By Alexander Grothendieck

ISBN-10: 3540054995

ISBN-13: 9783540054993

ISBN-10: 3540368574

ISBN-13: 9783540368571

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Extending extending This implies, that every v a l u a t i o n w in L from a v a l u a t i o n w~in a s~mms~Id of L ~ K j and K v i . 2. 1. Spec B at least normal one and U= S - D . , dim ~S,s = l) the local ring ~S,s is a discrete valuation rin~. 2. 2. A morphism f: X m ~ S (or by abuse of language X is a tamely ramified coverin G of S relative to the set D if: l) f is finite, 2) f is ~tale over U, 3) every irreducible component of X dominates an irreducible component of S, a) X is normal and 5) for s~D of codimension over ~S,s 1 in S we have that X is tamely ramified (see remark 3 below).

Definition A finite morphism f:~ = S p f ~ ramified coverin5 o f ~ we have that --~ Spec ~ , s is a tamely ramified covering of Spec ~ , s l) is called a tamely relative to D if for every s ~ Spec ~ s Remarks: --~ ~ relative to D s. By abuse of language we often call ~ itself a tamely ramified covering of ~ relative to D, or a covering of ~ tame over D. We say shortly: ~ 2) tame over ~ (relative to D). For s ~ supp D we have by definition that Spec ~ s --~ Spec 0~, s is an ~tale covering. 1. 7 .

2. 1. Let ~ be a locally noetherian, normal and connected formal scheme and (Di)i6 1 a locally finite set of regular divisors with normal crossinss on ~ . Put D= formal ~ - s c h e m e s E D i. Consider in the category of ieI the following full subcategories: Rev(#): the formal ~-schemes which are finite over ~, RevEt(#): the formal ~ -schem~ which are ~tale coverings of ~ RevD(~): the formal ~ -schemes which are tamely ramified over relative to D. We have the following inclusions: RevEt(J) ~ RevD(~)~ Rev(~).

### The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme by Alexander Grothendieck

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