Kummer type

Remarks a) Morphisms of coverings of Kummer type are defined as ft -morphisms of the oarerings. 

Consider in the Stale topology on the fibred category C(j ) (called the category of coverings of Kiimmer type relative to ( ) defined as follows  

Proposition 6.2.3 The fibred category C( ) of coverings of Kummer type relative to is a gerbe (see [4]) for the dtale topology on with lien Moreover, the 2-cohomology class 

Remark If D is a positive divisor on and (D) the corresponding Ideal then we write C(D) (resp. c(D) instead of C(J (D)) (resp.c(0 (D)). The proof of 6.2.3 is preceded by several lemmast 

Proof This is a local question the lemma follows from 1.2.5 and. 

Now if we consider such a homomorphism Parishi locaHy on X, then 

Using the fact that cp is compatible with the action of /Zn / we 

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Corollary 6.2.7. There is an equivalence between the category C(tf) (cf) of coverings of J of Kummer type relative to (f and the category of couples f>) as described in 6.2.4- with as morphisms such Oy>-morphisms of the Modules which are compatible with the f s... 

Start with 6 C( ) (S ), i.e., is a covering of of Kummer type relative to. We have the following situation ... 

Then there exists a fj T-+ / such that ( 3, ft ) is a covering of tf of Kummer type relative to the divisor... 

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