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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.
Artin's criterion. In mathematics, Artin's criteria[1][2][3][4] are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces [5] or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of ...
His work on the problem of characterising the representable functors in the category of schemes has led to the Artin approximation theorem in local algebra as well as the "Existence theorem".
15. Feb. 2024 · Artin approximation. 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 $\hat {A}$.
APPROXIMATION OF VERSAL DEFORMATIONS BRIAN CONRAD AND A.J. DE JONG 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.
13. Juni 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).
Artin’s approximation and algebraization theorems together with Artin’s cri- terion for algebraicity instruct us on how we should think of the local structure of algebraic stacks.
