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## On different notions of tameness in arithmetic geometry (2009)

Citations: | 16 - 4 self |

### Citations

629 |
Elements de geometrie algebrique. IV.. Etude locale des schemas et des morphismes des schemas IV.
- Grothendieck
- 1967
(Show Context)
Citation Context ... finite type over a field or over a Dedekind domain with infinitely many prime ideals is pure-dimensional. A proper scheme over a pure-dimensional universally catenary scheme is pure-dimensional, see =-=[EGA4]-=-, IV, 5.6.5. The affine line A1 over the ring of p-adic integers gives an Zp example of a regular scheme which is not pure-dimensional. Let from now on S be an integral, pure-dimensional, separated an... |

191 |
Revetements etales et groupe fondamental.
- Grothendieck
- 1971
(Show Context)
Citation Context ...nt notions of curve-, divisor-, discrete-valuation- and chain-tameness and tame covering means a finite, étale morphism which is tame. The tame coverings of X satisfy the axioms of a Galois category (=-=[SGA1]-=-, V, 4). After choosing a geometric point ¯x of X, we have the fibre functor (Y → X) ↦→ MorX( ¯x, Y) from the category of tame coverings of X to the category of sets, whose automorphism group is calle... |

145 |
Algebraic geometry and arithmetic curves, Oxford Graduate Texts in
- Liu
- 2002
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Citation Context ...d = dimS X. Using Lemma 3.5, we find a wildly ramified geometric discrete rank 1 valuation v on k(X). Choose any normal compactification ¯X of X. As Y → X is étale, the center of v lies in ¯X � X. By =-=[Liu]-=- §8, ex. 3.14, after blowing up ¯X in centers contained in ¯X � X and finally normalizing, we find a normal compactification ¯X of X such that v is the valuation associated to a point x ∈ ¯X � X of co... |

115 |
Cohomology of number fields,
- Neukirch, Schmidt, et al.
- 2008
(Show Context)
Citation Context ...ices i from the notation and call B|A tamely ramified if for every maximal ideal m ⊂ A the inertia group Tm(B|A) is of order prime to the characteristic of A/m. The following claims are standard, cf. =-=[NSW]-=-, Theorem 6.1.10. Claim 1: tr B|A : B → A is surjective if and only if B|A is cohomologically tamely ramified. Proof of claim 1. For a maximal ideal m ⊂ A, we denote the henselization of : B → A is Am... |

95 |
Desingularization of two-dimensional schemes,
- Lipman
- 1978
(Show Context)
Citation Context ... F. Pop for helpful discussions on the subject, and T. Chinburg for his proposal to consider cohomological tameness. 2 The Key Lemma We start by recalling the following desingularization results. See =-=[Lip]-=- for a proof of (i) and [Sha], Lecture 3, Theorem on p. 38 and Remark 2 on p. 43, for (ii). We denote the set of regular points of a scheme X by X reg . Proposition 2.1. Let X be a two-dimensional, no... |

49 |
Murre The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme. Springer Lect.
- Grothendieck, P
- 1971
(Show Context)
Citation Context ...ith respect to another regular compactification. In the case that the function field of S is absolutely finitely generated, we show the following tame variant of a finiteness theorem of Katz and Lang =-=[KL]-=-. Theorem 1.3 (see Theorem 7.1). Let S be an integral, pure-dimensional and excellent base scheme whose function field is absolutely finitely generated. Let f : Y → X be a smooth, surjective morphism ... |

29 |
Class field theory for curves over local fields
- Saito
- 1985
(Show Context)
Citation Context ...struct a similar example as in Example 2, but with arithmetic surfaces instead of varieties. This example assumes some familiarity with S. Saito’s class field theory for curves over local fields, see =-=[Sa]-=-. Let p ̸= 2 be a prime number, k|Qp a p-adic field, E → k an elliptic curve and E → P1 k the degree 2 covering defined by a Weierstraß model. Then the normalization W of P1 k in k(E) is a normal mode... |

17 |
Tame actions of group schemes: integrals and slices
- Chinburg, Erez, et al.
- 1996
(Show Context)
Citation Context ...he semi-local ring ¯Y,π −1 (x) is a cohomologically trivial G-module. The notion of numerical tameness has been considered in [Ab], [CE], [Sc1] and [Wi]. Cohomological tameness has been considered in =-=[CEPT]-=- (in the more general context of group scheme actions). Our second result is : Theorem 1.2 (see Theorems 5.3 and 6.2). Numerical tameness and cohomological tameness are equivalent and imply valuation ... |

11 |
Tame coverings of arithmetic schemes.
- Schmidt
- 2002
(Show Context)
Citation Context ...ion-tameness: every nonarchimedean valuation of k(X) with center outside X is tamely ramified in k(Y)|k(X). The notion of curve-tameness has been considered in [Wi] and the notion of chaintameness in =-=[Sc1]-=-. Curve-tameness is the maximal definition of tameness which is stable under base change and extends the given definition for curves. Valuation tameness is obviously stronger than divisor-tameness and... |

6 |
The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces
- Abbes
(Show Context)
Citation Context ...e residue field k(y). cohomological tameness: for every x ∈ ¯X � X the semi-local ring ¯Y,π −1 (x) is a cohomologically trivial G-module. The notion of numerical tameness has been considered in [Ab], =-=[CE]-=-, [Sc1] and [Wi]. Cohomological tameness has been considered in [CEPT] (in the more general context of group scheme actions). Our second result is : Theorem 1.2 (see Theorems 5.3 and 6.2). Numerical t... |

1 |
Singular homology of arithmetic schemes. Algebra & Number Theory 1
- Schmidt
- 1979
(Show Context)
Citation Context ...th respect to v2. Consider the polynomial F(X) = X 2 − f = X 2 − T 2 + 16. We have F(T) ≡ 0 mod 16 and the derivative F ′ (T) = 2T has the exact 2-valuation 1. By the usual approximation process (cf. =-=[Se]-=- 2.2. Theorem 1), we see that f has a square root in K2. Hence the ramification locus of XL → X is exactly D, and since D is the sum of horizontal prime divisors, the morphism U L → U is tamely ramifi... |