Spectral Spaces

Dickmann, Max; Schwartz, Niels; Tressl, Marcus

doi:10.1017/9781316543870

Cited by 66 publications

(101 citation statements)

References 3 publications

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“…Recall that, for r ∈ R, α r + denotes the prime cone consisting of all polynomials that are non-negative on some interval [r, r + ε), ε > 0; clearly, supp(α r + ) = {0}. Likewise, α r denotes the prime cone {F ∈ R[X] | F (r) ≥ 0}; we have supp(α r ) = (X − r), the ideal of R[X] generated by X − r. For more details, see [BCR,Example 7.1.4(b), p. 135], or [DST,13.1.8,.…”

Section: Remarkmentioning

confidence: 99%

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Fans in the theory of real semigroups

Dickmann

Petrovich

2020

Dissertationes Math.

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In [DP1] we introduced the notion of a real semigroup (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semireal rings. Two important classes of RSs were studied at length in [DP2], [DP3]. In this paper we introduce and develop the theory of RS-fans, a third class of RSs providing a vast generalization of homonymous notions previously existing in field theory and in the theories of abstract order spaces and of reduced special groups. The Introduction briefly reviews the existing theories of fans and describes the contents of the paper. Acknowledgements. The authors wish to thank F. Miraglia for his careful reading of and fruitful comments on an advanced draft of this paper, as well as for his invaluable help in preparing the final version of the manuscript. Thanks are also due to the referee for a meticulous examination of the submitted text and many useful suggestions, leading to a substantial improvement of the paper and, even, to some new results. The first author was partially supported by a "Projet de mobilité international Sorbonne Paris Cité-Argentine-Brésil 2014".

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Section: Remarkmentioning

confidence: 99%

“…Spectral spaces with this property are called hereditarily normal in [DP3, 1.1(B), p. 365], and spectral root systems in[DST, §8.5, pp. 290 ff].…”

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Fans in the theory of real semigroups

Dickmann

Petrovich

2020

Dissertationes Math.

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“…We work in the constructible topology throughout. Note that the finitely generated subfields of K of Kronecker dimension one are global fields; we freely use results on localities of global fields which are standard in number theory, see for instance [25,Chapter II,§8]. As S(K ) is the inverse limit of S(K 0 ) for the finitely generated subfields K 0 of K , and S abs (K 0 )\S val (K 0 ) is finite for all of these, the space S(K ) is compact Hausdorff.…”

Section: Kronecker Dimension One and Reduction To Finitely Generated mentioning

confidence: 99%

Approximation theorems for spaces of localities

2020

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The classical Artin–Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.

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“…Dickmann et al [2019]). A lattice homomorphism f : D → D ′ is surjective if and only if the spectral map f −1 : SpecD ′ → SpecD is a homeomorphism onto its image.…”

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confidence: 99%

More on a curious nucleus

Haykazyan

2020

Journal of Pure and Applied Algebra

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Spectral Spaces

Cited by 66 publications

References 3 publications

Fans in the theory of real semigroups

Fans in the theory of real semigroups

Approximation theorems for spaces of localities

More on a curious nucleus

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