Let be a Dedekind domain, let denote its quotient field, let be a finite separable field extension of degree , let be the integral closure of in . Then is in particular a Dedekind domain
be the induced map between the ring spectra called ramified covering
Let be a maximal prime ideal. Then the ideal in has a unique product decomposition
with different . It is custom to introduce some classifying vocabulary depending on the kind of this decomposition and the degree of the field extension : the are called ramification indices and the are called inertia degrees (see e.g. the German wikipedia for more details). These satisfy the fundamental identity and every point has preimages. If has preimages then is called ramified.
commutes. is an example of a cover automorphism (also called cover transformation or Deck transformation). Since is an equivalence of categories we have an isomorphism
where the object to the right we have the group of cover automorphism.
One can show that there is a maximal unramified extension ( and the extensions being separable) of and the scheme where denotes the integral closure of in . Then satisfies the axioms of a universal covering and consequently we define on the side of schemes the fundamental group