منابع مشابه
The Classical Artin Approximation Theorems
The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q : formal, analytic, algebraic, of systems of equations of the same quality Q, assuming the existence of power series solutions of a weaker quality Q’ < Q : approximated, formal. The results are frequently used in commutative algebra and algebraic geometry. We present a systematic argum...
متن کاملCorrigendum to “the Classical Artin Approximation Theorems”
The purpose of this note is to clarify and complement various places in the article The classical Artin approximation theorems, published in Bull. Amer. Math. Soc. 54 (2017), no. 4, 595–633. A few reasonings in the article may have been difficult to follow or were even problematic. The author is indebted to F. Castro-Jiménez, C. Chiu, R. Ephraim, B. Lamel, S. Perlega, G. Pfister, G. Rond, M. Si...
متن کاملThe Two Artin-Schreier Theorems
Proof. We prove by the following steps: y 1 Start with induction on n, for n = 1 the result is trivial. y 2 Let g be such that σ1(g) ̸= σ2(g) and consider ∑ aiσi(gx) = 0 and ∑ aiσ1(g)σi(x) = 0 y 3 Cancel one summand by showing a2 = 0 and eventually show all ai = 0 One can prove by induction. Let a1σ1 ≡ 0 then since σ1 does not map to 0 ∈ K one must have a1 = 0. Suppose now that for any linear co...
متن کاملArtin Approximation
In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of Płoski, he conjectured that this remains true when we replace the ring of convergent power series by a more general ring. This paper presents the state of the art on this problem, aimed at non-...
متن کاملArtin Approximation and Proper Base Change
and an abelian sheaf F of torsion abelian groups on X, the natural base change map gRf∗F ∼ −→ Rf ′ ∗(g F) is an isomorphism. By limit arguments and considerations with geometric stalks as discussed last time, we can arrange that F is a Z/nZ-sheaf for some n > 0 and it suffices to treat the case where S = Spec (A) for a strictly henselian local ring and S′ = Spec (k′) for a separably closed fiel...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 2017
ISSN: 0273-0979,1088-9485
DOI: 10.1090/bull/1579