Kummer coverings

The S-group jHn 5 (shortly tn) operates on Z over S. Again by general principles of EGA Oj-j-j- 8.2 it suffices to see that for a variable S-scheme T the group jnn(T) operates on the set Z(T) = Homg(T,2) 

One should remark here that a morphism of two couples (Yu G ) (i= 1,2), consisting of S-schemes Y and of S-group schemes (1 operating over S on Y (i= 1,2), always means a couple (u, p) with a S-group homomorphism 

It follows immediately that the notion of Rummer covering is stable with respect to arbitrary base change provided the inverse 

Lemma 1,2.5. Let S, a be as before with regular sections a - Let f Y —V S be a finite morphism and A = fw(Oj) the corresponding Og- Algebra. Then the following conditions are equivalent  

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G operates over S, is called a generalized Kummer covering of S. 

Remarks. a) If (Y,G) is a generalized Kummer covering of S, then it follows from 1.1.6 that G is an Stale surjective covering of S and f Y —> S is finite and flat (1.3.2 ), hence open and closed if S is locally noetherian. Also it is easily seen that G operates transitively on the fibers. 

Finally we want to determine the automorphisms of a (generalized) Kummer covering.For this purpose we have the following - somewhat more general - results... 

In this section we study generalized Kummer coverings under the assumption that the base and/or the sections have special properties. 

Proposition 1.7.2. Let S be a locally noetherian scheme and (Y,G) a generalized Kummer covering of S relative to the sections Then... 

In the following we are primarily concerned with generalized Kummer coverings over a locally noetherian scheme S relative to a set of divisors.(see 1.3.9 c), Moreover we assume that the divisors have "normal crossings". We first recall the definition. 

Y,G) is a Kummer covering then Y is regular above the points of... 

X is a union of generalized Kummer coverings and even that X... 

Since B is strict hensel (EGA IV 18.5 16), we have by 2.3.4- that such a tamely ramified covering is a disjoint union of generalized Kummer coverings. From this, and from the fact that in Spec B we have D= + Dp with and regular divisors with normal crossings, we get... 

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