Inertia groups and fibers

Abstract

Let X and Y be proper, normal, connected schemes over a field K, and let f : X → Y be a finite, flat K-morphism which is generically Galois (i.e., the extension of function fields K(Y ) ↪→ K(X) is Galois) with Galois group G. It is well-known that for the Zariski-open complement U ⊆ Y of the branch locus of f , the map f−1(U) → U is a (right) G-torsor. Thus, for any y ∈ U and x ∈ f−1(y), the extension of fields K(x)/K(y) is Galois and the stabilizer in G of x maps isomorphically to the Galois group Gal(K(x)/K(y)). In particular, when the fiber f−1(y) is irreducible, then Gal(K(x)/K(y)) = G. If K is a global field, it is natural to ask how the injection Gal(K(x)/K(y)) ↪→ G relates ‘arithmetic’ inertia groups in Gal(K(x)/K(y)) with ‘geometric’ inertia groups in G, corresponding to ramification in the map f . The same question can be asked more generally when K is the function field of a connected, normal, noetherian scheme S with positive dimension, where ‘arithmetic’ ramification in Gal(K(x)/K(y)) corresponds to ramification in K(x) of the valuations on K(y) arising from codimension 1 points of the normalization of S in K(y). A special case of this question was investigated by S. Beckmann. She considered the case when K is a number field (with integer ring OK), Y = PK , and X is a geometrically connected curve over K. Let a1, . . . , am be the finitely many branch points of f . Since K has characteristic 0, so f is tamely ramified over each ai, the inertia groups of f over the ai’s are cyclic subgroups of G. For any closed point y ∈ PK , it is not difficult to show that the scheme-theoretic closure {y} in P1OK , which is proper over OK , is also quasi-finite and therefore finite over OK . For example, if y, y′ ∈ PK are distinct closed points, then {y} ∩ {y′} is artinian. In particular, when y is a K-rational point distinct from the ai’s, the intersection {y} ∩ {ai} is an artinian closed subscheme of {y} ' Spec(OK). Let Ip(y, ai) ≥ 0 denote the length of the part of {y} ∩ {ai} which lies over p ∈ Spec(OK), so obviously Ip(y, ai) = 0 for all but finitely many p (depending on y and ai). Let Σf denote the finite set of primes p of OK at which one of the following occurs: • some K(ai)/K is ramified at p, • Ip(ai, aj) > 0 for some i 6= j (i.e., the closures {ai} and {aj} in P1OK meet over p), • the p[t]-adic valuation on K(PK) = K(t) is ramified in K(X), • p divides the degree of f . Note that Σf can be effectively determined and depends only on the the geometry of f and the arithmetic in some of the fibers of f . Beckmann proved the following result: