Additivity of homological dimensions for tensor products of some Banach algebras

It is proved that if $A=C(\Omega)$, where $\Omega$ is an infinite metrizable compact space such that some finite-order iterated derived set of $\Omega$ is empty, then for every unital Banach algebra $B$ the global dimensions and the bidimensions of the Banach algebras $A\mathop{\widehat{\otimes}} B$ and $B$ are related by $\mathop{\mathrm{dg}} A\mathop{\widehat{\otimes}} B=2+\mathop{\mathrm{dg}} B$ and $\mathop{\mathrm{db}} A\mathop{\widehat{\otimes}} B=2+\mathop{\mathrm{db}} B$. Thus, a partial extension of Selivanov's result is obtained.