The purpose of these lectures is to give a gentle introduction to Frobenius splitting, or more broadly “Frobenius techniques”, for beginners. Frobenius splitting has inspired a vast arsenal of techniques in commutative algebra, algebraic geometry, and representation theory. Many related techniques have been developed by different camps of researchers, often using different language and notation. Although there are great number of technical papers and books written over the past forty years, many of the most elegant ideas, and the connections between them, have coalesced only in the past decade. We wish to bring this emerging simplicity to the uninitiated.

Our story of Frobenius splitting begins in the 1970s, with the proof of the celebrated Hochster–Roberts theorem on the Cohen–Macaulayness of rings of invariants [Hochster and Roberts 1974]. This proof, in turn, was inspired by Peskine and Szpiro’s ingenious use [1973] of the iterated Frobenius—or p-th power—map to prove a constellation of “intersection conjectures” due to Serre. Mehta and Ramanathan [1985] coined the term “Frobenius splitting” a decade later in a beautiful paper which moved beyond the affine case to prove theorems about Schubert varieties and other important topics in the representation theory of algebraic groups. Although these “characteristic p techniques” are powerful also for proving theorems for algebras and varieties over fields of characteristic zero, we focus on the prime characteristic case, since the technique of reduction to characteristic p has now become fairly standard.