Generalized Kummer coverings

Consequently these quotients are flat over S and commute with arbitrary base change. 

Proof Consider the case Zn / K the case jpt n / K is analogous. First note that the existence of these quotients is guaranteed since 

B is the subsheaf of An consisting of the invariants under the group action, i.e., the kernel of the couple 

With the usual identifications D( Zn)= y n, etc. we have that q corresponds with the homomorphism p, from 1.1.4 as follows from the description of p, on the Og-Algebras given in 1.1.4. Consider on the other hand the homomorphism of Og-Algebras 

This identifies A with a sub-Algebra of A, . Furthermore it is easily checked that v is compatible with the action of Jftn, reap, and the homomorphism (p, . image of h under q 

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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. 

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... 

Generalized Kummer coverings over strict local rings... 

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