The definition of the dimension of a triangulated category T was given by Rouquier within the last decade. This number is the minimum among the generation times of all possible generators of T. In the case of the derived category of fintely generated modules over a ring R, the generation time of R in T equals the global dimension of R.

Generalizing this result to the dg or $A_\infty$-algebra setting is the first goal of the talk. Such algebras appear naturally as endomorphism algebras of generators when one starts with a pre-triangulated $A_\infty$ category whose homotopy category is T.

With this in hand, I will discuss the $A_\infty$ version of base change inequalities. The classical formula obtained through a change of base ring aquires an additional parameter in the $A_\infty$-algebra setting. This parameter can be directly tied to the formality, or lack thereof, of the algebras involved in the base change.

We will conclude with a test of this inequality for generators on the elliptic curve, where we will make some use of Homological Mirror Symmetry.