Generators for derived categories in positive characteristic

One of the fundamental ways of understanding an algebraic structure is to find basic elements which can build all others through allowed operations, ie generators. For triangulated categories, where the allowed operation is taking a mapping cone, such generation was introduced by Bondal and Van den Bergh and expanded upon by Rouquier and others. 

Identifying generators, in particular ones that are of an elementary form and generate fast, can be difficult. In this talk, I will discuss how the action of Frobenius in positive characteristic can be used. As an application, for an F-finite Noetherian algebra $R$, $F_\ast^e R$ is a generator for $D^b(\operatorname{mod} R)$ for sufficiently large $e$. 

We will discuss joint work with Pat Lank, Srikanth Iyengar, Josh Pollitz, and Alapan Mukhopadhyay from arXiv:2303.18085.