The global dimension of $\mathrm L$-$\mathrm R$ twisted smash products

We construct the new algebra $A\sharp H$ of an $H$-bimodule algebra $A$ called the $\mathrm L$-$\mathrm R$ twisted smash product, and give the duality theorem for $\mathrm L$-$\mathrm R$ twisted smash products which extends the duality theorem for smash products given by Blattner and Montgomery. Furthermore, by using the duality theorem for $\mathrm L$-$\mathrm R$ twisted smash products, we establish the relationship of global dimension between the $H$-bimodule algebra $A$ and its $\mathrm L$-$\mathrm R$ twisted smash product $A\sharp H$.