Generators of bounded derived categories in prime characteristics

Since Bondal- van den Bergh’s work on the representability of func-tors, proving existence of strong generators of the bounded derived category ofa scheme has been a central problem. While for a quasi-excellent, separatedscheme the existence of strong generators is established, explicit examples ofsuch generators are not common. In this talk, we show that explicit generatorscan be produced in prime characteristics using the Frobenius pushforward func-tor. As a consequence, we will see that for a domain R with finite Frobeniusendomorphism, R1/pn - for large enough n- generates the bounded derived cate-gory of R-modules. This recovers Kunz’s characterization of regularity in termsof flatness of Frobenius. Part of the talk is based on a joint work with MatthewBallard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.