# 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 $\hat{A}$. In fact, $A$ has the Artin approximation property if and only if for every finite system of polynomial equations $f$ over $A$ the set of its solutions in $A$ is dense, with respect to the $m$-adic topology, in the set of its solutions in $\hat{A}$. That is, for every solution $\widehat{y}$ of $f$ in $\hat{A}$ and every positive integer $c \in \bf N$ there exists a solution $y _ { c }$ of $f$ in $A$ such that $y _ { c } \cong \widehat { y }$ modulo $m ^ { c }\hat{ A}$. The study of Artin approximation started with the famous papers of M. Artin [a3], [a4], which state that the convergent power series rings over a non-trivial valued field of characteristic zero, the Henselization of a local ring essentially of finite type over a field, and an excellent Dedekind ring all have the Artin approximation property. The first result was extended by M. André [a1] to certain convergent formal power series rings over a field of non-zero characteristic.

The following assertion holds: A Noetherian local ring has AP if and only if it is excellent and Henselian.

The necessity is stated in [a24], a weaker result, namely that AP implies Henselian and universally Japanese, being proved in [a14], (5.4), and [a9]. The sufficiency gives a positive answer to Artin's conjecture [a5] and is a consequence (see [a21], (1.3), and [a27]) of the following theorem on general Néron desingularization ([a20], [a21], [a23], [a2], [a16], [a27], [a26]): A morphism $u : A \rightarrow A ^ { \prime }$ between Noetherian rings is regular (i.e. it is flat and for every field $K$ that is a finite $A$-algebra, the ring $K \otimes _ { A } A ^ { \prime }$ is regular) if and only if it is a filtered inductive limit of smooth algebras of finite type.

Roughly speaking, general Néron desingularization says in particular that if $u$ is a regular morphism of Noetherian rings, then every finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$ can be enlarged to a finite system of polynomial equations over $A$ having a solution in $A ^ { \prime }$, for which one may apply the implicit function theorem. Another consequence of general Néron desingularization says that a regular local ring containing a field is a filtered inductive limit of regular local rings essentially of finite type over $\mathbf{Z}$. This is a partial positive answer to the Swan conjecture and, using [a15], proves the Bass–Quillen conjecture in the equicharacteristic case (see also [a22], [a27]).

Let $( A , m )$ be a Noetherian local ring. $A$ has the strong Artin approximation property (in brief, $A$ has SAP) if for every finite system of equations $f$ in $Y = ( Y _ { 1 } , \dots , Y _ { s } )$ over $A$ there exists a mapping $\nu :\mathbf{N} \rightarrow \mathbf{N}$ with the following property: If $\tilde { y } \in A ^ { s }$ satisfies $f (\tilde{y}) \cong 0$ modulo $m ^ { \nu ( c ) }$, $c \in \bf N$, then there exists a solution $y \in A ^ { S }$ of $f$ with $y \cong \widetilde{y}$ modulo $m ^ { c }$.

M. Greenberg [a13] proved that excellent Henselian discrete valuation rings have the strong Artin approximation property and M. Artin [a4] showed that the Henselization of a local ring which is essentially of finite type over a field has the strong Artin approximation property.

The following assertion is true: A Noetherian complete local ring $A$ has the strong Artin approximation property. In particular, $A$ has AP if and only if it has SAP. A special case of this is stated in [a11], together with many other applications.

When $A$ contains a field, some weaker results were stated in [a29], [a30]. In the above form, the result appeared in [a17], but the proof there has a gap in the non-separable case, which was repaired in [a14], Chap. 2. In [a8] it was noted that SAP is more easily handled using ultraproducts. Let $D$ be a non-principal ultrafilter on $\mathbf{N}$ (i.e. an ultrafilter containing the filter of cofinite subsets of $\mathbf{N}$). The ultraproduct $A ^ { * }$ of $A$ with respect to $D$ is the factor of $A ^ {\bf N }$ by the ideal of all $( a _ { n } ) _ { n \in \mathbf{N} }$ such that the set $\{ n : a _ { n } = 0 \} \in D$. Assigning to $a \in A$ the constant sequence $( a , a , \dots )$ one obtains a ring morphism $A \rightarrow A ^ { * }$. Using these concepts, easier proofs of the assertion were given in [a19] and [a10]. The easiest one is given in [a21], (4.5), where it is noted that the separation $A _ { 1 } = A ^ { * } / \cap _ { i \in \mathbf{N} } m ^ { i } A ^ { * }$ of $A ^ { * }$ in the $m$-adic topology is Noetherian, that the canonical mapping $u : A \rightarrow A _ { 1 }$ is regular if $A$ is excellent and that $A$ is SAP if and only if for every finite system of polynomial equations $f$ over $A$, for every positive integer $c$ and every solution $\tilde{y}$ of $f$ in $A _ { 1 }$, there exists a solution $y _ { c }$ of $f$ in $A ^ { * }$ which lifts $\tilde{y}$ modulo $m ^ { c } A ^ { * }$. The result follows on applying general Néron desingularization to $u$ and using the implicit function theorem.

Theorems on Artin approximation have many direct applications in algebraic geometry (for example, to the algebraization of versal deformations and the construction of algebraic spaces; see [a6], [a5]), in algebraic number theory and in commutative algebra (see [a4], [a14], Chaps. 5, 6). For example, if $( R , m )$ is a Noetherian complete local domain and $( a _ { i } ) _ { i \in \mathbf{N} }$ is a sequence of elements from $R$ converging to an irreducible element $a$ of $R$, then G. Pfister proved that $a_i$ is irreducible for $i \gg 1$ (see [a14], Chap. 5). Using these ideas, a study of approximation of prime ideals in the $m$-adic topology was given in [a18]. Another application is that the completion of an excellent Henselian local domain $A$ is factorial if and only if $\hat{A}$ is factorial [a21], (3.4).

All these approximation properties were studied also for couples $( R , a )$, were $R$ is not necessarily local and $a$ is not necessarily maximal. A similar proof shows that the Artin approximation property holds for a Henselian couple $( R , a )$ if $R$ is excellent [a21], (1.3). If $R / a$ is not Artinian, then $R _ { 1 } = R ^ { * } / \cap _ { i \in \mathbf{N} } a ^ { i } R ^ { * }$ is not Noetherian and SAP cannot hold in this setting, because one cannot apply general Néron desingularization. Moreover, the SAP property does not hold for general couples, as noticed in [a25].

A special type of Artin approximation theory was required in singularity theory. Such types were studied in [a14], Chaps. 3, 4. However, the result holds even in the following extended form: Let $( A , m )$ be an excellent Henselian local ring, $\hat{A}$ its completion, $A \langle X \rangle $ the Henselization of $A[X]$, $X = ( X _ { 1 } , \ldots , X _ { n } )$, in $( X )$, $f$ a finite system of polynomial equations over $A \langle X \rangle $ and $\hat { y } = ( \hat { y } _ { 1 } , \dots , \hat { y } _ { n } ) \in \hat { A } [ [ X ] ] ^ { n }$ a formal solution of $f$ such that $\hat { y } _ { i } \in \hat { A } [ [ X _ { 1 } , \dots , X _ { s _ { i } } ] ]$, $1 \leq i \leq n$, for some positive integers $s _ { i } \leq n$. Then there exists a solution $y = ( y _ { 1 } , \dots , y _ { n } )$ of $f$ in $A \langle X \rangle $ such that $y _ { i } \in A \langle X _ { 1 } , \dots , X _ { s_i } \rangle$, $1 \leq i \leq n$, and $y _ { i } \cong \hat { y } _ { i }$ modulo $( m , X _ { 1 } , \dots , X _ { s_i } ) ^ { c }$, $1 \leq i \leq n$, for $c > 1$.

The proof is given in [a21], (3.6), (3.7), using ideas of H. Kurke and Pfister, who noticed that this assertion holds if $A ( X _ { 1 } , \dots , X _ { n } )$ has AP, where $A$ is an excellent Henselian local ring. If the sets of variables $X$ of $\hat{y}_ { i }$ are not "nested" (i.e. they are not totally ordered by inclusion), then the assertion does not hold, see [a7]. If $A = \mathbf{C} \{ Z _ { 1 } , \dots , Z _ { r } \}$ is the convergent power series ring over $\mathbf{C}$ and the algebraic power series rings $A ( X _ { 1 } , \dots , X _ { s _ { i } } )$ are replaced by $A \{ X _ { 1 } , \dots , X _ { s _ { i } } \}$, then the theorem does not hold, see [a12]. Extensions of this theorem are given in [a28], [a27].

#### References

[a1] | M. André, "Artin's theorem on the solution of analytic equations in positive characteristic" Manuscripta Math. , 15 (1975) pp. 314–348 MR379493 |

[a2] | M. André, "Cinq exposés sur la desingularization" École Polytechn. Féd. Lausanne (1991) (Handwritten manuscript) |

[a3] | M. Artin, "On the solution of analytic equations" Invent. Math. , 5 (1968) pp. 277–291 MR0232018 |

[a4] | M. Artin, "Algebraic approximation of structures over complete local rings" Publ. Math. IHES , 36 (1969) pp. 23–58 MR0268188 Zbl 0181.48802 |

[a5] | M. Artin, "Construction techniques for algebraic spaces" , Actes Congres Internat. Math. , 1 (1970) pp. 419–423 MR0427316 Zbl 0232.14003 |

[a6] | M. Artin, "Versal Deformations and Algebraic Stacks" Invent. Math. , 27 (1974) pp. 165–189 MR0399094 Zbl 0317.14001 |

[a7] | J. Becker, "A counterexample to Artin approximation with respect to subrings" Math. Ann. , 230 (1977) pp. 195–196 MR0480508 Zbl 0359.13007 |

[a8] | J. Becker, J. Denef, L. Lipshitz, L. van den Dries, "Ultraproducts and approximation in local rings I" Invent. Math. , 51 (1979) pp. 189–203 MR0528023 Zbl 0416.13004 |

[a9] | M. Cipu, D. Popescu, "Some extensions of Néron's $p$-desingularization and approximation" Rev. Roum. Math. Pures Appl. , 24 : 10 (1981) pp. 1299–1304 |

[a10] | J. Denef, L. Lipshitz, "Ultraproducts and approximation in local rings II" Math. Ann. , 253 (1980) pp. 1–28 MR0594530 Zbl 0426.13010 |

[a11] | R. Elkik, "Solutions d'equations à coefficients dans une anneau (!!) henselien" Ann. Sci. Ecole Norm. Sup. 4 , 6 (1973) pp. 533–604 MR345966 |

[a12] | A.M. Gabrielov, "The formal relations between analytic functions" Funkts. Anal. Prilozh. , 5 : 4 (1971) pp. 64–65 (In Russian) MR0302930 |

[a13] | M. Greenberg, "Rational points in Henselian discrete valuation rings" Publ. Math. IHES , 31 (1966) pp. 59–64 MR0207700 MR0191897 Zbl 0146.42201 Zbl 0142.00901 |

[a14] | H. Kurke, T. Mostowski, G. Pfister, D. Popescu, M. Roczen, "Die Approximationseigenschaft lokaler Ringe" , Lecture Notes Math. , 634 , Springer (1978) (Note: The proof of (3.1.1) is wrong) MR0485851 Zbl 0401.13013 |

[a15] | H. Lindel, "On the Bass–Quillen conjecture concerning projective modules over polynomial rings" Invent. Math. , 65 (1981) pp. 319–323 MR0641133 Zbl 0477.13006 |

[a16] | T. Ogoma, "General Néron desingularization based on the idea of Popescu" J. Algebra , 167 (1994) pp. 57–84 MR1282816 Zbl 0821.13003 |

[a17] | G. Pfister, D. Popescu, "Die strenge Approximationseigenschaft lokaler Ringe" Invent. Math. , 30 (1975) pp. 145–174 MR0379490 |

[a18] | G. Pfister, D. Popescu, "Die Approximation von Primidealen" Bull. Acad. Polon. Sci. , 27 (1979) pp. 771–778 MR603146 |

[a19] | D. Popescu, "Algebraically pure morphisms" Rev. Roum. Math. Pures Appl. , 26 : 6 (1979) pp. 947–977 MR0546539 Zbl 0416.13005 |

[a20] | D. Popescu, "General Néron desingularization" Nagoya Math. J. , 100 (1985) pp. 97–126 MR0818160 Zbl 0561.14008 |

[a21] | D. Popescu, "General Néron desingularization and approximation" Nagoya Math. J. , 104 (1986) pp. 85–115 MR0868439 Zbl 0592.14014 |

[a22] | D. Popescu, "Polynomial rings and their projective modules" Nagoya Math. J. , 113 (1989) pp. 121–128 MR0986438 Zbl 0663.13006 |

[a23] | D. Popescu, "Letter to the Editor: General Néron desingularization and approximation" Nagoya Math. J. , 118 (1990) pp. 45–53 MR1060701 Zbl 0685.14009 |

[a24] | C. Rotthaus, "Rings with approximation property" Math. Ann. , 287 (1990) pp. 455–466 MR1060686 Zbl 0702.13007 |

[a25] | M. Spivakovsky, "Non-existence of the Artin function for Henselian pairs" Math. Ann. , 299 (1994) pp. 727–729 MR1286894 Zbl 0803.13005 |

[a26] | M. Spivakovsky, "A new proof of D. Popescu's theorem on smoothing of ring homomorphisms" J. Amer. Math. Soc. , 294 (to appear) MR1647069 |

[a27] | R. Swan, "Néron–Popescu desingularization" , Proc. Internat. Conf. Algebra and Geometry, Taipei, Taiwan, 1995 , Internat. Press Boston (1998) Zbl 0954.13003 |

[a28] | B. Teissier, "Résultats récents sur l'approximation des morphisms en algèbre commutative,[d'après Artin, Popescu, André, Spivakovsky]" Sem. Bourbaki , 784 (1994) pp. 1–15 |

[a29] | M. Van der Put, "A problem on coefficient fields and equations over local rings" Compositio Math. , 30 : 3 (1975) pp. 235–258 Zbl 0304.13018 |

[a30] | J.J. Wavrik, "A theorem on solutions of analytic equations with applications to deformations of complex structures" Math. Ann. , 216 : 2 (1975) pp. 127–142 MR0387649 Zbl 0303.32018 |

**How to Cite This Entry:**

Artin approximation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Artin_approximation&oldid=55482