cex-an-unbounded-complex-of-projectives-that-is-not-k-projective.md

Set $R=\mathbb{Z}\mathrm{/}4$, and take $P^i=R$ and $d^i=2$ for every integer $i$. The terms are free of rank one and thus projective (a map out of $R$ lifts by lifting its value at $1$). Also $d^{i+1}{d}^i=4=0$, and $\ker d^i=2R=\mathrm{im}{d}^{i-1}$. Thus $P$ is an acyclic doubly infinite complex.

Every proposed homotopy $h^i:R\to R$ is multiplication by an element $a_i$. The identity-homotopy equation at degree $i$ would require $1=2a_i+2a_{i+1}$, impossible modulo two. Hence $1_P$ is nonzero in ${\mathrm{Hom}}_K(P,P)$. Since $P$ itself is acyclic, this contradicts the vanishing required for a K-projective source. The obstruction is an equation at every degree, not evidence from a finite truncation.