  Let $S$ denote the graded polynomial ring $\C[x_1,...,x_m]$. We interpret a
chain complex of free $S$-modules having finite length homology modules as an
$S^1$-equivariant map $\C^m\sm\{0\} \to X$, where $X$ is a moduli space of
exact sequences. By computing the cohomology of such spaces $X$ we obtain
obstructions to such maps, including a slight generalization of the
Herzog-K\"uhl equations.
