  For a topological space $X$, we introduce a criterion for the $\rm FI$ module
$H^i({\rm Conf}_n(X))$ to be finitely generated and give several applications.
For instance, if $C$ is a finite connected $CW$ complex, then $X = C \times
\mathbb{R}^2$ satisfies the criterion. Our main tool is a spectral sequence
that we derive from the homological algebra of the partition lattice.
