a : M^{-2} \to K^{-2}$ with $a \circ d_ M^{-3} = 0$, then $\alpha $ can be represented by a map of complexes $a^\bullet : M^\bullet \to K^\bullet $ with $a^{-2} = a$.

In (2) for any second map of complexes $(a')^\bullet : M^\bullet \to K^\bullet $ representing $\alpha $ with $a = (a')^{-2}$ there exist $h' : M^0 \to K^{-1}$ and $h : M^{-1} \to K^{-2}$ such that

\[ h \circ d_ M^{-2} = 0, \quad (a')^{-1} = a^{-1} + d_ K^{-2} \circ h + h' \circ d_ M^{-1},\quad (a')^0 = a^0 + d_ K^{-1} \circ h' \]

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