Speaker: Alapan Mukhopadhyay
Affiliation: EPFL
Title of Talk: Frobenius and Homological Algebra
Date: May 20, 2026
Time: 11:14:00 Hours
Venue: AG-77

Abstract: The Frobenius endomorphism 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 the Frobenius endomorphism witnesses this homological obstruction to smoothness. This provides an explanation for 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. Time permitting, we will discuss examples hinting relationship between generators of derived categories and the global geometry of the underlying projective variety. Part of the talk will report a joint work with Matthew Ballard, Patrick Lank, Srikanth Iyengar and Josh Pollitz.