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In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k.
13 de jun. de 2017 · The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q (i.e., 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 (i.e., approximated, formal).
15 de feb. de 2024 · Let $( A , m )$ be a Noetherian local ring and $\hat{A}$ its completion. $A$ has the Artin approximation property (in brief, $A$ has AP) if every finite system of polynomial equations over $A$ has a solution in $A$ if it has one in $\hat{A}$.
Artin’s approximation theorems [A68, A69] are powerful tools in analytic and algebraic geometry for finding solutions of systems of analytic or algebraic equations whenever a given formal solution exists.
1 de dic. de 2023 · The first – Artin approximation – asserts that object associated to the completion of a local ring of a finite type superscheme are approximated to any fixed finite order by some object over some finite type étale cover, depending on the order. Theorem 1.1Artin approximation.
Contents. Introduction 2. Lecture 1: Artin approximation 3. 1.1. N ́eron–Popescu desingularization 3. 1.2. Artin approximation 4. 1.3. Alternative formulations of Artin approximation 5. 1.4. A first application of Artin approximation 6. 1.5. Proof of Artin approximation 7. 1.6.
1. Introduction In Artin’s work on algebraic spaces and algebraic stacks [A2], [A3], a crucial ingredient is the use of his approximation theorem to prove the algebraizability of formal deformations under quite general conditions.
