In this article we survey the basic properties of p −e -linear endomorphisms of coherent OX -modules, i.e. of OX -linear maps F * F − → G where F , G are OX -modules and F is the Frobenius of a variety of finite type over a perfect field of characteristic p > 0. We emphasize their relevance to commutative algebra, local cohomology and the theory of test ideals on the one hand, and global geometric applications to vanishing theorems and lifting of sections on the other. p −1 -LINEAR MAPS IN ALGEBRA AND GEOMETRY 5 Exercise 2.7. Suppose that X is a projective variety with ample line bundle L and suppose that F is a coherent sheaf on X. Set S = i∈Z H 0 (X, L i ) to be the section ring with respect to L and set M = i∈Z H 0 (X, F ⊗