Milne - Étale cohomology

Algebraic geometry

"The purpose of this book is to provide a comprehensive introduction ot the étale topology, sheaf theory, and cohomology... The étale topology was initially defined by A. Grothendieck and developed by him with the aid of M. Artin and J.-L. Verdier in order to explain Weil's insight that, for polynomial equations with integer coefficients, the complex topology of the set of complex solutions of the equations should profoundly influence the number of solutions of the equations modulo a prime number. In this, the étale topology has been brilliantly successful."

Chapter I. Étale Morphisms

Section I.1. Finite and Quasi-Finite Morphisms

• If, moreover, $\Gamma(f^{-1}(U), \mathcal{O}_y)$ is a finite $\Gamma(U, \mathcal{O}_X)$-algebra...: i.e. if the former is finitely generated as a module over the latter.

• ... and let L be a finite field extension of R(x)...: i.e. a finite-dimensional extension.