Homology of perfect complexes

Yes. Let $... 0\to P_r \to ... \to P_0 \to 0 ...$ be a complex of projective modules of finite type and denote by $Z_*$ the cycles. If $n=0$ it is clear. If not, $0\to Z_1\to P_1\to P_0\to 0$ is exact and so $Z_1$ is projective and of finite type. Then if $n=1$, $H_1(P)$ is of finite type. If $n\neq 1$, $0\to Z_2\to P_2\to Z_1\to 0$ is exact. And so on.

So the "last" nonzero homology module is of finite type.