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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202601.png" /> be a Noetherian [[Local ring|local ring]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202602.png" /> its completion. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202603.png" /> has the Artin approximation property (in brief, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202604.png" /> has AP) if every finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202605.png" /> has a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202606.png" /> if it has one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202607.png" />. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202608.png" /> has the Artin approximation property if and only if for every finite system of polynomial equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a1202609.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026010.png" /> the set of its solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026011.png" /> is dense, with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026012.png" />-adic topology, in the set of its solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026013.png" />. That is, for every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026016.png" /> and every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026017.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026020.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026021.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026022.png" />. The study of Artin approximation started with the famous papers of M. Artin [[#References|[a3]]], [[#References|[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 [[#References|[a1]]] to certain convergent formal power series rings over a field of non-zero characteristic.
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Let $( A , m )$ be a Noetherian [[Local ring|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 [[#References|[a3]]], [[#References|[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 [[#References|[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 following assertion holds: A Noetherian local ring has AP if and only if it is excellent and Henselian.
  
The necessity is stated in [[#References|[a24]]], a weaker result, namely that AP implies Henselian and universally Japanese, being proved in [[#References|[a14]]], (5.4), and [[#References|[a9]]]. The sufficiency gives a positive answer to Artin's conjecture [[#References|[a5]]] and is a consequence (see [[#References|[a21]]], (1.3), and [[#References|[a27]]]) of the following theorem on general Néron desingularization ([[#References|[a20]]], [[#References|[a21]]], [[#References|[a23]]], [[#References|[a2]]], [[#References|[a16]]], [[#References|[a27]]], [[#References|[a26]]]): A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026023.png" /> between Noetherian rings is regular (i.e. it is flat and for every field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026024.png" /> that is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026025.png" />-algebra, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026026.png" /> is regular) if and only if it is a filtered inductive limit of smooth algebras of finite type.
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The necessity is stated in [[#References|[a24]]], a weaker result, namely that AP implies Henselian and universally Japanese, being proved in [[#References|[a14]]], (5.4), and [[#References|[a9]]]. The sufficiency gives a positive answer to Artin's conjecture [[#References|[a5]]] and is a consequence (see [[#References|[a21]]], (1.3), and [[#References|[a27]]]) of the following theorem on general Néron desingularization ([[#References|[a20]]], [[#References|[a21]]], [[#References|[a23]]], [[#References|[a2]]], [[#References|[a16]]], [[#References|[a27]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026027.png" /> is a regular morphism of Noetherian rings, then every finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026028.png" /> having a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026029.png" /> can be enlarged to a finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026030.png" /> having a solution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026031.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026032.png" />. This is a partial positive answer to the Swan conjecture and, using [[#References|[a15]]], proves the Bass–Quillen conjecture in the equicharacteristic case (see also [[#References|[a22]]], [[#References|[a27]]]).
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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 [[#References|[a15]]], proves the Bass–Quillen conjecture in the equicharacteristic case (see also [[#References|[a22]]], [[#References|[a27]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026033.png" /> be a Noetherian local ring. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026034.png" /> has the strong Artin approximation property (in brief, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026035.png" /> has SAP) if for every finite system of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026037.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026038.png" /> there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026039.png" /> with the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026040.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026041.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026043.png" />, then there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026045.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026046.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026047.png" />.
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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 [[#References|[a13]]] proved that excellent Henselian discrete valuation rings have the strong Artin approximation property and M. Artin [[#References|[a4]]] showed that the Henselization of a local ring which is essentially of finite type over a field has the strong Artin approximation property.
 
M. Greenberg [[#References|[a13]]] proved that excellent Henselian discrete valuation rings have the strong Artin approximation property and M. Artin [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026048.png" /> has the strong Artin approximation property. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026049.png" /> has AP if and only if it has SAP. A special case of this is stated in [[#References|[a11]]], together with many other applications.
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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 [[#References|[a11]]], together with many other applications.
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026050.png" /> contains a field, some weaker results were stated in [[#References|[a29]]], [[#References|[a30]]]. In the above form, the result appeared in [[#References|[a17]]], but the proof there has a gap in the non-separable case, which was repaired in [[#References|[a14]]], Chap. 2. In [[#References|[a8]]] it was noted that SAP is more easily handled using ultraproducts. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026051.png" /> be a non-principal [[Ultrafilter|ultrafilter]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026052.png" /> (i.e. an ultrafilter containing the filter of [[cofinite subset]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026053.png" />). The ultraproduct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026055.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026056.png" /> is the factor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026057.png" /> by the ideal of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026058.png" /> such that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026059.png" />. Assigning to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026060.png" /> the constant sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026061.png" /> one obtains a ring morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026062.png" />. Using these concepts, easier proofs of the assertion were given in [[#References|[a19]]] and [[#References|[a10]]]. The easiest one is given in [[#References|[a21]]], (4.5), where it is noted that the separation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026064.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026065.png" />-adic topology is Noetherian, that the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026066.png" /> is regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026067.png" /> is excellent and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026068.png" /> is SAP if and only if for every finite system of polynomial equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026069.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026070.png" />, for every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026071.png" /> and every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026072.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026073.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026074.png" />, there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026075.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026076.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026077.png" /> which lifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026078.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026079.png" />. The result follows on applying general Néron desingularization to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026080.png" /> and using the implicit function theorem.
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When $A$ contains a field, some weaker results were stated in [[#References|[a29]]], [[#References|[a30]]]. In the above form, the result appeared in [[#References|[a17]]], but the proof there has a gap in the non-separable case, which was repaired in [[#References|[a14]]], Chap. 2. In [[#References|[a8]]] it was noted that SAP is more easily handled using ultraproducts. Let $D$ be a non-principal [[Ultrafilter|ultrafilter]] on $\mathbf{N}$ (i.e. an ultrafilter containing the filter of [[cofinite subset]]s 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 [[#References|[a19]]] and [[#References|[a10]]]. The easiest one is given in [[#References|[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|algebraic geometry]] (for example, to the algebraization of versal deformations and the construction of algebraic spaces; see [[#References|[a6]]], [[#References|[a5]]]), in algebraic number theory and in commutative algebra (see [[#References|[a4]]], [[#References|[a14]]], Chaps. 5, 6). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026081.png" /> is a Noetherian complete local domain and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026082.png" /> is a sequence of elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026083.png" /> converging to an irreducible element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026084.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026085.png" />, then G. Pfister proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026086.png" /> is irreducible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026087.png" /> (see [[#References|[a14]]], Chap. 5). Using these ideas, a study of approximation of prime ideals in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026088.png" />-adic topology was given in [[#References|[a18]]]. Another application is that the completion of an excellent Henselian local domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026089.png" /> is factorial if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026090.png" /> is factorial [[#References|[a21]]], (3.4).
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Theorems on Artin approximation have many direct applications in [[Algebraic geometry|algebraic geometry]] (for example, to the algebraization of versal deformations and the construction of algebraic spaces; see [[#References|[a6]]], [[#References|[a5]]]), in algebraic number theory and in commutative algebra (see [[#References|[a4]]], [[#References|[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 [[#References|[a14]]], Chap. 5). Using these ideas, a study of approximation of prime ideals in the $m$-adic topology was given in [[#References|[a18]]]. Another application is that the completion of an excellent Henselian local domain $A$ is factorial if and only if $\hat{A}$ is factorial [[#References|[a21]]], (3.4).
  
All these approximation properties were studied also for couples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026091.png" />, were <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026092.png" /> is not necessarily local and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026093.png" /> is not necessarily maximal. A similar proof shows that the Artin approximation property holds for a Henselian couple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026094.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026095.png" /> is excellent [[#References|[a21]]], (1.3). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026096.png" /> is not Artinian, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026097.png" /> 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 [[#References|[a25]]].
+
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 [[#References|[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 [[#References|[a25]]].
  
A special type of Artin approximation theory was required in singularity theory. Such types were studied in [[#References|[a14]]], Chaps. 3, 4. However, the result holds even in the following extended form: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026098.png" /> be an excellent Henselian local ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a12026099.png" /> its completion, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260100.png" /> the Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260102.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260104.png" /> a finite system of polynomial equations over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260106.png" /> a formal solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260109.png" />, for some positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260110.png" />. Then there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260112.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260113.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260115.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260116.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260118.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260119.png" />.
+
A special type of Artin approximation theory was required in singularity theory. Such types were studied in [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260101.png"/>, $X = ( X _ { 1 } , \ldots , X _ { n } )$, 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 $c &gt; 1$.
  
The proof is given in [[#References|[a21]]], (3.6), (3.7), using ideas of H. Kurke and Pfister, who noticed that this assertion holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260120.png" /> has AP, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260121.png" /> is an excellent Henselian local ring. If the sets of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260122.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260123.png" /> are not "nested" (i.e. they are not totally ordered by inclusion), then the assertion does not hold, see [[#References|[a7]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260124.png" /> is the convergent power series ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260125.png" /> and the algebraic power series rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260126.png" /> are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260127.png" />, then the theorem does not hold, see [[#References|[a12]]]. Extensions of this theorem are given in [[#References|[a28]]], [[#References|[a27]]].
+
The proof is given in [[#References|[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 [[#References|[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 [[#References|[a12]]]. Extensions of this theorem are given in [[#References|[a28]]], [[#References|[a27]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. André, "Artin's theorem on the solution of analytic equations in positive characteristic" ''Manuscripta Math.'' , '''15''' (1975) pp. 314–348 {{MR|379493}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. André, "Cinq exposés sur la desingularization" ''École Polytechn. Féd. Lausanne'' (1991) (Handwritten manuscript)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Artin, "On the solution of analytic equations" ''Invent. Math.'' , '''5''' (1968) pp. 277–291 {{MR|0232018}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Artin, "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58 {{MR|0268188}} {{ZBL|0181.48802}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Artin, "Construction techniques for algebraic spaces" , ''Actes Congres Internat. Math.'' , '''1''' (1970) pp. 419–423 {{MR|0427316}} {{ZBL|0232.14003}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Artin, "Versal Deformations and Algebraic Stacks" ''Invent. Math.'' , '''27''' (1974) pp. 165–189 {{MR|0399094}} {{ZBL|0317.14001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Becker, "A counterexample to Artin approximation with respect to subrings" ''Math. Ann.'' , '''230''' (1977) pp. 195–196 {{MR|0480508}} {{ZBL|0359.13007}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Becker, J. Denef, L. Lipshitz, L. van den Dries, "Ultraproducts and approximation in local rings I" ''Invent. Math.'' , '''51''' (1979) pp. 189–203 {{MR|0528023}} {{ZBL|0416.13004}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Cipu, D. Popescu, "Some extensions of Néron's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120260/a120260128.png" />-desingularization and approximation" ''Rev. Roum. Math. Pures Appl.'' , '''24''' : 10 (1981) pp. 1299–1304</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J. Denef, L. Lipshitz, "Ultraproducts and approximation in local rings II" ''Math. Ann.'' , '''253''' (1980) pp. 1–28 {{MR|0594530}} {{ZBL|0426.13010}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R. Elkik, "Solutions d'equations à coefficients dans une anneau (!!) henselien" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''6''' (1973) pp. 533–604 {{MR|345966}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A.M. Gabrielov, "The formal relations between analytic functions" ''Funkts. Anal. Prilozh.'' , '''5''' : 4 (1971) pp. 64–65 (In Russian) {{MR|0302930}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> M. Greenberg, "Rational points in Henselian discrete valuation rings" ''Publ. Math. IHES'' , '''31''' (1966) pp. 59–64 {{MR|0207700}} {{MR|0191897}} {{ZBL|0146.42201}} {{ZBL|0142.00901}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> 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) {{MR|0485851}} {{ZBL|0401.13013}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> H. Lindel, "On the Bass–Quillen conjecture concerning projective modules over polynomial rings" ''Invent. Math.'' , '''65''' (1981) pp. 319–323 {{MR|0641133}} {{ZBL|0477.13006}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> T. Ogoma, "General Néron desingularization based on the idea of Popescu" ''J. Algebra'' , '''167''' (1994) pp. 57–84 {{MR|1282816}} {{ZBL|0821.13003}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> G. Pfister, D. Popescu, "Die strenge Approximationseigenschaft lokaler Ringe" ''Invent. Math.'' , '''30''' (1975) pp. 145–174 {{MR|0379490}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> G. Pfister, D. Popescu, "Die Approximation von Primidealen" ''Bull. Acad. Polon. Sci.'' , '''27''' (1979) pp. 771–778 {{MR|603146}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D. Popescu, "Algebraically pure morphisms" ''Rev. Roum. Math. Pures Appl.'' , '''26''' : 6 (1979) pp. 947–977 {{MR|0546539}} {{ZBL|0416.13005}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D. Popescu, "General Néron desingularization" ''Nagoya Math. J.'' , '''100''' (1985) pp. 97–126 {{MR|0818160}} {{ZBL|0561.14008}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D. Popescu, "General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''104''' (1986) pp. 85–115 {{MR|0868439}} {{ZBL|0592.14014}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> D. Popescu, "Polynomial rings and their projective modules" ''Nagoya Math. J.'' , '''113''' (1989) pp. 121–128 {{MR|0986438}} {{ZBL|0663.13006}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> D. Popescu, "Letter to the Editor: General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''118''' (1990) pp. 45–53 {{MR|1060701}} {{ZBL|0685.14009}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> C. Rotthaus, "Rings with approximation property" ''Math. Ann.'' , '''287''' (1990) pp. 455–466 {{MR|1060686}} {{ZBL|0702.13007}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> M. Spivakovsky, "Non-existence of the Artin function for Henselian pairs" ''Math. Ann.'' , '''299''' (1994) pp. 727–729 {{MR|1286894}} {{ZBL|0803.13005}} </TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> M. Spivakovsky, "A new proof of D. Popescu's theorem on smoothing of ring homomorphisms" ''J. Amer. Math. Soc.'' , '''294''' (to appear) {{MR|1647069}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> R. Swan, "Néron–Popescu desingularization" , ''Proc. Internat. Conf. Algebra and Geometry, Taipei, Taiwan, 1995'' , Internat. Press Boston (1998) {{MR|}} {{ZBL|0954.13003}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> M. Van der Put, "A problem on coefficient fields and equations over local rings" ''Compositio Math.'' , '''30''' : 3 (1975) pp. 235–258 {{MR|}} {{ZBL|0304.13018}} </TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> 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 {{MR|0387649}} {{ZBL|0303.32018}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> M. André, "Artin's theorem on the solution of analytic equations in positive characteristic" ''Manuscripta Math.'' , '''15''' (1975) pp. 314–348 {{MR|379493}} {{ZBL|}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. André, "Cinq exposés sur la desingularization" ''École Polytechn. Féd. Lausanne'' (1991) (Handwritten manuscript)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> M. Artin, "On the solution of analytic equations" ''Invent. Math.'' , '''5''' (1968) pp. 277–291 {{MR|0232018}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> M. Artin, "Algebraic approximation of structures over complete local rings" ''Publ. Math. IHES'' , '''36''' (1969) pp. 23–58 {{MR|0268188}} {{ZBL|0181.48802}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> M. Artin, "Construction techniques for algebraic spaces" , ''Actes Congres Internat. Math.'' , '''1''' (1970) pp. 419–423 {{MR|0427316}} {{ZBL|0232.14003}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M. Artin, "Versal Deformations and Algebraic Stacks" ''Invent. Math.'' , '''27''' (1974) pp. 165–189 {{MR|0399094}} {{ZBL|0317.14001}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> J. Becker, "A counterexample to Artin approximation with respect to subrings" ''Math. Ann.'' , '''230''' (1977) pp. 195–196 {{MR|0480508}} {{ZBL|0359.13007}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> J. Becker, J. Denef, L. Lipshitz, L. van den Dries, "Ultraproducts and approximation in local rings I" ''Invent. Math.'' , '''51''' (1979) pp. 189–203 {{MR|0528023}} {{ZBL|0416.13004}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> M. Cipu, D. Popescu, "Some extensions of Néron's $p$-desingularization and approximation" ''Rev. Roum. Math. Pures Appl.'' , '''24''' : 10 (1981) pp. 1299–1304</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> J. Denef, L. Lipshitz, "Ultraproducts and approximation in local rings II" ''Math. Ann.'' , '''253''' (1980) pp. 1–28 {{MR|0594530}} {{ZBL|0426.13010}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> R. Elkik, "Solutions d'equations à coefficients dans une anneau (!!) henselien" ''Ann. Sci. Ecole Norm. Sup. 4'' , '''6''' (1973) pp. 533–604 {{MR|345966}} {{ZBL|}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> A.M. Gabrielov, "The formal relations between analytic functions" ''Funkts. Anal. Prilozh.'' , '''5''' : 4 (1971) pp. 64–65 (In Russian) {{MR|0302930}} {{ZBL|}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> M. Greenberg, "Rational points in Henselian discrete valuation rings" ''Publ. Math. IHES'' , '''31''' (1966) pp. 59–64 {{MR|0207700}} {{MR|0191897}} {{ZBL|0146.42201}} {{ZBL|0142.00901}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> 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) {{MR|0485851}} {{ZBL|0401.13013}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> H. Lindel, "On the Bass–Quillen conjecture concerning projective modules over polynomial rings" ''Invent. Math.'' , '''65''' (1981) pp. 319–323 {{MR|0641133}} {{ZBL|0477.13006}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> T. Ogoma, "General Néron desingularization based on the idea of Popescu" ''J. Algebra'' , '''167''' (1994) pp. 57–84 {{MR|1282816}} {{ZBL|0821.13003}} </td></tr><tr><td valign="top">[a17]</td> <td valign="top"> G. Pfister, D. Popescu, "Die strenge Approximationseigenschaft lokaler Ringe" ''Invent. Math.'' , '''30''' (1975) pp. 145–174 {{MR|0379490}} {{ZBL|}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> G. Pfister, D. Popescu, "Die Approximation von Primidealen" ''Bull. Acad. Polon. Sci.'' , '''27''' (1979) pp. 771–778 {{MR|603146}} {{ZBL|}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> D. Popescu, "Algebraically pure morphisms" ''Rev. Roum. Math. Pures Appl.'' , '''26''' : 6 (1979) pp. 947–977 {{MR|0546539}} {{ZBL|0416.13005}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> D. Popescu, "General Néron desingularization" ''Nagoya Math. J.'' , '''100''' (1985) pp. 97–126 {{MR|0818160}} {{ZBL|0561.14008}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> D. Popescu, "General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''104''' (1986) pp. 85–115 {{MR|0868439}} {{ZBL|0592.14014}} </td></tr><tr><td valign="top">[a22]</td> <td valign="top"> D. Popescu, "Polynomial rings and their projective modules" ''Nagoya Math. J.'' , '''113''' (1989) pp. 121–128 {{MR|0986438}} {{ZBL|0663.13006}} </td></tr><tr><td valign="top">[a23]</td> <td valign="top"> D. Popescu, "Letter to the Editor: General Néron desingularization and approximation" ''Nagoya Math. J.'' , '''118''' (1990) pp. 45–53 {{MR|1060701}} {{ZBL|0685.14009}} </td></tr><tr><td valign="top">[a24]</td> <td valign="top"> C. Rotthaus, "Rings with approximation property" ''Math. Ann.'' , '''287''' (1990) pp. 455–466 {{MR|1060686}} {{ZBL|0702.13007}} </td></tr><tr><td valign="top">[a25]</td> <td valign="top"> M. Spivakovsky, "Non-existence of the Artin function for Henselian pairs" ''Math. Ann.'' , '''299''' (1994) pp. 727–729 {{MR|1286894}} {{ZBL|0803.13005}} </td></tr><tr><td valign="top">[a26]</td> <td valign="top"> M. Spivakovsky, "A new proof of D. Popescu's theorem on smoothing of ring homomorphisms" ''J. Amer. Math. Soc.'' , '''294''' (to appear) {{MR|1647069}} {{ZBL|}} </td></tr><tr><td valign="top">[a27]</td> <td valign="top"> R. Swan, "Néron–Popescu desingularization" , ''Proc. Internat. Conf. Algebra and Geometry, Taipei, Taiwan, 1995'' , Internat. Press Boston (1998) {{MR|}} {{ZBL|0954.13003}} </td></tr><tr><td valign="top">[a28]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a29]</td> <td valign="top"> M. Van der Put, "A problem on coefficient fields and equations over local rings" ''Compositio Math.'' , '''30''' : 3 (1975) pp. 235–258 {{MR|}} {{ZBL|0304.13018}} </td></tr><tr><td valign="top">[a30]</td> <td valign="top"> 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 {{MR|0387649}} {{ZBL|0303.32018}} </td></tr></table>

Latest revision as of 17:45, 1 July 2020

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 , $X = ( X _ { 1 } , \ldots , X _ { n } )$, 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
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[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=39476
This article was adapted from an original article by D. Popescu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article