  The conditions on a Banach space, $E$, under which the algebra,
$\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically
unital are investigated. These homological properties are related to
factorization in the algebra and it is shown that, for $\mathcal{K}(E)$, they
are closely associated with the approximation property for $E$. The class of
spaces, $E$, such that $\mathcal{K}(E)$ is known to be right flat and
homologically unital is extended to include spaces which do not have the
bounded compact approximation property.
