Tamely ramified coverings of formal schemes

Let cf be a locally noetherlan. normal formal scheme and D a closed subset on. Note that for every s / Spec Oy is a normal scheme. We say that D has codimension at least one if for every seCf the corresponding closed subset in Spec 2 g (see 3 1. ) Nas codimension at least one. Note that it may happen that D is of codimension at least one and that supp(D)= i Let furthermore f jE— be a finite morphism and write f (Oj )= by we denote the stalk at s clearly is a finite Oy g-algebra (cf.3.2.3 ). 

Definition A finite morphism f X = Spf— is called a tamely ramified covering of S relative to D if for every we have that 

Remarks 1) By abuse of language we often callZ itself a tamely 

2) For s supp D we have by definition that Spec — Spec 0  

Lemma 4.1.3. Let d = Spf A, with A a J-adic ring (always noetherlan). 

Remarks 1) By abuse of language we often callZ itself a tamely ramified covering of relative to D, or a covering of tame over D. We say shortly . tame over (relative to D). 

Proof First note that S is normal (3.1.3 ) Let self consider the 

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