Constructive Néron Desingularization of algebras with big smooth locus

Kosar, Zunaira; Pfister, Gerhard; Popescu, Dorin

doi:10.48550/arxiv.1702.01867

Cited by 3 publications

(7 citation statements)

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“…is in the kernel of ϕ as in the proof of [8,Proposition 3]. Set E = D[Y, T ]/(I, g, h) and let ψ : E → A ′ be the map induced by ϕ.…”

Section: Proof By Hypothesis There Exists Anmentioning

confidence: 99%

“…Using [15, Lemma 2.4] we may replace B i by an A-algebra C of finite type with H C/A = C, that is C is a smooth algebra over A. We may choose C to be standard smooth over A using [15,Lemma 3.4] (see also [8,Lemma 4]).…”

Section: Proof By Hypothesis There Exists Anmentioning

confidence: 99%

“…Other authors gave similar proofs (see [23], [21], [9]) but most of them were not constructive. In [14], [12], [13], [5], [6], [7], [8], there exist some constructive proofs with algorithms in some special cases. We mention that all these proofs are simpler when we deal with Q-algebras because in this case all residue field extensions are separable and it is easier to handle the regularity of the morphisms.…”

Section: Introductionmentioning

confidence: 99%

“…The proof was based on the following proposition (see [8,Proposition 3], where the conditions "A ′ is Henselian, and A, A ′ have dimensions m are superfluous" and we remove them here). Proposition 3.…”

Section: Introductionmentioning

confidence: 99%

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Simple General Neron Desingularization in local Q-algebras

Popescu

2019

Communications in Algebra

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“…is in the kernel of ϕ as in the proof of [8,Proposition 3]. Set E = D[Y, T ]/(I, g, h) and let ψ : E → A ′ be the map induced by ϕ.…”

Section: Proof By Hypothesis There Exists Anmentioning

confidence: 99%

Section: Proof By Hypothesis There Exists Anmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Simple General Neron Desingularization in local Q-algebras

Popescu

2019

Communications in Algebra

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“…For this, we express A ′ as a filtered direct limit of smooth A-algebras S i , note that S ′ descends to a smooth S i -algebra S ′ i for some i, and conclude that S ′ is then the filtered direct limit of the smooth A-algebras S ′ j = S j ⊗ S i S ′ i with j ≥ i. The following lemma originates in [24, (7.1)] and its variants have appeared, for instance, in [33, 18.1], [26, 7.2], [32,07CT], [16,Proposition 3], and [28, Proposition 5]. The version below differs in two aspects: we do not assume Noetherianness and do not require the elements a or b to come from the base ring A.…”

Section: A Reduction To the Case Of Complete Valuation Rings Of Rankmentioning

confidence: 99%

Néron desingularization of extensions of valuation rings with an Appendix by Kęstutis Česnavičius

Popescu¹

2019

Preprint

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Zariski's local uniformization, a weak form of resolution of singularities, implies that every valuation ring containing Q should be a filtered direct limit of smooth Q-algebras. Given an extension of valuation rings V ⊂ V ′ containing Q we give conditions when V ′ is a filtered direct limit of smooth V -algebras. This corrects a paper of us [23] where we thought that we may reduce to the case when the value groups are finitely generated. For this we use an infinite tower of ultrapowers construction that rests on results from model theory. stated somehow by mistake in [23] without condition (2), and assuming that the value group Γ ′ /Γ is finitely generated when the fraction field extension of V ⊂ V ′ is finitely generated. This is not the case as shows [2, Theorem 7.1]. For solving the general case, that is Γ ′ /Γ infinitely generated we had to study the so called the cross-sections with the help of Model Theory.3. The method of proof. To achieve the desingularization claimed in Theorem 2, we replace the initial V by the limit Ṽ of a certain countable tower of iterated ultrapowers of V , constructed in such a way that Ṽ would, in turn, be an immediate extension of a filtered increasing union of valuation rings for which one knows local uniformization. To then conclude, we argue that large immediate extensions and ultrapowers interact well with desingularization.Desingularizing immediate extensions is in fact the first major step. We show that if V ⊂ V ′ is a local extension of valuation rings containing Q that induces isomorphisms on value groups and residue fields (that is, is immediate) then V ′ is a filtered direct limit of smooth V -algebras.The techniques we use for include extensions of steps of the General Néron desingularization, notably, Lemma 7 that is also key for reductions to complete rank 1 cases. In the purely transcendental case, Kaplansky's classification [12] of Ostrowski's pseudo-convergent sequences plays an important role.The utility of desingularizing immediate extensions is evident already in the case when V ′ is complete of rank 1 with a finitely generated value group Γ ′ . Such a V ′ has a coefficient field k, so, by choosing a presentation Γ ′ ∼ = Zval(x 1 )⊕• • •⊕Zval(x n ), one obtains the immediate extension V ′ ∩ k(x 1 , . . . , x n ) ⊂ V ′ . To show that V = k ⊂ V ′ is ind-smooth, it remains to observe that a local uniformization of V ′ ∩ k(x 1 , . . . , x n ) may be constructed using Perron's algorithm in the style of Zariski.The goal of the tower of ultrapowers argument given in Appendix is to overcome the obstacle that in general Γ may not be finitely generated and there may not even be a group section s : Γ → K * to val : K * → Γ (roughly, such an s suffices). Nevertheless, s can always be arranged for any finitely generated submonoid of Γ, and the idea is to then use a following fact from model theory: for a system of equations whose finite subsystems have solutions in V , the entire system has a solution in a well-chosen ultrapower of V (see Appendix).This fact, which rests o...

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