Frobenius or the $p$-th power map is crucial in defining singularity classes in characteristic $p > 0$, especially those appearing in the birational classification of algebraic varieties. On the other hand, the obstruction to smoothness is homological, according to a celebrated theorem of Serre. In this talk, we will show that Frobenius witnesses this homological obstruction to smoothness. This will explain the effectiveness of Frobenius in detecting singularities, from a homological point of view. The key will be to produce (explicit) generators of the bounded derived category of a variety in characteristic $p > 0$ from perfect complexes using the Frobenius pushforward functor. Our results recover earlier characterizations of smoothness using Frobenius, such as Kunz’s theorem. Part of the talk will report a joint work with Matthew Ballard, Patrick Lank, Srikanth Iyengar and Josh Pollitz.