Additivity of Homological Dimensions for a Class of Banach Algebras

Let $\Omega$ be a metrizable compact space. Suppose that its derived set of some finite order is empty. Let $B$ be a unital Banach algebra, and let $\widehat{\otimes}$ stand for the projective tensor product. We prove the additivity formulas $\operatorname{dg}C(\Omega)\widehat{\otimes}B=\operatorname{dg}C(\Omega)+\operatorname{dg}B$ and $\operatorname{db}C(\Omega)\widehat{\otimes}B=\operatorname{db}C(\Omega)+\operatorname{db}B$ for the global homological dimension and the homological bidimension. Thus these formulas are true for a new class of commutative Banach algebras in addition to those considered earlier by Selivanov.