is the morphism completing the short exact sequence to an exact triangle in the derived category, which we again call the Atiyah class $\mathit{At}({\mathscr{F}}) \in \mathit{Hom}_{D({\mathscr{R}})}({\mathscr{F}}, \Omega^1_{\mathscr{R}}\otimes {\mathscr{F}}[1])$. To get a more hands-on description, assume that each of the modules ${\mathscr{F}}^i$ forming our complex already comes equipped with a connection. We call this a [*pre-connection*]{} on ${\mathscr{F}}$, and denote it by ${\nabla\mkern-12mu/\mkern2mu}_{\mathscr{F}}$. Its failure to commute with the differential $d_{\mathscr{F}}$ gives rise to a chain map $$\label{eq:cochain-atiyah}
\begin{aligned}
& \mathit{at}({\nabla\mkern-12mu/\mkern2mu}_{\mathscr{F}}): {\mathscr{F}}\longrightarrow \Omega^1_{\mathscr{R}}\otimes {\mathscr{F}}[1], \\
& \mathit{at}({\nabla\mkern-12mu/\mkern2mu}_{\mathscr{F}})({\mathscr{f}}) = (\mathit{id}_{\Omega^1_{\mathscr{R}}} \otimes d_{\mathscr{F}})({\nabla\mkern-12mu/\mkern2mu}_{{\mathscr{F}}} {\mathscr{f}}) - {\nabla\mkern-12mu/\mkern2mu}_{{\mathscr{F}}}(d_{\mathscr{F}}{\mathscr{f}}),
\end{aligned}$$ which is the boundary homomorphism for , hence represents $\mathit{At}({\mathscr{F}})$. We call $\mathit{at}({\nabla\mkern-12mu/\mkern2mu}_{\mathscr{F}})$ the [*Atiyah cocycle*]{} of ${\nabla\mkern-12mu/\mkern2mu}_{\mathscr{F}}$. Suppose that ${\mathscr{F}}_0,{\mathscr{F}}_1$ are complexes of modules, with ${\mathscr{F}}_0$ consisting of projective modules. Then we have a short exact sequence [@dold Theorem VI.10.11] $$\label{eq:dold}
0 \rightarrow \mathit{Ext}^1(H({\mathscr{F}}_0),H({\mathscr{F}}_1))[-1] \longrightarrow H(\mathit{hom}({\mathscr{F}}_0,{\mathscr{F}}_1)) \longrightarrow \mathit{Hom}(H({\mathscr{F}}_0),H({\mathscr{F}}_1)) \rightarrow 0.$$ In particular:

\[th:proj\] If ${\mathscr{F}}_0$ is a complex of projective ${\mathscr{R}}$-modules, and ${\mathscr{F}}_1$ an acyclic complex of ${\mathscr{R}}$-modules, then $\mathit{hom}({\mathscr{F}}_0,{\mathscr{F}}_1)$ is again acyclic.

In the terminology of [@spaltenstein88], this means that (unbounded) complexes of projective ${\mathscr{R}}$-modules are $K$-projective.

\[th:homotopy-idempotent\] Let ${\mathscr{F}}$ be a complex of projective ${\mathscr{R}}$-modules, and $c: {\mathscr{F}}\rightarrow {\mathscr{F}}$ a chain map which 