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].

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