Homological dimension of cyclic Banach modules and homological characterization of metrizable compacta

It is proved that if the relative homological dimension of the $A$ module $A/I$ is less than two, then the spectrum (which is not necessarily complementable) of the closed ideal $I$ of the Banach algebra $A$ is paracompact. As an application in the realm of relative homological theory a criterion for metrizability of compacta is obtained.