Showing that chain complex of projectives homotopic to zero chain complex

Question

If I have a chain complex $ \dots\rightarrow P_{-2} \rightarrow P_{-1} \rightarrow P_{0} \rightarrow 0 \rightarrow \dots$ of $R$-modules, for some ring $R$, where every $P_i$ is projective how can I show that this chain complex is homotopic to the zero chain complex? Is this even true? Sorry for the nonexistent effort but I'm really stumped by this problem.

Answer 1

This is obviously false: a complex of projective modules normally does not have trivial (co)homology. Think for example of algebraic topology, where all the calculations involve complexes of free modules.

If your complex is exact (i.e. $H^\bullet (P^\bullet) = 0$), then this is true. The null homotopy may be constructed inductively, using the lifting property that characterizes projective objects.