I apologize in advance if this question is basic. 


If $P\_{\bullet}$ is a perfect complex over say a ring $R$ such that 


1. $H\_{i}(P\_{\bullet})=0 $ if $i\neq n$
2. $H\_{i}(P\_{\bullet})=E$ if $i=n$


is $E$ a finitely generated $R$-module ? 


What can we say about the homology of a generic perfect complex in general? 



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.


