In this chapter,
is a fixed prime number, and all rings are assumed to have characteristic
, unless explicitly mentioned otherwise. We review the notion of tight closure due toHochster and Huneke (as a general reference, we will use ). The main protagonist in this elegant theory is the
-th power Frobenius map. We will focus on five key properties of tight closure, which will enable us to prove, virtually effortlessly, several beautiful theorems. Via these five properties, we can give a more axiomatic treatment, which lends itself nicely to generalization, and especially to a similar theory in characteristic zero (see Chapters 6 and 7).