A virtually ample field that is not ample

Srinivasan, Padmavathi

doi:10.1007/s11856-019-1934-y

Cited by 4 publications

(4 citation statements)

References 3 publications

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“…(1) L is large, (2) the E L -topology on L is not discrete, (3) the E L -topology on V (L) is not discrete when V is an L-variety with V (L) infinite. By [Sri19] there are non-large fields with large finite extensions. If L is large and K is not then the E K -topology on K d is discrete, hence the Ext L/K (E K )-topology on L is discrete, and the E L -topology on L is not discrete.…”

Section: Behaviour Of E K Under Field Extensionmentioning

confidence: 99%

When is the étale open topology a field topology?

Dittmann¹,

Walsberg²,

Ye³

2022

Preprint

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We investigate the following question: Given a field K, when is the étale open topology E K induced by a field topology? On the positive side, when K is the fraction field of a local domain R = K, using a weak form of resolution of singularities due to Gabber, we show that E K agrees with the R-adic topology when R is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gthenselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology τ, the étale open topology is induced by τ if and only if τ is gt-henselian and some non-empty étale image is τ-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field K, E K is never induced by a field topology.

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Section: Behaviour Of E K Under Field Extensionmentioning

confidence: 99%

When is the étale open topology a field topology?

Dittmann¹,

Walsberg²,

Ye³

2022

Preprint

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“…There is a field K and a finite extension L/K such that L is large and K is not large [10]. Fact 4 is [7,Theorem 5.8].…”

Section: Algebraic Extensionsmentioning

confidence: 99%

Topological proofs of results on large fields

Walsberg¹

2022

Comptes Rendus. Mathématique

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“…There is a field K and a finite extension L/K such that L is large and K is not large [Sri19]. Fact 2.2 is [JTWY, Theorem 4.10].…”

Section: Algebraic Extensionsmentioning

confidence: 99%

Topological proofs of results on large fields

Walsberg¹

2021

Preprint

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We use the recently introduced étale open topology to prove several facts on large fields. We show that these facts lift to a very general topological setting.Throughout K, L are fields, L is infinite, and A m K , A m L is m-dimensional affine space over K, L, respectively. A K-variety is a separated K-scheme of finite type, not assumed to be reducedbe the coordinate ring of V , and K(V ) be the function field of V when V is integral.L is large if every smooth L-curve with an L-point has infinitely many L-points. Finitely generated fields are not large. Most other fields of particular interest are either large, or are function fields over large fields, or have unknown status. Local fields, real closed fields, separably closed fields, fields which admit Henselian valuations, quotient fields of Henselian domains, pseudofinite fields, infinite algebraic extensions of finite fields, PAC fields, p-closed fields, and fields that satisfy a local-global principle are all large. Function fields are not large. It is an open question whether the maximal abelian or maximal solvable extension of Q is large. See [Pop] and [BSF14] for more background on large fields.We will give topological proofs of Facts A,B, and C below. Fact A is [Pop, Proposition 2.6].Fact A. Suppose that L is large and V is an irreducible L-variety with a smooth L-point. Then V (L) is Zariski dense in V .Facts B and C are due to Fehm. Fact B is proven in [Feh10]. Note that Fehm uses "ample" for "large" (this is one of a surprisingly large number of names used in the literature.) Fact B. Suppose that L is large, K is a proper subfield of L, and V is a positive-dimensional irreducible K-variety with a smooth K-point. Then |V (L) \ V (K)| = |L|.

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A virtually ample field that is not ample

Abstract: A field K is called ample if for every geometrically integral K-variety V with a smooth K-point, V (K) is Zariski-dense in V . A field K is virtually ample if some finite extension of K is ample. We prove that there exists a virtually ample field that is not ample.Date: October 15, 2018.

Cited by 4 publications

References 3 publications

When is the étale open topology a field topology?

When is the étale open topology a field topology?

Topological proofs of results on large fields

Topological proofs of results on large fields

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