Homological properties of the Algebra of compact operators on a banach space

The conditions on a Banach space E under which the algebra 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 K(E), they are closely associated with the approximation property for E. The class of spaces E such that 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.