We study the scaling dimension (or the similarity dimension) of the perfect derived category of a smooth compact dg algebra called the Serre dimension. It is expected that the infimum of the Ikeda-Qiu’s global dimsion function on the space of stability conditions also gives another “good” notion of dimension. One of our results is that its infimum is always greater than or equal to the Serre dimension. Motivated by the ADE classification of the 2-dimensional N=2 SCFT with \widehat{c}<1, we also give a characterization of the derived category of Dynkin quivers in terms of the Serre dimension and the global dimension function. This is a joint work in progress with Kohei Kikuta and Genki Ouchi.