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

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