The idempotency of the real spectrum implies the extension theorem for Nash functions

Quarez, Ronan

doi:10.1007/pl00004395

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…Then, we wait ten years to really put everything in order and produce complete solutions. This final part of the story started with [RzSh], where the methods of Commutative Algebra were fetched to the stage, and finally the series [CoRzSh1], [CoRzSh2], [Qz2], [Qz3], [CoSh2], [CoRzSh3], which appeared from 1995 to 2001 and fixed everything.…”

Section: Brief History Of the Topicmentioning

confidence: 99%

“…It is instructive for our topic the way that gap was discovered. After [CoRzSh1], Quarez found ( [Qz2]) an argument that deduced extension from idempotency, by making the ideas in the compact case available in the non-compact one. This lead to a protocolary revision of the quoted articles on idempotency, and to the unexpected finding that the proofs were incomplete.…”

Section: Abstract Nash Functionsmentioning

confidence: 99%

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Global Problems on Nash Functions.

Shiota

Coste²,

Sancho

2004

Rev. Mat. Complut.

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This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.Nash functions appeared for the first time at the early fifties, in a seminal paper by John Nash. Soon other mathematicians were interested on them, and some relevant applications found, but also soon their bad cohomological behaviour was realized. This put an end to a possible systematic development of the theory of Nash functions. Since that moment many specialists have devoted time and efforts to understand the nature of these functions, but only after fifty years we start to understand what is behind their failures, and how the problems generated by those failures should be settled. During this fifty years process the notion of Nash function has evolved in progressively more general settings, with an appealing feedback from concrete to abstract and viceversa. Moreover, a maze of surprising links have been revealed among all the questions involved. These are depicted in the Nash Labyrinth shown below, where each box contains a solved problem, each arrow is an implication, and the doubled boxes are the sources of the flow of arguments. We will try to explain the problems and their solutions in the following sections: §1. Brief history of the topic. §2. The notion of Nash function. §3. Global properties and cohomological failures. §4. Formulation of the problems. §5. Solutions in the compact case. §6. Solutions in the non-compact case. §7. Nash functions over arbitrary real closed fields. §8. Abstract Nash functions.

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Section: Brief History Of the Topicmentioning

confidence: 99%

Section: Abstract Nash Functionsmentioning

confidence: 99%

Global Problems on Nash Functions.

Shiota

Coste²,

Sancho

2004

Rev. Mat. Complut.

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“…It was claimed in [Ro] that the idempotency was true for every commutative ring A, but the proof contained a gap. On the other hand, it was proved in [Qu1] that the idempotency would give a rather direct proof of the Extension theorem for Nash manifolds over the reals, both compact and noncompact. In this discussion, [Qu1] shows also that the Artin-Mazur property implies the idempotency for arbitrary rings, and consequently we can now conclude (Section 7) that idempotency holds for every commutative ring.…”

Section: Theorem 3 (Extension) Let I Be An Ideal Of N (ω) Then Everymentioning

confidence: 99%

“…On the other hand, it was proved in [Qu1] that the idempotency would give a rather direct proof of the Extension theorem for Nash manifolds over the reals, both compact and noncompact. In this discussion, [Qu1] shows also that the Artin-Mazur property implies the idempotency for arbitrary rings, and consequently we can now conclude (Section 7) that idempotency holds for every commutative ring. This is a big detour to prove the idempotency of the real spectrum, and in some sense goes in the unexpected direction, but so far there is no other approach that we know.…”

Section: Theorem 3 (Extension) Let I Be An Ideal Of N (ω) Then Everymentioning

confidence: 99%

Uniform bounds on complexity and transfer of global properties of Nash functions

Coste¹,

Sancho²,

Shiota³

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian .

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On the factoriality of some rings of complex Nash functions

Fatabbi

Tancredi

2002

Bulletin des Sciences Mathématiques

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The idempotency of the real spectrum implies the extension theorem for Nash functions

Cited by 3 publications

References 8 publications

Global Problems on Nash Functions.

Global Problems on Nash Functions.

Uniform bounds on complexity and transfer of global properties of Nash functions

On the factoriality of some rings of complex Nash functions

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