2 Arquile varieties

In this section, we collect the basic material on arquile varietes needed in later sections. To ease the orientation of the reader, some overlap with Section 1 is admitted.

Our ground field $\mathbf {k}$ will be a perfect field of arbitrary characteristic. It will be assumed to be equipped with a valuation $\vert -\vert :\mathbf {k}\rightarrow \mathbb R$ whenever we talk about convergent power series. To ease notation we often suppress the dependence of power series $y=y(\mathrm {t})$ on the variable $\mathrm {t}$ . Vectors $f=f(\mathrm {t},\mathrm {y})$ will be written as columns of length k, vectors of variables $\mathrm {y}$ and $\mathrm {z}$ and of power series $y=y(\mathrm {t})$ as rows of length m. Accordingly, the relative Jacobian matrix ${\partial }_{\mathrm {y}} f$ of f will be a $(k\times m)$ -matrix.

Spaces of power series. We denote by $\mathcal A_{\circ }$ the ring of formal, respectively convergent or algebraic, power series $y(\mathrm {t})$ in a single variable $\mathrm {t}$ over $\mathbf {k}$ and without constant term, $y(0)=0$ .Footnote ¹⁵ This is a commutative ring without one element $1$ . For $m\in {\mathbb N}$ we will consider vectors $y=(y_1,\ldots ,y_m)\in \mathcal A_{\circ }^m$ of power series without constant terms and call accordingly $y_{ij}=(y_{ij},\, 1\leq i \leq m,\, j\geq 1)\in (\mathbf {k}^{{\mathbb N}_{>0}})^m$ their coefficient vector. We will interpret $\mathcal A_{\circ }^m$ as an infinite-dimensional affine space over $\mathbf {k}$ . In the setting of formal power series, its ‘coordinate ring’ is the polynomial ring $\mathbf {k}[y_{m,\infty }]=\mathbf {k}[\mathrm {y}_{ij},\, 1\leq i \leq m,\, j\geq 1]$ in countably many variables.

Arquile maps. For variables $\mathrm {y}=(\mathrm {y}_1,\ldots ,\mathrm {y}_m)$ , let ${\mathcal B}$ denote one of the rings $\mathbf {k}[[\mathrm {t},\mathrm {y}]]$ , $\mathbf {k}\{\mathrm {t},\mathrm {y}\}$ or $\mathbf {k}\langle \mathrm {t},\mathrm {y}\rangle $ of power series in $\mathrm {t}$ and $\mathrm {y}$ . If f is in ${\mathcal B}$ and $y=y(\mathrm {t})\in \mathcal A_{\circ }^m$ , we may replace in f the $\mathrm {y}$ -variables by the vector $y(t)$ , which we abbreviate by $f(y)=f(\mathrm {t},y(\mathrm {t}))$ . Here and in the sequel, the qualities of $\mathcal A_{\circ }$ and ${\mathcal B}$ (formal, convergent or algebraic) are tacitly accorded. Every vector $f\in {\mathcal B}^k$ induces an arquile map

$$ \begin{align*} \begin{array}{rcl} {f_{\infty}}: \mathcal A_{\circ}^m& \longrightarrow & {\mathcal A}^k,\\ y & \longrightarrow & f(y) \end{array} \end{align*} $$

by evaluation at y. If f is a linear polynomial in $y_1,\ldots ,y_m$ with coefficients in $\mathcal A_{\circ }$ , the induced map ${f_{\infty }}$ is $\mathcal A_{\circ }$ -linear. If ${\mathcal S}$ is a subset of $\mathcal A_{\circ }^m$ , we may restrict ${f_{\infty }}$ to ${\mathcal S}$ to get an arquile map ${f_{\infty }} :{\mathcal S}\rightarrow {\mathcal A}^k$ .

Theorem 2.1. Inverse function theorem

For any vector $p(\mathrm {t},\mathrm {y})\in {\mathcal B}^m$ whose components are all of order at least $2$ in $\mathrm {t},\mathrm {y}$ , the arquile map

$$ \begin{align*} \begin{array}{rcl} \varphi: \mathcal A_{\circ}^m& \longrightarrow & \mathcal A_{\circ}^m,\\ y & \longrightarrow & y+p(y), \end{array} \end{align*} $$

is an arquile isomorphism.

Arquile varieties. Every ideal I of ${\mathcal B}$ defines a subset ${\mathcal Y}$ of $\mathcal A_{\circ }^m$ by

$$ \begin{align*} {\mathcal Y}={\mathcal Y}(I)=\{y\in\mathcal A_{\circ}^m,\, f(y)=0 \text{ for all } f\in I\}. \end{align*} $$

We call such sets arquile subvarieties of $\mathcal A_{\circ }^m$ , or simply arquile varieties. We also say that these sets are arquile closed in $\mathcal A_{\circ }^m$ . They are the fibres $f_{\infty }^{-1}(0)$ of arquile maps $f_{\infty }:\mathcal A_{\circ }^m\rightarrow \mathcal A_{\circ }^k$ induced by a vector $f\in {\mathcal B}^k$ with $f(0)=0$ , where the components of f generate the ideal I in ${\mathcal B}$ . Note that arquile open subsets of $\mathcal A_{\circ }^m$ are $\mathrm {t}$ -adically open.

Given two ideals I and J of ${\mathcal B}$ , we have ${\mathcal Y}(I\cap J)={\mathcal Y}(I)\cup {\mathcal Y}(J)$ and ${\mathcal Y}(I+ J)={\mathcal Y}(I)\cap {\mathcal Y}(J)$ . It is clear that if $\sqrt I$ denotes the radical of I, then ${\mathcal Y}(I)={\mathcal Y}(\sqrt I)$ . So we may assume without loss of generality that arquile subvarieties ${\mathcal Y}(I)$ are defined by radical ideals $I=\sqrt I$ .Footnote ¹⁶ For ${\mathcal Y}$ an arbitrary subset of $\mathcal A_{\circ }^m$ we define

$$ \begin{align*} I_{{\mathcal Y}}=\{f\in{\mathcal B},\, f(y)=0 \text{ for all } y\in {\mathcal Y}\} \end{align*} $$

as the ideal of ${\mathcal B}$ of power series vanishing upon the substitution of the variables $\mathrm {y}$ by elements y of ${\mathcal Y}$ . The ideal I is called saturated if I equals its saturation $I_{{\mathcal Y}(I)}$ . The saturation of an arbitrary ideal I is always radical; for the converse in the convergent case, see Theorem 2.8.

The arquile varieties ${\mathcal Y}(I_i)$ will be called the irreducible components of ${\mathcal Y}$ (they are the irreducible components with respect to the arquile topology). While the prime components of a saturated ideal are again saturated, the saturation of a prime ideal need not be prime; cf. the example after the proof.

Textile maps. A map $\tau : U\subset \mathcal A_{\circ }^m\rightarrow {\mathcal A}^k$ is called textile if each coefficient of the power series expansion of the images $z(\mathrm {t})=\tau (y(\mathrm {t}))$ of vectors $y(\mathrm {t})=(y_1(\mathrm {t}),\ldots ,y_m(\mathrm {t}))\in \mathcal A_{\circ }^m$ depends polynomially on finitely many of the coefficients of the expansions of the vectors y: There exist, for $\ell \geq 0$ , polynomial vectors $F_{\ell }=(F_{1\ell },\ldots ,F_{k\ell })\in \mathbf {k}[\mathrm {y}_{m,\infty }]^k$ in countably many variables $\mathrm {y}_{m,\infty }=\{\mathrm {y}_{ij},\, 1\leq i\leq m,\,j\geq 1\}$ , each of them only depending on finitely many of the variables, such that, writing $y_j(\mathrm {t})=\sum _{i\geq 0} y_{ij}\cdot \mathrm {t}^i$ , we have

$$ \begin{align*} z(\mathrm{t})=\tau(y(\mathrm{t}))=\sum_{\ell\geq 0} F_{\ell}(y_{m,\infty})\cdot \mathrm{t}^{\ell}. \end{align*} $$

Textile maps induce ring homomorphisms

$$ \begin{align*} \alpha_{\tau}: \mathbf{k}[\mathrm{z}_{k,\infty}]\rightarrow \mathbf{k}[\mathrm{y}_{m,\infty}] \end{align*} $$

defined by $\alpha _{\tau }(\mathrm {z}_{n\ell })=F_{n\ell }(\mathrm {y}_{m,\infty })$ , for $n=1,\ldots ,k$ , and $\ell \geq 0$ . Conversely, any $\mathbf {k}$ -algebra homomorphism $\alpha :\mathbf {k}[\mathrm {z}_{k,\infty }]\rightarrow \mathbf {k}[\mathrm {y}_{m,\infty }]$ induces, in the case of formal power series $\mathcal A_{\circ }=\mathbf {k}_{\circ }[[\mathrm {t}]]$ , a textile map $\tau _{\alpha }:\mathcal A_{\circ }^m\rightarrow {\mathcal A}^k$ .Footnote ¹⁸

Every arquile map ${f_{\infty }}: \mathcal A_{\circ }^m\rightarrow {\mathcal A}^k,\, y(\mathrm {t})\rightarrow f(\mathrm {t},y(\mathrm {t}))$ , associated to a vector $f\in {\mathcal B}^k$ is textile. Also, if $y(\mathrm {t})=a(\mathrm {t})\cdot h(\mathrm {t})+z(\mathrm {t})$ denotes the componentwise division of $y\in \mathcal A_{\circ }^m$ by a power series $h(\mathrm {t})$ of order d, the associated division maps $y\rightarrow v=a\cdot h(\mathrm {t})$ and $y\rightarrow z$ are textile; see the section on Weierstrass division. These two types of maps together with their compositions and inversions will be the main instances of textile maps in this article. A textile isomorphism is a bijective textile map whose inverse is again textile. See [Reference Bruschek and HauserBH] for more information on textile maps.

A map $\tau : {\mathcal U}\subset \mathcal A_{\circ }^m\rightarrow {\mathcal A}^k$ defined on a subset ${\mathcal U}$ of $\mathcal A_{\circ }^m$ is called rationally textile if each coefficient of the power series expansion of the images $\tau (y)$ of vectors $y\in \mathcal A_{\circ }^m$ is a rational function of finitely many of the coefficients of the expansions of the vectors y. The denominators of the rational functions are not allowed to vanish when evaluating them on the coefficients of vectors in ${\mathcal U}$ . So a rationally textile map is defined on whole ${\mathcal U}$ . If $y(\mathrm {t})=a(\mathrm {t})\cdot h(\mathrm {t}) +z(\mathrm {t})$ denotes the componentwise Weierstrass division of $y\in \mathcal A_{\circ }^m$ by a series $h(\mathrm {t})$ of order d, the map $y\rightarrow a$ is rationally textile; cf. Theorem 3.1.

Topologies. The affine space $\mathcal A_{\circ }^m$ over $\mathcal A_{\circ }$ will be equipped with three topologies. The $\mathrm {t}$ -adic topology defined by the ideals $\langle \mathrm {t}\rangle ^d=\mathrm {t}^d\cdot \mathcal A_{\circ }$ of $\mathcal A_{\circ }$ , for $d\geq 0$ , as a basis of neighbourhoods of $0\in \mathcal A_{\circ }^m$ ; the textile topology, whose closed sets are the zerosets of (possibly infinitely many) polynomial equations in the coefficient vectors of the expansions of vectors $y\in \mathcal A_{\circ }^m$ ; and the arquile topology, whose closed sets are the zerosets of arquile maps. All three topologies will appear in the sequel. The arquile topology is Noetherian since a descending chain of arquile varieties ${\mathcal Y}_1\supset {\mathcal Y}_2\supset \ldots $ corresponds to an increasing chain of saturated ideals $I_1\subset I_2\subset \ldots $ of ${\mathcal B}$ with ${\mathcal Y}_j={\mathcal Y}(I_j)$ .

A subset of $\mathcal A_{\circ }^m$ is textile locally closed if it is the set-theoretic difference of two textile closed sets. Arquile subvarieties ${\mathcal Y}(f)$ of $\mathcal A_{\circ }^m$ as defined above are textile closed and $\mathrm {t}$ -adically closed, since arquile maps ${f_{\infty }}$ are textile continuous and $\mathrm {t}$ -adically continuous. Textile maps are textile continuous by definition. They are also $\mathrm {t}$ -adically continuous, since each coefficient of a power series vector in the image depends only on finitely many coefficients of the power series vector in the source. In particular, textile isomorphisms are $\mathrm {t}$ -adic homeomorphisms.

Arquile maps ${f_{\infty }}:\mathcal A_{\circ }^m\rightarrow {\mathcal A}^k$ associated to vectors $f\in {\mathcal B}^k$ map the $\mathrm {t}$ -adic neighbourhood $y+\langle \mathrm {t}\rangle ^c\cdot \mathcal A_{\circ }^m$ of a vector $y\in \mathcal A_{\circ }^m$ into the neighbourhood ${f_{\infty }}(y)+\langle \mathrm {t}\rangle ^c\cdot {\mathcal A}^k$ . They are hence uniformly Lipschitz continuous with Lipschitz constant 1 with respect to the $\mathrm {t}$ -adic metric on $\mathcal A_{\circ }$ . For division maps $y\rightarrow v=a\cdot h(\mathrm {t})$ and $y\rightarrow z$ sending $y\in \mathcal A_{\circ }^m$ to the summands $y=v+z$ of the componentwise division $y=a\cdot h(\mathrm {t})+ z$ of y by a series $h(\mathrm {t})$ of order d, the neighbourhood $\langle \mathrm {t}\rangle ^c\cdot \mathcal A_{\circ }^m$ is mapped into $\langle \mathrm {t}\rangle ^c\cdot \mathcal A_{\circ }^m$ . In contrast, the map $y\rightarrow a$ sending y to the quotient a maps $\langle \mathrm {t}\rangle ^c\cdot \mathcal A_{\circ }^m$ only into $\langle \mathrm {t}\rangle ^{c-d}\cdot \mathcal A_{\circ }^m$ , for all $c\geq d$ .

A textile closed subset ${\mathcal Z}$ of a subset ${\mathcal U}$ of $\mathcal A_{\circ }^m$ is cofinite in ${\mathcal U}$ if it can be defined (as a subset of ${\mathcal U}$ ) by finitely many polynomial equations in (finitely many) coefficients of the power series expansions of vectors of $\mathcal A_{\circ }^m$ . In the literature, such subsets are also known as cylindrical or generically stable. The analogous definition applies to textile locally closed subsets.

Singular locus. Let $y=y(\mathrm {t})\in \mathcal A_{\circ }^m$ be a given power series vector. We may associate to y the evaluation map $\varepsilon _y:{\mathcal B}\rightarrow \mathcal A_{\circ }$ sending the variable $\mathrm {y}$ to y. Its kernel $\mathrm {Ker}(\varepsilon _y)$ is a prime ideal of ${\mathcal B}$ which will be denoted by ${\mathcal P}_y$ ,

$$ \begin{align*} {\mathcal P}_y=\{f\in{\mathcal B},\, f(\mathrm{t},y(\mathrm{t}))=0\}. \\[-20pt] \end{align*} $$

It is called the ideal of relations between the components of y. This is, in contrast to the finite-dimensional situation of points in affine space $\mathbb A^m_{\mathbf {k}}$ over a field, never a maximal ideal. The map sending y to ${\mathcal P}_y$ is injective, since $y=y(\mathrm {t})$ can be reconstructed from the elements $\mathrm {y}_i-y_i(\mathrm {t})$ of ${\mathcal P}_y$ .Footnote ¹⁹ We denote by ${\mathcal B}_{{\mathcal P}_y}$ the localisation of ${\mathcal B}$ at ${\mathcal P}_y$ .

Denote by $\overline {\mathcal P}_y$ the image of ${\mathcal P}_y$ in ${\mathcal B}/I$ . We then say that y is an arq.regular point of ${\mathcal Y}$ if the localisation $({\mathcal B}/I)_{\overline {\mathcal P}_y}={\mathcal B}_{{\mathcal P}_y}/I\cdot {\mathcal B}_{{\mathcal P}_y}$ is a regular local ring. By the above, this is equivalent to saying that $p_y$ is a regular point of the scheme Y. The locus of arq.regular points of ${\mathcal Y}$ is denoted by $\mathrm {Arq.Reg}({\mathcal Y})$ , and we set $\mathrm {Arq.Sing}({\mathcal Y})={\mathcal Y}\setminus \mathrm {Arq.Reg}({\mathcal Y})$ , the arq.singular locus of ${\mathcal Y}$ . If y is an arq.regular point of ${\mathcal Y}$ , it necessarily lies in a single irreducible component of ${\mathcal Y}$ ; cf. Proposition 2.3 and the theorem below. For local investigations at arq.regular points we may therefore assume that I is prime. The following version of the Jacobian criterion will be used repeatedly; cf. [Reference de Jong and PfisterdJP], Theorem 4.3.15, [Reference RuizRu], Proposition 4.3, and [Reference ZariskiZa], Theorem 7 and its corollary (recall that the ground field $\mathbf {k}$ is assumed to be perfect).

As a corollary, we get the following.

Theorem 2.6. Arq.regular and arq.singular locus

Let ${\mathcal Y}={\mathcal Y}(I)$ be an arquile variety in $\mathcal A_{\circ }^m$ defined by a saturated ideal I of ${\mathcal B}$ , and let $y=y(\mathrm {t})\in {\mathcal Y}$ be a point of ${\mathcal Y}$ with prime ideal ${\mathcal P}_y$ . Denote by r the height of the ideal $I\cdot {\mathcal B}_{{\mathcal P}_y}$ .

(a) The point y is an arq.regular point of ${\mathcal Y}$ , if and only if there exist elements $f_1,\ldots ,f_r$ in I and an $(r\times r)$ -minor g of the relative Jacobian matrix ${\partial }_{\mathrm {y}}f$ of $f=(f_1,\ldots ,f_r)$ with respect to $\mathrm {y}$ which does not lie in ${\mathcal P}_y$ ; that is, which satisfies $g(y)\neq 0$ .

(b) For every arq.regular point y of ${\mathcal Y}$ , one can choose $f_1,\ldots ,f_r\in I$ and the minor $g\in {\mathcal B}\setminus {\mathcal P}_y$ as in assertion (a) such that $y\in {\mathcal Y}(f)\setminus {\mathcal Y}(g)$ and

$$ \begin{align*} {\mathcal Y}\setminus {\mathcal Y}(g) ={\mathcal Y}(f)\setminus {\mathcal Y}(g). \end{align*} $$

(c) The arq.regular locus $\mathrm {Arq.Reg}({\mathcal Y})$ of ${\mathcal Y}$ is covered by finitely many arquile open subsets of the form ${\mathcal Y}(f)\setminus {\mathcal Y}(g)$ as in assertion (b), for varying values r.

(d) The intersections ${\mathcal Y}_i\cap {\mathcal Y}_j$ of irreducible components of ${\mathcal Y}$ lie inside $\mathrm {Arq.Sing}({\mathcal Y})$ .

(e) The arq.singular locus $\mathrm {Arq.Sing}({\mathcal Y})$ is an arquile closed and proper subset of ${\mathcal Y}$ . The height of its defining ideal J in ${\mathcal B}$ is larger than the height of I.

Proof. Let us show (a). By the Jacobian criterion, there exist elements $f_1,\ldots ,f_r$ in I such that the entire Jacobian matrix ${\partial }_{\mathrm {t}\mathrm {y}}f$ of $f=(f_1,\ldots ,f_r)$ has rank r modulo ${\mathcal P}_y$ . So there is an $(r\times r)$ -minor g of ${\partial }_{\mathrm {t}\mathrm {y}}f$ which satisfies $g(y)\neq 0$ . Deriving the equation $f(\mathrm {t},y(\mathrm {t}))=0$ with respect to $\mathrm {t}$ gives ${\partial }_{\mathrm {t}} f(y(\mathrm {t}))=-{\partial }_{\mathrm {y}} f(y(\mathrm {t}))\cdot {\partial }_{\mathrm {t}} y(\mathrm {t})$ . This shows that g can actually be chosen from the relative Jacobian matrix ${\partial }_{\mathrm {y}}f$ of f with respect to $\mathrm {y}$ . This proves the ‘only if’ part. The converse implication follows directly from the Jacobian criterion.

To prove (b), the strategy is as follows: select the irreducible component ${\mathcal Y}_1$ of ${\mathcal Y}$ passing through y (anticipating here assertion (d)) and represent it as an irreducible component of a complete intersection defined by some elements $f_1,\ldots ,f_r$ in ${\mathcal B}$ . From there, new elements $f^{\prime }_1,\ldots ,f^{\prime }_r$ will be constructed multiplying $f_1,\ldots ,f_r$ with the square $h^2$ of an element h of ${\mathcal B}$ whose zero-locus ${\mathcal Y}(h)$ contains all components of ${\mathcal Y}$ but ${\mathcal Y}_1$ . The chosen minor $g'$ of the Jacobian matrix of $f'=(f^{\prime }_1,\ldots ,f^{\prime }_r)$ will then be a multiple of h from which we infer that $f'$ and $g'$ have the required properties.

Denote by $I_0\subset I$ the ideal of ${\mathcal B}$ generated by elements $f_1,\ldots ,f_r$ of I as indicated in assertion (a). The height of $I_0\cdot {\mathcal B}_{{\mathcal P}_y}$ is at most r. By the choice of $f_1,\ldots f_r$ and the Jacobian criterion it is exactly r. This implies that y is an arq.regular point of ${\mathcal Y}(I_0)$ ; that is, that the ring $({\mathcal B}/I_0)_{\overline {\mathcal P}_y}={\mathcal B}_{{\mathcal P}_y}/I_0\cdot {\mathcal B}_{{\mathcal P}_y}$ is regular. In particular, $I_0\cdot {\mathcal B}_{{\mathcal P}_y}$ is a prime ideal in ${\mathcal B}_{{\mathcal P}_y}$ . It is the localisation $I_1\cdot {\mathcal B}_{{\mathcal P}_y}$ of a prime ideal $I_1$ of ${\mathcal B}$ , the ideal defining the irreducible component ${\mathcal Y}_1$ of ${\mathcal Y}$ passing through y. Therefore, $I_1$ is an associated prime of both I and $I_0$ . We can write $I_1=(I_0:h)$ as a colon ideal with some element $h\in {\mathcal B}\setminus {\mathcal P}_y$ .

We now set $f_i'=h^2\cdot f_i$ , for $i=1,\ldots ,r$ , and let $g'$ denote the minor of ${\partial }_{\mathrm {y}} f'$ defined by the same columns as those which were taken for g. Observe that h divides $g'$ . As $g(y)\neq 0$ and $h(y)\neq 0$ , we also have $g'(y)\neq 0$ . Clearly, ${\mathcal Y}(h)\subset {\mathcal Y}(g')$ . Let $I'=h^2\cdot I_0$ denote the ideal of ${\mathcal B}$ generated by $f_1',\ldots ,f_r'$ . As $I'\subset I$ , we have ${\mathcal Y}(I)\subset {\mathcal Y}(I')$ . Conversely, $I_1=(I_0:h)$ implies that $h\cdot I_1\subset I_0$ and then $h^3\cdot I_1\subset I'$ . We get

$$ \begin{align*} {\mathcal Y}(I')\subset{\mathcal Y}(h^3\cdot I_1)={\mathcal Y}(h)\cup{\mathcal Y}(I_1)\subset {\mathcal Y}(h)\cup{\mathcal Y}(I) \end{align*} $$

and thus

$$ \begin{align*} {\mathcal Y}(I')\setminus{\mathcal Y}(h)\subset {\mathcal Y}(I)\setminus {\mathcal Y}(h), \end{align*} $$

from which, by $I'\subset I$ , follows the equality

$$ \begin{align*} {\mathcal Y}(I)\setminus {\mathcal Y}(h)={\mathcal Y}(I')\setminus{\mathcal Y}(h). \end{align*} $$

The inclusion ${\mathcal Y}(h)\subset {\mathcal Y}(g')$ finally implies

$$ \begin{align*} {\mathcal Y}(I')\setminus{\mathcal Y}(g')={\mathcal Y}(I)\setminus {\mathcal Y}(g'), \end{align*} $$

as was claimed in (b). Assertion (c) is obvious since the arquile topology is Noetherian.

To prove (d), let $I=I_1\cap \ldots \cap I_s$ be the irredundant prime decomposition of I. Order the ideals $I_i$ such that y is contained in ${\mathcal Y}(I_i)$ for $i\leq t$ and not contained in ${\mathcal Y}(I_i)$ for $i>t$ , where $1\leq t\leq s$ . Then $I\cdot {\mathcal B}_{{\mathcal P}_y}= I_1\cdot {\mathcal B}_{{\mathcal P}_y}\cap \ldots \cap I_t\cdot {\mathcal B}_{{\mathcal P}_y}$ . If y is contained in two irreducible components of ${\mathcal Y}$ , then $I\cdot {\mathcal B}_{{\mathcal P}_y}$ is the intersection of at least two prime ideals and hence itself is not a prime ideal. Then ${\mathcal B}_{{\mathcal P}_y}/I\cdot {\mathcal B}_{{\mathcal P}_y}$ is not regular so that $y\in \mathrm {Arq.Sing}({\mathcal Y})$ .

We conclude with the proof of assertion (e). The arq.regular locus $\mathrm {Arq.Reg}({\mathcal Y})$ of ${\mathcal Y}$ is open by (c). For the remaining assertions, assume first that ${\mathcal Y}$ is irreducible, say, that I is prime. Then $({\mathcal B}/I)_I$ is a field, in particular regular. By the Jacobian criterion there exist elements $f_1,\ldots ,f_r$ in I, where r is the height of $I\cdot {\mathcal B}_I$ , and a minor g of ${\partial }_{\mathrm {t}\mathrm {y}}f$ of $f=(f_1,\ldots ,f_r)$ which is not contained in I. But I is saturated and ${\mathcal Y}={\mathcal Y}(I)$ is nonempty, so there is a point $y\in {\mathcal Y}={\mathcal Y}(I)$ for which $g(y)\neq 0$ . This implies by the Jacobian criterion that $({\mathcal B}/I)_{\overline {\mathcal P}_y}$ is regular, so that y is an arq.regular point of ${\mathcal Y}$ . Hence, $\mathrm {Arq.Reg}({\mathcal Y})$ is nonempty and $\mathrm {Arq.Sing}({\mathcal Y})$ is a proper subset of ${\mathcal Y}$ . Using the argument from the proof of assertion (a), we may choose for g a minor of the relative Jacobian matrix ${\partial }_{\mathrm {y}} f$ of f with respect to $\mathrm {y}$ . Let J denote the ideal of ${\mathcal B}$ defining $\mathrm {Arq.Sing}({\mathcal Y})$ . By definition, $I\subset J$ . But $g\in J$ and $g\not \in I$ , so that I is strictly contained in J. As I is prime, Krull’s principal ideal theorem ensures that the height of J is larger than the height of I. This establishes assertion (e) for prime ideals.

Let now I be arbitrary. It is easy to see that $\mathrm {Arq.Sing}({\mathcal Y})$ is composed of the union of the arq.singular loci $\mathrm {Arq.Sing}({\mathcal Y}_i)$ of the irreducible components ${\mathcal Y}_i$ of ${\mathcal Y}$ together with the union of the pairwise intersections ${\mathcal Y}_i\cap {\mathcal Y}_j$ of these components, for $i\neq j$ . So it is defined by the saturation J of the intersection of the ideals $(I_i+I_j)\cap J_i$ over all $i\neq j$ , where $J_i$ denotes the ideal of ${\mathcal B}$ defining $\mathrm {Arq.Sing}({\mathcal Y}_i)$ . We already know by the preceding paragraph that the height of $J_i$ is larger than the height of $I_i$ . And as $I=I_1\cap \ldots \cap I_s$ is irredundant, the height of $I_i+ I_j$ is also larger than the heights of both $I_i$ and $I_j$ , for all $i\neq j$ . It follows that the height of J is larger than the height of I. This concludes the proof of the theorem.

It is possible to characterise saturated prime ideals in terms of arq.regularity, at least for convergent power series over $\mathbf {k} = {\mathbb C}$ , using Gabrielov’s theorem [Reference GabrielovGa, Reference IzumiIz].

3 Weierstrass division

We will give two versions of this theorem, one with coefficients in a field and one with coefficients in a complete local ring, both in the case of series in just one variable $\mathrm {t}$ . The main point is to describe how the quotient and the remainder of the division depend on the divisor and the series to be divided. In the first case, this requires to work on the strata where the divisor has a fixed order as a power series in $\mathrm {t}$ ; in the second case one considers deformations of a given divisor obtained by varying its coefficients by elements in the complete local ring. All results in this section are classical and hold equally for formal, convergent and algebraic power series.Footnote ²⁰

(1) Division on strata. The results apply to any of the rings $\mathbf {k}_{\circ }[[\mathrm {t}]]$ , $\mathbf {k}_{\circ }\{\mathrm {t}\}$ , $\mathbf {k}_{\circ }\{\mathrm {t}\}_s$ or $\mathbf {k}_{\circ }\langle \mathrm {t}\rangle $ of power series without constant term.Footnote ²¹ Denote by $\mathbf {k}_{\circ }[\mathrm {t}]_{\leq d}$ the vector space of polynomials in $\mathrm {t}$ of degree $\leq d$ without constant term.

Theorem 3.1. Weierstrass division on strata of constant order

Assume given a power series $g\in {\mathcal B}$ in variables $\mathrm {t}$ and $\mathrm {y}=(\mathrm {y}_1,\ldots ,\mathrm {y}_m)$ .

(a) Let $y=y(\mathrm {t})\in \mathcal A_{\circ }^m$ be a power series vector without constant term such that $g(y)=g(\mathrm {t},y(\mathrm {t}))$ is nonzero of order d. For every $w\in \mathcal A_{\circ }$ there exist a unique power series $a\in \mathcal A_{\circ }$ and a unique polynomial $z\in \mathbf {k}_{\circ }[\mathrm {t}]_{\leq d}$ of degree $\leq d$ such that

$$ \begin{align*} w=g(y)\cdot a+z. \end{align*} $$

(b) Denote by ${\mathcal S}_d \subset \mathcal A_{\circ }^m$ the textile locally closed stratum of vectors $y\in \mathcal A_{\circ }^m$ for which $g(y)$ is nonzero of order d. The maps

$$ \begin{align*} \psi:{\mathcal S}_d\times \mathcal A_{\circ}\rightarrow \mathbf{k}_{\circ}[\mathrm{t}]_{\leq d}\times \langle\mathrm{t}\rangle^d\cdot\mathcal A_{\circ} \end{align*} $$

$$ \begin{align*} (y,w)\rightarrow (z,v) \end{align*} $$

and

$$ \begin{align*} \chi:{\mathcal S}_d\times \mathcal A_{\circ}\rightarrow \mathbf{k}_{\circ}[\mathrm{t}]_{\leq d}\times \mathcal A_{\circ} \end{align*} $$

$$ \begin{align*} (y,w)\rightarrow (z, a) \end{align*} $$

given by the division $w=v+z=g(y)\cdot a+z$ are textile, respectively rationally textile.

Note that letting g in assertion (a) be independent of $\mathrm {y}$ , say, taking $g\in {\mathcal A}$ a series in $\mathrm {t}$ of order d, one obtains the usual version of the division. The requirement that the quotient $a\in \mathcal A_{\circ }$ has no constant term is only imposed for technical reasons that will simplify the exposition later on. This convention implies that the remainder of the division is a polynomial of degree $\leq d$ instead of degree $\leq d-1$ .

Note also that in assertion (b) the map $\psi $ does depend on y since the divisor $g(y)$ in $w=v+z= g(y)\cdot a+z$ depends on y. The map $\chi $ is only rationally textile as can be seen by taking $g(\mathrm {y})=\mathrm {y}$ , $y=c\cdot t^d$ , $c\in \mathbf {k}^*$ and $w=t^d$ . Then $a=\frac {1}{c}$ is rational in the coefficient of y.

We will apply the Weierstrass division in the case where the series w which is divided by $g(y)$ varies over the components $y_j$ of the vector y itself. In this case, it will be possible to determine explicitly the sets of quotients and remainders of the division as y varies in ${\mathcal S}_d$ ; see Theorem 3.4. This description will become crucial for the proof of the factorisation theorem 7.1. We first need an auxiliary result. Recall that ${\mathcal B}$ denotes one of the rings $\mathbf {k}[[\mathrm {t},\mathrm {y}]]$ , $\mathbf {k}\{\mathrm {t},\mathrm {y}\}$ or $\mathbf {k}\langle \mathrm {t},\mathrm {y}\rangle $ . For later applications we write $\mathrm {z}$ instead of $\mathrm {y}$ .

Lemma 3.2. Let $g\in {\mathcal B}$ be a formal power series in variables $\mathrm {t}$ and $\mathrm {z}=(\mathrm {z}_1,\ldots ,\mathrm {z}_m)$ . For every choice of vectors z in $\mathcal A_{\circ }^m$ and $v=g(z)\cdot a\in \langle g(z)\rangle \cdot \mathcal A_{\circ }^m$ , the ideals of $\mathcal A_{\circ }$ generated by $g(z)$ and $g(z+v)$ coincide,

$$ \begin{align*} \langle g(z)\rangle =\langle g(z+v)\rangle. \end{align*} $$

Remark 3.3. The lemma implies that, whenever $g(z)$ is nonzero, the quotient $g(z)/g(z+g(z)\cdot a)$ is an invertible power series in $\mathcal A$ . It is then easy to see that, denoting by ${\mathcal S}\subset \mathcal A_{\circ }^m$ the set of vectors z such that $g(z)\neq 0$ , the map

$$ \begin{align*} {\mathcal S}\times \mathcal A_{\circ}^m\rightarrow \mathcal A, \end{align*} $$

$$ \begin{align*} (z,a)\rightarrow \frac{g(z)}{g(z+g(z)\cdot a)}, \end{align*} $$

is an arquile map.

Proof. Write $v=g(z)\cdot a$ for some $a\in \mathcal A_{\circ }^m$ . Taylor expansion gives

$$ \begin{align*} g(z+v)=g(z+g(z)\cdot a)=g(z)+ r(z,g(z)\cdot a) \end{align*} $$

with some power series r in $\mathrm {t}$ , $\mathrm {z}$ and new variables $\mathrm {w}=(\mathrm {w}_1,\ldots ,\mathrm {w}_m)$ so that r is at least of order $1$ in $\mathrm {w}$ . Factoring $g(z)$ from the right-hand side we get $g(z+v)=g(z)\cdot u(z,a)$ for some series u in $\mathrm {t}$ , $\mathrm {z}$ and $\mathrm {w}$ . The series $u(\mathrm {z},\mathrm {w})$ is invertible since its constant term is $1$ . As z and a vanish at $\mathrm {t}=0$ , the evaluation $u(z,a)$ of u is again invertible, now as a series in $\mathcal A$ . The required equality $\langle g(z)\rangle =\langle g(z+v)\rangle $ of ideals in $\mathcal A_{\circ }$ follows.

Theorem 3.4. Let $g\in {\mathcal B}$ be a power series in $\mathrm {t}$ and $\mathrm {y}$ . Denote by ${\mathcal R}_d\subset \mathcal A_{\circ }^m$ the $\mathbf { k}$ -vector space of polynomial vectors in $\mathrm {t}$ of degree $\leq d$ and without constant term,

$$ \begin{align*} {\mathcal R}_d=(\mathbf{k}_{\circ}[\mathrm{t}]_{\leq d})^m, \end{align*} $$

and by ${\mathcal V}_d=\langle \mathrm {t}\rangle ^d\cdot \mathcal A_{\circ }^m$ the $\mathcal A_{\circ }$ -submodule of $\mathcal A_{\circ }^m$ generated in each component by $\mathrm {t}^d$ . Denote furthermore by ${\mathcal Z}_d$ the intersection ${\mathcal Z}_d={\mathcal S}_d\cap {\mathcal R}_d$ , with ${\mathcal S}_d=\{y\in \mathcal A_{\circ }^m,\, g(y)\neq 0,\, \mathrm {ord}\, g(y)=d\}$ as above.

(a) For every vector $y\in {\mathcal S}_d$ there exist unique vectors $a\in \mathcal A_{\circ }^m$ and $z\in {\mathcal Z}_d$ such that

$$ \begin{align*} y=g(y)\cdot a+z \end{align*} $$

is in each component of y the Weierstrass division of $y_j$ by $g(y)$ .

(b) The map

$$ \begin{align*} \psi_d:{\mathcal S}_d\rightarrow {\mathcal Z}_d\times {\mathcal V}_d, \end{align*} $$

$$ \begin{align*} y\rightarrow (z,v), \end{align*} $$

given by the division $y=v+z=g(y)\cdot a +z$ is a textile isomorphism, and

$$ \begin{align*} \widetilde \psi_d:{\mathcal S}_d\rightarrow {\mathcal Z}_d\times \mathcal A_{\circ}^m, \end{align*} $$

$$ \begin{align*} y\rightarrow (z,a), \end{align*} $$

is a rationally textile isomorphism.

Proof. For assertion (a), the Weierstrass division theorem 3.1 ensures the existence and uniqueness of $v\in {\mathcal V}_d$ and $z\in {\mathcal R}_d$ . So we only have to show that $z\in {\mathcal S}_d$ . But this is just the content of the preceding lemma, whence (a). Let us prove assertion (b). The maps $\psi $ and $\widetilde \psi $ are textile, respectively rationally textile, maps by Theorem 3.1. The map

$$ \begin{align*} {\mathcal Z}_d\times{\mathcal V}_d\rightarrow \mathcal A_{\circ}^m:(z,v)\rightarrow v+z \end{align*} $$

sends, again by the lemma, ${\mathcal Z}_d\times {\mathcal V}_d$ into ${\mathcal S}_d$ and is therefore the inverse of $\psi $ . Finally, as $z\in {\mathcal Z}_d$ , the ideal of $\mathcal A_{\circ }$ generated by $g(z)$ equals $\langle \mathrm {t}\rangle ^d$ . We may therefore write $v\in {\mathcal V}_d$ as $v=g(z)\cdot b$ with a unique $b\in \mathcal A_{\circ }^m$ . The map associating $b=v/g(z)$ to a pair $(z,v)\in {\mathcal Z}_d\times {\mathcal V}_d$ is clearly rationally textile. By Lemma 3.2, both $g(z)$ and $g(v+z)$ generate the same ideal in $\mathcal A_{\circ }$ ; hence, there is a unique $a\in \mathcal A_{\circ }^m$ such that $v=g(v+z)\cdot a $ . Setting $y=v+z$ we see that $v=g(y)\cdot a$ , say, $\widetilde \psi (y)= (z,a)$ and $\widetilde \psi $ is bijective. As we have

$$ \begin{align*} a={v\over g(y)}={v\over g(v+z)} = {v\over g(z)}\cdot {g(z)\over g(v+z)}, \end{align*} $$

the map associating a to a pair $(z,v)$ is again rationally textile. This gives (b) and finishes the proof.

(2) Division of deformations: To have an isomorphism $\psi _d$ as in assertion (b) of the preceding theorem it was necessary to restrict the set of vectors y to the stratum ${\mathcal S}_d$ . Otherwise, for arbitrary y, the order of $g(y)$ may jump, so that the space of remainders could not be defined uniformly. Instead of working with a stratification and the Cartesian product decomposition ${\mathcal S}_d\cong {\mathcal Z}_d\times {\mathcal V}_d$ of the stratum ${\mathcal S}_d$ , one may also perform the division for series whose coefficients lie in a complete local ring; that is, for deformations of given power series vectors in $\mathcal A_{\circ }^m$ . In this case, no stratification is necessary. This kind of division will be explained in the second part of this section. First we need a couple of definitions.

Throughout the remainder of this section, only formal power series will be considered, so that $\mathcal A_{\circ }=\mathbf {k}_{\circ }[[\mathrm {t}]]$ , ${\mathcal B}=\mathbf {k}[[\mathrm {t},\mathrm {y}]]$ . The convergent case can be treated similarly but requires considerably more efforts; cf. Theorem 2, A.II.4, in [Reference HauserHa1].

Let S be a local ring, not necessarily Noetherian, with maximal ideal $M_S$ and residue field $S/M_S=\mathbf {k}$ . It will always be equipped with a filtred topology defined by a decreasing sequence of ideals $J_k$ satisfying $J_k\cdot J_{\ell }\subset J_{k+\ell }$ . We assume that $M_S^k\subset J_k$ for all k and that S is complete. Further, to avoid complications, we assume that $\mathbf {k}$ embeds into S as a field of coefficients.Footnote ²² Let us denote by ${\mathcal A}_{S,\circ }=S_{\circ }[[\mathrm {t}]]$ the ring of formal power series in $\mathrm {t}$ with coefficients in S and without constant term (in S). Series in ${\mathcal A}_{S,\circ }$ will be distinguished from series in $\mathcal A_{\circ }$ by a tilde. Taking the coefficients of a power series vector $\widetilde y\in {\mathcal A}_{S,\circ }^m$ modulo $M_S$ gives a power series vector y in $\mathcal A_{\circ }^m$ , the reduction of $\widetilde y$ modulo $M_S$ . A deformation of a vector $y\in \mathcal A_{\circ }^m$ over S is, by definition, a vector $\widetilde y\in {\mathcal A}_{S,\circ }^m$ whose reduction modulo $M_S$ equals y. The set of all deformations of y parametrised by S is denoted by $({\mathcal A}_{S,\circ }^m,y)$ .Footnote ²³

Theorem 3.6. Weierstrass division for deformations

We restrict to the case of formal power series. For a given power series $g\in {\mathcal B}$ , denote by $w=g(y)\cdot a+z$ the Weierstrass division of a series $w\in \mathcal A_{\circ }$ by $g(y)$ for some $y\in \mathcal A_{\circ }^m$ for which $g(y)$ is nonzero of order d, with $a\in \mathcal A_{\circ }$ and $z\in \mathbf {k}_{\circ }[\mathrm {t}]_{\leq d}$ . Let S be a complete local ring as before with maximal ideal $M_S$ and residue field $S/M_S=\mathbf {k}$ embedding into S.

(a) For every vector $\widetilde y \in {\mathcal A}_{S,\circ }^m$ and every series $\widetilde w\in {\mathcal A}_{S,\circ }$ , deformations of y and w respectively, there exist a unique power series $\widetilde a \in {\mathcal A}_{S,\circ }$ and a unique polynomial $\widetilde z \in S[\mathrm {t}]_{\leq d}$ of degree $\leq d$ without constant term, both deformations of a and z respectively, such that

$$ \begin{align*} \widetilde w = g(\widetilde y) \cdot \widetilde a +\widetilde z. \end{align*} $$

(b) The coefficients of $\widetilde a$ and $\widetilde z$ are the evaluations in the coefficients of $\,\widetilde y$ and $\widetilde w$ of formal power series.

The statement also holds for convergent series but is then more delicate to prove; see Theorem 2, A.II.4, in [Reference HauserHa1].

Proof. We show that the map

$$ \begin{align*} \alpha:({\mathcal A}_{S,\circ}^m,a)\times (S[\mathrm{t}]_{\leq d},z) \rightarrow ({\mathcal A}_{S,\circ},w) \end{align*} $$

$$ \begin{align*} (\widetilde a, \widetilde z)\rightarrow \widetilde w = g(\widetilde y) \cdot \widetilde a +\widetilde z \end{align*} $$

is an isomorphism. Using that $\mathbf {k}\subset S$ , we write $\widetilde y= y+y^{\circ }$ with $y^{\circ }\in M_S\cdot {\mathcal A}_{S,\circ }^m$ and, accordingly, $g(\widetilde y)= g(y)+h(y,y^{\circ })$ with $h\in \mathbf {k}_{\circ }[[\mathrm {t},\mathrm {y},\mathrm {y}^{\circ }]]$ , where $\mathrm {y}^{\circ }=(\mathrm {y}_1^{\circ },\ldots ,\mathrm {y}_m^{\circ })$ denotes a new vector of variables. Observe here that $h(y,y^{\circ })\in M_S\cdot {\mathcal A}_{S,\circ }$ . By the assumption on y we can write $g(y)=\mathrm {t}^d\cdot u$ with a unit $u\in \mathbf {k}[[\mathrm {t}]]$ . Without loss of generality, we may assume that $u=1$ , say $g(y)=\mathrm {t}^d$ . The map

$$ \begin{align*} \gamma(\widetilde a, \widetilde z)\rightarrow [ g(y)-g(\widetilde y)] \cdot \widetilde a \end{align*} $$

is the multiplication of $\widetilde a$ with $\mathrm {t}^d-g(\widetilde y)= g(y)-g(\widetilde y)=h(y,y^{\circ })\in M_S\cdot {\mathcal A}_{S,\circ }$ . The map $\beta $ given by $\beta (\widetilde a, \widetilde z)= \mathrm {t}^d\cdot \widetilde a + \widetilde z$ is clearly an isomorphism. It therefore suffices to show that

$$ \begin{align*} \alpha\circ\beta^{-1} =\mathrm{Id}_{({\mathcal A}_{S,\circ},w)} - \gamma\circ\beta^{-1} \end{align*} $$

is an isomorphism. To see this, observe first that the map $\gamma \circ \beta ^{-1}$ sends the maximal ideal $M_S\cdot {\mathcal A}_{S,\circ }$ of ${\mathcal A}_{S,\circ }$ into $M_S^2\cdot {\mathcal A}_{S,\circ }$ . Therefore, the geometric series in $\gamma \circ \beta ^{-1}$ , say

$$ \begin{align*} \delta=\sum_{k=0}^{\infty} (\gamma\circ\beta^{-1})^k, \end{align*} $$

induces, by the completeness of S, a well-defined map $\delta :{\mathcal A}_{S,\circ }\rightarrow {\mathcal A}_{S,\circ }$ which is the inverse to $\alpha \circ \beta ^{-1}$ . This proves the theorem.

We will apply the preceding theorem to the division of the components $\widetilde y_j$ of a deformation $\widetilde y$ of $y\in \mathcal A_{\circ }^m$ by $g(\widetilde y)$ analogously as we did in the first part of this section with y itself. Let $y\in \mathcal A_{\circ }^m$ be given with $g(y)\neq 0$ of order d and induced division $y=g(y)\cdot a+z$ with $a\in \mathcal A_{\circ }^m$ and $z\in {\mathcal Z}_d$ as in Theorem 3.4.

Theorem 3.8. For a given power series $g(\mathrm {t},\mathrm {y})\in {\mathcal B}$ denote by $y=g(y)\cdot a+z$ the componentwise Weierstrass division of a vector $y\in \mathcal A_{\circ }^m$ by $g(y)$ for some y for which $g(y)$ is nonzero of order d, with $a\in \mathcal A_{\circ }^m$ and $z\in (\mathbf {k}_{\circ }[\mathrm {t}]]_{\leq d})^m$ .

(a) For every deformation $\widetilde y \in ({\mathcal A}_{S,\circ }^m,y)$ of y there exist a unique power series vector $\widetilde a \in {\mathcal A}_{S,\circ }^m$ and a unique polynomial vector $\widetilde z \in (S[\mathrm {t}]_{\leq d})^m$ without constant term, both deformations of a and z respectively, such that

$$ \begin{align*} \widetilde y = g(\widetilde y) \cdot \widetilde a +\widetilde z. \end{align*} $$

(b) The coefficients of $\widetilde a$ and $\widetilde z$ are formal power series in the coefficients of $\widetilde y$ .

4 Division modules

This section can be skipped on first reading. The proofs of the factorisation and linearisation theorems are based on the decomposition of a power series vector $y\in \mathcal A_{\circ }^m$ into a sum $y=v+z$ where z belongs to a finite-dimensional space of polynomial vectors and where v is the part of y on which the linearisation can be achieved. The decomposition is done by the componentwise Weierstrass division of y by a suitable divisor. This section characterises possible choices of such divisors.

Definition 4.1. A division module for a vector $f=f(\mathrm {t},\mathrm {y})\in {\mathcal B}^k$ is a $\mathcal B_{\circ }$ -submodule ${\mathcal G}$ of $\mathcal B_{\circ }^m$ , with $\mathcal B_{\circ }={\mathcal B}\cap \mathbf {k}_{\circ }[[\mathrm {t},\mathrm {y}]]$ , which is a Cartesian product

$$ \begin{align*} {\mathcal G}= \langle g_1\rangle \times \ldots\times \langle g_m\rangle \subset\mathcal B_{\circ}^m \end{align*} $$

of nonzero principal ideals $\langle g_i\rangle :=g_i\cdot \mathcal B_{\circ }$ of $\mathcal B_{\circ }$ , with $g_i\in {\mathcal B}$ , satisfying the following conditions:

1. (i) For all $z\in \mathcal A_{\circ }^m$ and $v\in {\mathcal G}(z):= g_1(z)\cdot \mathcal A_{\circ } \times \ldots \times g_m(z)\cdot \mathcal A_{\circ }\subset \mathcal A_{\circ }^m$ we have

   $$ \begin{align*} {\mathcal G}(z)={\mathcal G}(z+v). \end{align*} $$

2. (ii) Write elements v of ${\mathcal G}(z)$ as $v=\Delta _g(z)\cdot a$ with column vector $a\in \mathcal A_{\circ }^m$ and a diagonal matrix $\Delta _g=\mathrm {diag}(g_1,\ldots ,g_m)\in {\mathcal B}^{m\times m}$ . There exists a (formal, convergent, respectively algebraic) power series vector $p\in \mathbf {k}[[\mathrm {t},\mathrm {z},\mathrm {a}]]^m$ in three sets of variables $\mathrm {t}$ , $\mathrm {z}=(\mathrm {z}_1,\ldots ,\mathrm {z}_m)$ and $\mathrm {a}=(\mathrm {a}_1,\ldots ,\mathrm {a}_m)$ , which is at least of order $2$ in $\mathrm {a}$ and satisfies for all z, $a\in \mathcal A_{\circ }^m$ ,

   $$ \begin{align*} f(z+\Delta_g(z)\cdot a) = f(z)+{\partial}_{\mathrm{y}}f(z)\cdot \Delta_g(z)\cdot a+{\partial}_{\mathrm{y}}f(z)\cdot \Delta_g(z)\cdot p(z,a). \end{align*} $$

Remarks 4.2. (a) Condition (i) can be rewritten as an equality of ideals in $\mathcal A_{\circ }$ , say $\langle g_i(z)\rangle =\langle g_i(z+\Delta _g(z)\cdot a)\rangle $ , for all z and a in $\mathcal A_{\circ }^m$ , and $i=1,\ldots ,m$ . Phrased differently, there exists an invertible matrix $U\in \mathrm {Gl}_m(\mathbf {k}[[\mathrm {t},\mathrm {z},\mathrm {a}]])$ (with formal, convergent, respectively algebraic entries) such that

$$ \begin{align*} \Delta_g(z) = \Delta_g(z+\Delta_g(z)\cdot a)\cdot U(z,a) \end{align*} $$

holds for all z and a.

(b) Condition (i) appears in Lemma 3.2, which is used in the proof of Theorem 3.4 and in the proof of Proposition 5.2.

(c) Condition (ii) ensures that the terms of order $\geq 2$ of the Taylor expansion of $f(z+\Delta _g(z)\cdot a)$ belong to the module generated by the linear terms ${\partial }_{\mathrm {y}}f(z)\cdot \Delta _g(z)\cdot a$ . This module equals the $\mathcal A_{\circ }$ -submodule ${\partial }_{\mathrm {y}}f(z)\cdot {\mathcal G}(z)$ of $\mathcal A_{\circ }^k$ which is the image of ${\mathcal G}(z)$ under the tangent map at z of the arquile map ${f_{\infty }}:\mathcal A_{\circ }^m\rightarrow \mathcal A_{\circ }^k$ induced by f. In the terminology of [Reference Bruschek and HauserBH], condition (ii) signifies that for each $z \in \mathcal A_{\circ }^m$ , the map

$$ \begin{align*} \alpha_z:{\mathcal G}(z)\rightarrow \mathcal A_{\circ}^k,\,v=\Delta_g(z)\cdot a\rightarrow f(z+\Delta_g(z)\cdot a) \end{align*} $$

is a quasi-submersion; that is, that its image is contained in the image of the tangent map. More generally speaking, it is a map of constant rank in the sense of [Reference Hauser and MüllerHM]. This is a necessary condition to be ‘locally linearisable’. To make this precise and to prove that with the appropriate assumptions this is also a sufficient condition – that is, that the map $\alpha _z$ is indeed linearisable (in a specific sense) – will be the subject of Section 5.

(d) Note that $\Delta _g(z)\cdot a$ is just the vector $(g_1(z)\cdot a_1,\ldots ,g_m(z)\cdot a_m)\in \mathcal A_{\circ }^m$ and that the series $g_i$ may have nonzero constant term $g_i(0)$ .

(e) The name ‘division module’ is motivated by the later decomposition of vectors $y\in \mathcal A_{\circ }^m$ into $y=\Delta _g(y)\cdot \widetilde a+z=\Delta _g(z)\cdot a+z$ given by the Weierstrass division of the components $y_i$ by $g_i(y)$ ; see Proposition 5.2.

Example. Typically, one could take for ${\mathcal G}$ the m-fold Cartesian product

$$ \begin{align*} {\mathcal G}=\langle g^{o_1}\rangle\times \ldots\times \langle g^{o_m}\rangle \end{align*} $$

of ideals of $\mathcal B_{\circ }$ for some chosen nonzero series $g\in {\mathcal B}$ and nonnegative integers $o_i$ . It is easy to see that such a ${\mathcal G}$ satisfies condition (i) of the definition. However, condition (ii) will not hold without extra assumptions on g. These are specified in the next result.

Proposition 4.3. Let $f\in {\mathcal B}^k$ be given. Assume that $k\leq m$ and that some $(k\times k)$ -minor g of the relative Jacobian matrix ${\partial }_{\mathrm {y}} f$ of f is not zero. For any choice of nonnegative integers $o_1,\ldots ,o_m$ satisfying

$$ \begin{align*} o_i+o_j\geq o_l+1 \end{align*} $$

for all $i,j\in \{1,\cdots ,m\}$ and $\ell \in \{1,\ldots ,k\}$ , the $\mathcal B_{\circ }$ -module

$$ \begin{align*} {\mathcal G}=\langle g^{o_1}\rangle\times\ldots\times\langle g^{o_m}\rangle\subset\mathcal B_{\circ}^m \end{align*} $$

is a division module for f.

Proof. For condition (i), it suffices to show, since ${\mathcal G}$ is a Cartesian product of principal ideals, that for any $z\in \mathcal A_{\circ }^m$ and $v\in {\mathcal G}(z)$ we have the equality of ideals

$$ \begin{align*} \langle g(z)\rangle=\langle g(z+v)\rangle \end{align*} $$

in $\mathcal A_{\circ }$ . This equality does not use that g is a minor – it is the content of Lemma 3.2.

To prove that ${\mathcal G}$ fulfils condition (ii), let $v=\Delta _g(z)\cdot a\in {\mathcal G}(z)$ with $a\in \mathcal A_{\circ }^m$ . Taylor expansion gives

$$ \begin{align*} f(z+\Delta_g(z)\cdot a)=f(z)+{\partial}_{\mathrm{y}} f(z)\cdot \Delta_g(z)\cdot a + q(z,\Delta_g(z)\cdot a) \end{align*} $$

with some vector $q\in \mathbf {k}[[\mathrm {t},\mathrm {z},\mathrm {w}]]^k$ which is of order at least 2 in $\mathrm {w}=(\mathrm {w}_1,\ldots ,\mathrm {w}_m)$ . We have to find a power series vector $p\in \mathbf {k}[[\mathrm {t},\mathrm {z},\mathrm {a}]]^k$ of order at least two in $\mathrm {a}=(\mathrm {a}_1,\ldots ,\mathrm {a}_m)$ such that

$$ \begin{align*} q(z,\Delta_g(z)\cdot a) = {\partial}_y f(z)\cdot \Delta_g(z)\cdot p(z,a) \end{align*} $$

holds for all z and a in $\mathcal A_{\circ }^m$ . First observe that each component of $q(z,\Delta _g(z)\cdot a)$ belongs to the ideal of $\mathcal A_{\circ }$ generated by the powers $g(z)^{o_i+o_j}$ for $i,j\in \{1,\ldots ,m\}$ . It therefore suffices to show that any vector of the form $(0,\ldots ,0,g^{o_i+o_j},0,\ldots ,0)$ , with $i,j\in \{1,\cdots ,m\}$ , can be written as a $\mathcal B_{\circ }$ -linear combination of the vectors ${\partial }_{\mathrm {y}_1}f\cdot g^{o_1},\ldots ,{\partial }_{\mathrm {y}_k}f\cdot g^{o_k}$ .

To prove this, set $\mathrm {y}^1=(\mathrm {y}_1,\ldots ,\mathrm {y}_k)$ , so that g is without loss of generality the determinant of the $(k\times k)$ -submatrix ${\partial }_{\mathrm {y}^1}f$ of ${\partial }_{\mathrm {y}} f$ . Denote by ${\partial }^*_{\mathrm {y}^1}f$ the adjoint matrix of ${\partial }_{\mathrm {y}^1}f$ . We then have ${\partial }_{\mathrm {y}^1}f\cdot {\partial }^*_{\mathrm {y}^1}f=g\cdot {1\!\! 1}_k$ and, accordingly, $g^o\cdot {\partial }_{\mathrm {y}^1}f\cdot {\partial }^*_{\mathrm {y}^1}f=g^{o+1}\cdot {1\!\! 1}_k$ for any integer o. Taking now for o the values $o_i+o_j-1$ and using the assumption that $o_i+o_j\geq o_l+1$ for all $i,j\in \{1,\cdots ,m\}$ and $\ell \in \{1,\ldots ,k\}$ , we get the claim. This establishes condition (ii).

5 Linearisation theorem

This is the central section of the article. To prepare the setting of the linearisation theorem below, let $f=(f_1,\ldots ,f_k)\in {\mathcal B}^k$ be a vector of power series which admits a division module ${\mathcal G}=\langle g_1\rangle \times \ldots \times \langle g_m\rangle \subset \mathcal B_{\circ }^m$ . Recall that by Proposition 4.3 the existence of such a module is ensured, for instance, in the case where $k\leq m$ and if there exists a nonzero $(k\times k)$ -minor g of ${\partial }_{\mathrm {y}} f$ . Up to a permutation of the variables we may assume that g is given by the first k columns of ${\partial }_{\mathrm {y}} f$ . By Proposition 4.3, the $\mathcal B_{\circ }$ -module ${\mathcal G}=\langle g^{o_1}\rangle \times \ldots \times \langle g^{o_m}\rangle $ with $o_i+o_j\geq o_l+1$ for all $i,j\in \{1,\cdots ,m\}$ and $\ell \in \{1,\ldots ,k\}$ is a division module for f.

The following constructions (until the next lemma) hold for any vector $g\in {\mathcal B}^m$ . For a chosen m-tuple $\mathbf { d}=(d_1,\ldots ,d_m)\in {\mathbb N}^m$ , set

$$ \begin{align*} {\mathcal S}_{\mathbf{d}}=\{y\in \mathcal A_{\circ}^m,\, g_i(y)\neq 0,\, \mathrm{ord}\,g_i(y)=d_i \text { for all } i\}. \end{align*} $$

Since $\mathrm {t}$ is just a single variable, the condition $\mathrm {ord}\,g_i(y)=d_i$ is equivalent to saying that $g_i(y)$ equals the monomial $\mathrm {t}^{d_i}$ up to the multiplication by a unit in $\mathcal A$ ; that is, that $\langle g_i(y)\rangle =\langle \mathrm {t}^{d_i}\rangle $ as ideals of $\mathcal A_{\circ }$ , for each $y\in {\mathcal S}_{\mathbf {d}}$ . This implies that ${\mathcal S}_{\mathbf {d}}$ is a cofinite textile locally closed subset of $\mathcal A_{\circ }^m$ . It is $\mathrm {t}$ -adically open since ${\mathcal S}_{\mathbf {d}}$ contains with every element y the $\mathrm {t}$ -adic neighbourhood $y+\langle \mathrm {t}^{d_1}\rangle \times \ldots \times \langle \mathrm {t}^{d_m}\rangle $ in $\mathcal A_{\circ }^m$ . We agree to set $\mathrm {ord}\,g_i(y)=\infty $ if $g_i(y)=0$ and then define accordingly ${\mathcal S}_{\mathbf {d}}$ also if some $d_i=\infty $ . This gives a countable partition of $\mathcal A_{\circ }^m$ into

$$ \begin{align*} \mathcal A_{\circ}^m=\bigcup_{\mathbf{d}\in({\mathbb N}\cup\{\infty\})^m} {\mathcal S}_{\mathbf{d}}. \end{align*} $$

Restrict now to m-tuples $\mathbf {d}\in {\mathbb N}^m$ and vectors $y\in {\mathcal S}_{\mathbf {d}}$ . We set

$$ \begin{align*} {\mathcal V}_{\mathbf{d}}={\mathcal G}(y)=\langle\mathrm{t}^{d_1}\rangle\times \ldots\times \langle\mathrm{t}^{d_m}\rangle\subset\mathcal A_{\circ}^m. \end{align*} $$

We consider ${\mathcal V}_{\mathbf {d}}$ as an $\mathcal A_{\circ }$ -submodule of $\mathcal A_{\circ }^m$ . Let ${\mathcal R}_{\mathbf {d}}$ be the direct monomial complement of ${\mathcal V}_{\mathbf {d}}$ in $\mathcal A_{\circ }^m$ consisting of vectors $z\in \mathcal A_{\circ }^m$ whose ith component $z_i$ is a polynomial in $\mathbf {k}_{\circ }[\mathrm {t}]$ of degree $\leq d_i$ ,

$$ \begin{align*} {\mathcal R}_{\mathbf{d}}=\mathbf{k}_{\circ}[\mathrm{t}]_{\leq d_1}\times \ldots\times \mathbf{k}_{\circ}[\mathrm{t}]_{\leq d_m}, \end{align*} $$

with direct sum decomposition

$$ \begin{align*} {\mathcal V}_{\mathbf{d}}\oplus{\mathcal R}_{\mathbf{d}}=\mathcal A_{\circ}^m. \end{align*} $$

Accordingly, we decompose vectors $y\in {\mathcal S}_{\mathbf {d}}$ into $y=v+z$ with $v\in {\mathcal V}_{\mathbf {d}}$ and $z\in {\mathcal R}_{\mathbf {d}}$ . The ith component $z_i$ of z is the remainder of the Weierstrass division of the ith component $y_i$ of y with respect to $\mathrm {t}^{d_i}$ ; cf. Theorem 3.4. Denote by

$$ \begin{align*} \pi_{\mathbf{d}}:{\mathcal S}_{\mathbf{d}}\rightarrow {\mathcal R}_{\mathbf{d}},\,y\rightarrow z, \end{align*} $$

the induced textile map. Set

$$ \begin{align*} {\mathcal Z}_{\mathbf{d}}={\mathcal S}_{\mathbf{d}}\cap {\mathcal R}_{\mathbf{d}}=\{z\in {\mathcal R}_{\mathbf{d}},\, g_i(z)\neq 0 \text{ and }\mathrm{ord}\,g_i(z)=d_i \text { for all } i\}. \end{align*} $$

Note that ${\mathcal R}_{\mathbf {d}}$ is a finite-dimensional $\mathbf {k}$ -vector space of dimension $d_1+\ldots +d_m$ which will be considered as affine space ${\mathcal R}_{\mathbf {d}}\cong \mathbb A_{\mathbf {k}}^{d_1+\ldots +d_m}$ , and ${\mathcal Z}_d$ is a Zariski locally closed subset. Both ${\mathcal V}_{\mathbf {d}}$ and ${\mathcal R}_{\mathbf {d}}$ only depend on $\mathbf {d}=(d_1,\ldots ,d_m)$ . The sets ${\mathcal S}_{\mathbf {d}}$ and ${\mathcal Z}_{\mathbf {d}}$ will depend on f in the situation of Proposition 4.3 where the series $g_i=g^{o_i}$ are chosen as powers of a minor g of ${\partial }_{\mathrm {y}} f$ .

Said differently, if $y\in \mathcal A_{\circ }^m$ satisfies $\mathrm {ord}\, g_i(y)=d_i$ for all i, then its remainder z satisfies the same condition.

Proposition 5.2. Let $f\in {\mathcal B}^k$ be a power series vector with division module ${\mathcal G}\subset \mathcal B_{\circ }^m$ ; diagonal matrix $\Delta _g$ ; associated sets ${\mathcal S}_{\mathbf {d}}$ , ${\mathcal V}_{\mathbf {d}}$ , ${\mathcal R}_{\mathbf {d}}$ and ${\mathcal Z}_{\mathbf {d}}$ ; and projection $\pi _{\mathbf {d}}:{\mathcal S}_{\mathbf {d}}\rightarrow {\mathcal Z}_{\mathbf {d}},\,y\rightarrow z$ , as defined above, for $\mathbf {d}\in {\mathbb N}^m$ .

(a) The map

$$ \begin{align*} \psi_{\mathbf{d}}:{\mathcal S}_{\mathbf{d}}\rightarrow {\mathcal Z}_{\mathbf{d}}\times{\mathcal V}_{\mathbf{d}}, \, y=v+z\rightarrow (z,v), \end{align*} $$

given by the division of y by ${\mathcal G}(y)$ is a textile isomorphism, restricting to the identity on ${\mathcal Z}_{\mathbf {d}}$ , with inverse

$$ \begin{align*} \psi_{\mathbf{d}}^{-1}:{\mathcal Z}_{\mathbf{d}}\times{\mathcal V}_{\mathbf{d}}\rightarrow{\mathcal S}_{\mathbf{d}},\, (z,v)\rightarrow v+z. \end{align*} $$

(b) For $(z,v)\in {\mathcal Z}_{\mathbf {d}}\times {\mathcal V}_{\mathbf {d}}$ , let $a\in \mathcal A_{\circ }^m$ be the unique vector such that $v=\Delta _g(z)\cdot a$ . Let $p\in \mathbf {k}[[\mathrm {t},\mathrm {z},\mathrm {a}]]^m$ be a vector which is of order at least $2$ in $\mathrm {a}$ . The map

$$ \begin{align*} \varphi_{\mathbf{d}}: {\mathcal Z}_{\mathbf{d}}\times{\mathcal V}_{\mathbf{d}}\rightarrow {\mathcal Z}_{\mathbf{d}}\times{\mathcal V}_{\mathbf{d}}, \end{align*} $$

$$ \begin{align*} (z,v)\rightarrow (z, \Delta_g(z)\cdot (a+p(z,a))) \end{align*} $$

is a rationally textile isomorphism, restricting to the identity on ${\mathcal Z}_{\mathbf {d}}\times 0$ .