Fibrés vectoriels sur un polydisque ultramétrique

Gruson, Laurent

doi:10.24033/asens.1160

Cited by 35 publications

(19 citation statements)

References 7 publications

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“…It represents an affinoid approximation of the nonquasi-compact rigid analytic space (A 1 A ) an since lim t →πt A t = H 0 ((A 1 A ) an , O). Note that the latter non-affinoid K-algebra is harder to control, compare [10,Ch. 5] and [3].…”

Section: Introductionmentioning

confidence: 99%

Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Kerz¹,

Saito²,

Tamme³

2019

J. Inst. Math. Jussieu

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We suggest an analog of the Bass-Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles K0.

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Section: Introductionmentioning

confidence: 99%

Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Kerz¹,

Saito²,

Tamme³

2019

J. Inst. Math. Jussieu

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“…3 below). Here we just outline an even simpler proof (not appealing to Tate algebras and reduction to algebraically closed fields) based on the method of passing to graded objects proposed in [6] and on the new concept of a v-approximable norm considered in the next section.…”

Section: 33 Proof Of Prop 2])mentioning

confidence: 98%

Stability preservation theorems

Ershov

2008

Algebra Logic

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The basic result of the paper is the main theorem worded as follows. Let F = F, R be a valued field such that F R has characteristic p > 0 and let F 0 ≥ F be an extension of valued fields satisfying the following conditions:. Then the property of being stable for F implies being stable for F 0 .I. The definitions of a valued field, of a Henselization, etc., can be found in [1, Chap. 1]. For the notion of a stable valued field and its properties needed in what follows, we ask the reader to consult [2].The basic result of the paper is the following:MAIN THEOREM. Let F = F, R be a valued field such that F R has characteristic p > 0 and let F 0 ≥ F be an extension of valued fields satisfying the following conditions:(iv) F 0 is algebraic over F (B 0 ∪ B 1 ). Then the property of being stable for F implies being stable for F 0 .Remark. In proving the theorem, we will establish that B B 0 ∪ B 1 is a transcendence basis of the field F 0 over F .We have COROLLARY. If F is stable, and F 0 ≥ F is a tame extension (see [3]), then F 0 is stable. The proof of the main theorem is derived by reducing it to three special versions (Thms.

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“…In the present paper we come up with an extension of the notion of a (ultrametric) norm on groups, rings, algebras, and vector spaces, as defined in [5], to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using -as in the general theory of valuations -the version of a logarithmic norm; see [6]). We define the concepts of being orthogonal and of being Cartesian, preserving all the basic properties specified in [5].…”

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confidence: 99%

Stable valued fields

Ershov

2007

Algebra Logic

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We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Güntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using -as in the general theory of valuations -the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields.In the paper we deal with a class of valued fields, which have different names (for definition of a valued field, see [1]). In [2], such fields are termed defectless; in [3], stable. Here, we opt for the latter term.For the case of Henselian valued fields, the corresponding property in [4] was called algebraic completeness. In [5], stable fields were defined (for valued fields with Archimedean value groups) in terms of Cartesian spaces with norm, and it was proved that this concept is equivalent to being defectless (see [5, Sec. 3.6, Prop. 6]).In the present paper we come up with an extension of the notion of a (ultrametric) norm on groups, rings, algebras, and vector spaces, as defined in [5], to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using -as in the general theory of valuations -the version of a logarithmic norm; see [6]). We define the concepts of being orthogonal and of being Cartesian, preserving all the basic properties specified in [5].Our main result is Theorem 2, which extends [5, Sec. 3.6, Prop. 6] to the general case. This theorem allows us to use the machinery of Cartesian spaces propounded in [5] to obtain further results on stable valued fields.I. Let Γ, +, , 0 be a linearly ordered Abelian group and Γ ω = Γ ∪ {ω} be the semigroup obtained by adjoining an element ω, for which we put a + ω = ω + a = ω, a ω for all a ∈ Γ, and ω + ω = ω. If G, +, 0 is an Abelian group, then a (ultrametric) (Γ-)norm on G is any mapping v : G → Γ ω such that:(1) v(g) = ω ⇔ g = 0, g ∈ G;(2) v(g 0 − g 1 ) min{v(g 0 ), v(g 1 )}. If R = R, +, ·, 0, 1 is a ring then a (Γ-)norm on R is any mapping v : R → Γ ω such that v is a norm on the group R, +, 0 , and the following conditions hold: *

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Fibrés vectoriels sur un polydisque ultramétrique

Cited by 35 publications

References 7 publications

Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Towards a Non-Archimedean Analytic Analog of the Bass–quillen Conjecture

Stability preservation theorems

Stable valued fields

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