Topology of local rings

A complete local ring is a local ring which is complete with respect to the adic topology defined by its maximal ideal, that is, a basis of neighborhoods of zero is given by the powers of the maximal ideal.

Note, in particular, that a formal power series ring over an infinite number of indeterminates, though it may be obtained by an adic completion of a local ring, is not a complete local ring in the previous sense (see Remark 2.2 below). Thus, we shall consider the category $\operatorname {\mathrm {\mathbf {TopLoc}}}$ (respectively, $\operatorname {\mathrm {\mathbf {TopLoc}}}_k$ , k a field), whose objects are the topological local rings (respectively, topological local k-algebras) which are isomorphic to the adic completion of a local ring (respectively of a local k-algebra with residue field k-isomorphic to k) and whose morphisms are continuous morphisms of rings (respectively of k-algebras). The complete local rings (respectively, the complete local k-algebras with residue field k-isomorphic to k) form a full subcategory $\operatorname {\mathrm {\mathbf {CplLoc}}}$ (respectively, $\operatorname {\mathrm {\mathbf {CplLoc}}}_k$ ) of $\operatorname {\mathrm {\mathbf {TopLoc}}}$ (respectively, $\operatorname {\mathrm {\mathbf {TopLoc}}}_k$ ). The following lemma will be useful (for the proof, see, e.g.,[Reference Bourqui and SebagBS17a, Section 2.1]).

Remark 2.2. Let k be a field and $\{X_i\}_{i\in I}$ be a family of indeterminates indexed by a set I (finite or infinite). The topological k-algebra $k{[[ {(X_i)_{i\in I}} ]]}$ may be defined as the completion of the localization of $k[(X_i)_{i\in I}]$ at the maximal ideal ${\langle {X_i} \rangle }_{i\in I}$ . In case I is finite, it is a complete local ring. However, as soon as I is infinite, the local ring $k{[[ {(X_i)_{i\in I}} ]]}$ is by definition complete for the topology of the inverse limit, yet it is not a complete local ring in the above sense (see, e.g., [Reference HaiechHai20] or [Sta21, 05JA]).

In any case, from the point of view of the previous lemma, $k{[[ {(X_i)_{i\in I}} ]]}$ may be seen as the object of $\operatorname {\mathrm {\mathbf {TopLoc}}}_k$ representing the functor

$$\begin{align*}\begin{array}{rcl} {\operatorname{\mathrm{\mathbf{CplLoc}}}_k} & {\longrightarrow} & {\operatorname{\mathrm{Sets}}} \\ {(A,{\mathfrak{M}}_A)} & \longmapsto & {{\mathfrak{M}}_A^I} \end{array}. \end{align*}$$

We shall also consider the full subcategories $\operatorname {\mathrm {\mathbf {NthCplLoc}}}_k$ and $\operatorname {\mathrm {\mathbf {NthCplLoc}}}_k^{\text {alg}}$ of $\operatorname {\mathrm {\mathbf {CplLoc}}}_k$ consisting of those objects of $\operatorname {\mathrm {\mathbf {CplLoc}}}_k$ which are Noetherian (respectively, Noetherian and “algebraizable,” that is, isomorphic to the completion of a local algebra essentially of finite type over k).

Reminder on arc spaces

2.3 Arc spaces and the universal arc

Let V be an algebraic variety over k and ${\mathscr {L}_{\infty }}(V)$ be the associated arc space. For more details on arc spaces, see, for example, [Reference Chambert-Loir, Nicaise and SebagCLNS18]. For the sake of simplicity, and since we are primarily interested in local properties of the arc space, we assume that V is affine. Recall, in particular, that ${\mathscr {L}_{\infty }}(V)$ is an affine k-scheme, such that, for any k-algebra A, one has a functorial bijection between the set of A-points of ${\mathscr {L}_{\infty }}(V)$ and the set of $A{[[ {t} ]]}$ -points of V.

We denote the k-algebra of regular functions on V (respectively, on ${\mathscr {L}_{\infty }}(V)$ ) by $\Gamma (V)$ (respectively, $\Gamma (V)_{\infty }$ ). The universal arc on V is the unique morphism of k-algebras $\Delta _{V}\colon \Gamma (V)\to \Gamma (V)_{\infty }{[[ {t} ]]},\,f\mapsto \sum _{j\ge 0}\Delta _{V,j}(f)\cdot t^j$ with the following property: for any k-algebra A and any A-point of ${\mathscr {L}_{\infty }}(V)$ , that is, any morphism of k-algebras $\theta \colon \Gamma (V)_{\infty }\to A$ , the corresponding $A{[[ {t} ]]}$ -point of V is obtained by composing $\Delta _V$ with the morphism $\Gamma (V)_{\infty }{[[ {t} ]]}\to A{[[ {t} ]]},\,\sum _{j\ge 0}f_j\cdot t^j\mapsto \sum _{j\ge 0}\theta (f_j)\cdot t^j$ . We will often write $\Delta =\Delta _V$ if the involved variety is clear from the context.

In case $V={\mathbf {A}}^n_k=\operatorname {\mathrm {Spec}}(k[X_1,\dots ,X_n])$ , we set $X_{i,j}:=\Delta _{{\mathbf {A}}^n_k,j}(X_i)$ . The $X_{i,j}$ ’s are algebraically independent over k and one has $\Gamma (V)_{\infty }=k[(X_{i,j})_{\substack {1\le i\le n\\j\in {\mathbf {N}}}}]$ . Moreover, for any $F\in k[X_1,\dots ,X_n]$ , one has

$$\begin{align*}\sum_{j\ge 0}\Delta_{{\mathbf{A}}^n_k,j}(F)\cdot t^j=F\left(\left(\sum_{j\ge 0}X_{i,j}\cdot t^j\right)_{1\le i\le n}\right). \end{align*}$$

In case $\Gamma (V)$ is presented as the quotient of $k[X_1,\dots ,X_n]$ by an ideal ${\mathfrak {i}}$ , then $\Gamma (V)_{\infty }$ is presented as the quotient of $k[(X_{i,j})_{\substack {1\le i\le n\\j\in {\mathbf {N}}}}]$ by the ideal ${\mathfrak {i}}_{\infty }$ generated by $\{\Delta _{{\mathbf {A}}^n,j}(F)\}_{\substack {F\in {\mathcal G}\\j\in {\mathbf {N}}}}$ , where ${\mathcal G}$ is any generating family of ${\mathfrak {i}}$ . Moreover, the following natural diagram is commutative:

2.4 Nondegenerate and constant arcs

Let $\gamma \in {\mathscr {L}_{\infty }}(V)$ , with residue field $\kappa (\gamma )$ . The image of the generic point (respectively of the closed point) of $\operatorname {\mathrm {Spec}}(\kappa (\gamma ){[[ {t} ]]})$ by the induced morphism $\operatorname {\mathrm {Spec}}(\kappa (\gamma ){[[ {t} ]]})\to V$ is called the generic point $\gamma (\eta )$ (respectively, the special point or the origin $\gamma (0)$ ) of $\gamma $ . Let $V^{\text {sing}}$ be the singular locus of V. The arc $\gamma $ is said to be nondegenerate if it does not belong to ${\mathscr {L}_{\infty }}(V^{\text {sing}})\subset {\mathscr {L}_{\infty }}(V)$ . In other words, $\gamma $ is nondegenerate if and only if its generic point does not belong to $V^{\text {sing}}$ . The arc $\gamma $ is said to be pseudo-constant if $\gamma (0)=\gamma (\eta )$ , and constant if the induced morphism $\Gamma (V)\to \kappa (\gamma ){[[ {t} ]]}$ has its image in $\kappa (\gamma )$ . Any constant arc is pseudoconstant; the arc $k[X_1]\to k(X_{1,j})_{j\in {\mathbf {N}}}{[[ {t} ]]}$ defined by $X_1\mapsto \sum _{j\in {\mathbf {N}}}X_{1,j}\cdot t^j$ is an example of a pseudoconstant yet not constant arc on ${\mathbf {A}}^1_k$ . Note, however, that if $\gamma $ is such that $\gamma (0)$ is a closed point of V, then $\gamma $ is constant if and only if $\gamma $ is pseudoconstant, since then the image of $\Gamma (V)\to \kappa (\gamma ){[[ {t} ]]}$ is algebraic over k, hence contained in $\kappa (\gamma )$ . In particular, in case V is a curve, any nonconstant arc on V is necessarily nondegenerate.

The Drinfeld-Grinberg-Kazhdan theorem ([Reference Grinberg and KazhdanGK00, Reference DrinfeldDri02]) states that in case $\dim (V)\ge 1$ , any nondegenerate k-rational arc $\gamma $ on V admits a finite formal model.

Moreover, any object ${\mathcal A}'$ of $\operatorname {\mathrm {\mathbf {NthCplLoc}}}_k$ , such that $\widehat {{\mathscr {O}}_{{{\mathscr {L}_{\infty }}(V),\gamma }}}$ is isomorphic to ${\mathcal A}'{[[ {(v_i)} ]]}_{i\in {\mathbf {N}}}$ in the category $\operatorname {\mathrm {\mathbf {TopLoc}}}_k$ is in fact an object of $\operatorname {\mathrm {\mathbf {NthCplLoc}}}_k^{\text {alg}}$ : indeed, if ${\mathcal A}$ is a finite formal model of $ \gamma $ , by Gabber’s cancellation theorem (see [Reference Bourqui and SebagBS17a, Theorem 7.2]), there exist nonnegative integers n and m, such that ${\mathcal A}'{[[ {v_1,\dots ,v_n} ]]}$ and ${\mathcal A}{[[ {u_1,\dots ,u_m} ]]}$ are isomorphic. On the other hand, if ${\mathcal A}'\in \operatorname {\mathrm {\mathbf {NthCplLoc}}}_k$ is such that ${\mathcal A}'{[[ {u} ]]}\in \operatorname {\mathrm {\mathbf {NthCplLoc}}}_k^{\text {alg}}$ , then ${\mathcal A}'\in \operatorname {\mathrm {\mathbf {NthCplLoc}}}_k^{\text {alg}}$ ; this is a consequence of Zariski’s simplification lemma (see [Reference Bourqui and SebagBS18, Section 5.6]).

Some facts about curve singularities

2.6

Let us now consider a pointed k-curve $({\mathcal C},c)$ , that is, ${\mathcal C}$ is a k-curve and $c\in {\mathcal C}(k)$ . Since we are only interested in local properties, without restricting the generality of the arguments, we may in any circumstances replace ${\mathcal C}$ with a Zariski open set of ${\mathcal C}$ containing c. In particular, we may assume that ${\mathcal C}$ is affine and that c is the only singular point of ${\mathcal C}$ .

2.7 Primitive parametrizations and multiplicity of a germ of curve

We, hereafter, assume that the singularity at c is analytically irreducible, in other words, that the normalization of the local ring ${\mathscr {O}}_{{{\mathcal C},c}}$ is a domain.

In particular, any choice of an isomorphism $\theta $ between the normalization of $\widehat {{\mathscr {O}}_{{{\mathcal C},c}}}$ and $k{[[ {t} ]]}$ in turn induces an injective morphism $p_{\theta }\colon \widehat {{\mathscr {O}}_{{{\mathcal C},c}}5}\to k{[[ {t} ]]}$ . Such a morphism will be called a primitive parametrization of the germ $({\mathcal C},c)$ . The integer $\min \big (\big \{\operatorname {\mathrm {ord}}_t(p_{\theta }(f))\;:\; f\in \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\big \}\setminus \{0\}\big )$ does not depend on $\theta $ and is called the multiplicity $\mu ({\mathcal C},c)$ of the germ. A morphism $p\colon \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\to k{[[ {t} ]]}$ is a primitive parametrization if and only if

$$\begin{align*}\min \left(\left\{\operatorname{\mathrm{ord}}_t(p(f))\;:\; f\in \widehat{{\mathscr{O}}_{{{\mathcal C},c}}}\right\}\setminus \{0\}\right)=\mu({\mathcal C},c). \end{align*}$$

More precisely, for any two primitive parametrizations $p_1$ , $p_2$ of the germ $({\mathcal C},c)$ , there exists an automorphism $\phi $ of $k{[[ {t} ]]}$ , such that $\phi \circ p_1=p_2$ .

2.8 Multiplicity of an arc centered at a singular point of a curve

Let $p\colon \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\to k{[[ {t} ]]}$ be a primitive parametrization of the germ $({\mathcal C},c)$ and $ \widehat {{\mathfrak {M}}_{{\mathcal C},c}} $ be the maximal ideal of $ \widehat {{\mathscr {O}}_{{{\mathcal C},c}}} $ . For any k-extension K and any morphism $\gamma ^{\ast }:\widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\to K{[[ {t} ]]}$ , such that $\gamma ^{\ast }(\widehat {{\mathfrak {M}}_{{\mathcal C},c}})\neq \{0\}$ , there exists a unique positive integer N and a unique automorphism $\psi $ of $K{[[ {t} ]]}$ , such that $\gamma ^{\ast }$ coincides with the composition of $ \psi \circ p $ with the endomorphism of $K{[[ {t} ]]}$ induced by $t\mapsto t^N$ . The integer N is called the multiplicity of $\gamma ^{\ast }$ and denoted by $\mu (\gamma ^{\ast })$ . Note that the datum of a local morphism $\gamma ^{\ast }\colon \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\to K{[[ {t} ]]}$ , such that $\gamma ^{\ast }(\widehat {{\mathfrak {M}}_{{\mathcal C},c}})\neq \{0\}$ is equivalent with the datum of a K-arc on ${\mathcal C}$ , centered at c, and not entirely contained in c. This allows us to define the multiplicity $\mu (\gamma )$ of an arc $\gamma \in {\mathscr {L}_{\infty }}({\mathcal C})^{c,\circ }:={\mathscr {L}_{\infty }}({\mathcal C})^{c}\setminus \{c\}$ (here, $ {\mathscr {L}_{\infty }}({\mathcal C})^{c} $ denotes the set of arcs centered at $ c $ , and we still denote the constant arc at c by c) by considering the multiplicity of the $\kappa (\gamma )$ -arc induced by $\gamma $ .

For any positive integer N, we set

$$\begin{align*}\Theta_N:=\left\{\gamma\in {\mathscr{L}_{\infty}}({\mathcal C})^{c,\circ}\;:\;\mu(\gamma)=N\right\}. \end{align*}$$

Note that any two elements $\gamma _1,\gamma _2\in \Theta _N(K)$ are connected by an automorphism of $K{[[ {t} ]]}$ . Then, the arguments in the proof of [Reference Bourqui and SebagBS17b, Lemma 3.2] can be directly adapted to prove the following more general result.

2.10

Let us choose a presentation of the curve ${\mathcal C}\cong \operatorname {\mathrm {Spec}}\left (k[Y_0,Y_1,\dots ,Y_\ell ]/{\mathfrak {i}}\right )$ , that is, the ideal ${\mathfrak {i}}$ of $k[Y_0,Y_1\dots ,Y_\ell ]$ defines ${\mathcal C}$ as a closed subscheme of the affine space $\boldsymbol {A}_k^{\ell +1}$ . We may and shall assume that the singular point $ c $ is the origin ${\mathfrak {o}}$ of $\boldsymbol {A}_k^{\ell +1}$ , and that any element of the set $\{Y_0,Y_1,\dots ,Y_{\ell }\}$ induces a nonzero regular function on ${\mathcal C}$ .

The closed embedding of ${\mathcal C}$ into the affine space $\boldsymbol {A}_k^{\ell +1}$ induces a surjective morphism $\widehat {{\mathscr {O}}_{{\boldsymbol {A}_k^{\ell +1},{\mathfrak {o}}}}}=k{[[ {Y_0,Y_1,\dots ,Y_\ell } ]]}\to \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}$ with kernel generated by ${\mathfrak {i}}$ . Composing this morphism with a primitive parametrization $p\colon \widehat {{\mathscr {O}}_{{{\mathcal C},c}}}\to k{[[ {t} ]]}$ gives an $\ell +1$ -tuple of power series $(Y_{p,0}(t),Y_{p,1}(t),\dots ,Y_{p,\ell }(t))\in k{[[ {t} ]]}^{\ell +1}$ , such that $\min _{0\le i\le \ell } \{\operatorname {\mathrm {ord}}_t(Y_{p,i}(t))\}=\mu ({\mathcal C},c)$ and

$$\begin{align*}\forall F\in {\mathfrak{i}},\quad F(Y_{p,i}(t))_{0\le i\le \ell}=0. \end{align*}$$

Conversely, any $\ell +1$ -tuple $(Y_{0}(t),Y_{1}(t),\dots ,Y_{\ell }(t))\in k{[[ {t} ]]}^{\ell +1}$ , such that $\min _{0\le i\le \ell } \{\operatorname {\mathrm {ord}}_t(Y_{i}(t))\}=\mu ({\mathcal C},c)$ and

$$\begin{align*}\forall F\in {\mathfrak{i}},\quad F(Y_{i}(t))_{0\le i\le \ell}=0 \end{align*}$$

is induced by a primitive parametrization. More generally, one has the following result.

Proposition 2.11. Let $ k $ be a field, $\widehat {({\mathcal C},c)}$ a germ of curve, and $ N $ a positive integer. Let p be a primitive parametrization of $\widehat {({\mathcal C},c)}$ . Let K be an extension of k and $(Y_{0}(t),Y_{1}(t),\dots ,Y_{\ell }(t))\in K{[[ {t} ]]}^{\ell +1}$ , such that $\min _{0\le i\le \ell } \{\operatorname {\mathrm {ord}}_t(Y_{i}(t))\}=N\cdot \mu ({\mathcal C},c)$ and

$$\begin{align*}\forall F\in {\mathfrak{i}},\quad F(Y_{i}(t))_{0\le i\le \ell}=0. \end{align*}$$

Then there exists an automorphism $\psi $ of $K{[[ {t} ]]}$ , such that for $0\le i\le \ell $ , one has $Y_{i}(t)=\psi (Y_{p,i}(t^N))$ .

From now on, we denote by $ n:=\mu ({\mathcal C},c) $ the multiplicity of the germ. We set $ X:=Y_0 $ and, up to a permutation of the coordinates, we may assume that $ n=\operatorname {\mathrm {ord}}_t(X(t)) $ . We also denote $ m_i=\operatorname {\mathrm {ord}}_t(Y_{i}(t)) $ for $ 1\le i \le \ell $ . Let us note that, by a suitable choice of a primitive parametrization p, we may assume that the associated $ \ell +1 $ -tuple of power series $(X_{p}(t),Y_{p,1}(t),\dots ,Y_{p,\ell }(t))\in k{[[ {t} ]]}^{\ell +1}$ satisfies $X_{p}(t)=t^n$ .

With this notation, we observe that, for $ N\in {\mathbf {N}} $ , if the $ \ell +1 $ -tuple $(X(t),Y_{1}(t),\dots ,Y_{\ell }(t))$ is as in Proposition 2.11, then $ \operatorname {\mathrm {ord}}_t(X(t))=N\cdot n $ and $ \operatorname {\mathrm {ord}}_t(Y_{i}(t))=N\cdot m_i $ , for $ 1\le i \le \ell $ .

3.1 The semivaluation associated with an arc

Let $ V $ be an affine $ k $ -variety. In this paper, we adopt the following terminology regarding (semi)valuations.

Let $\gamma \in {\mathscr {L}_{\infty }}(V)$ be an arc, with residue field $\kappa (\gamma )$ . Then $\gamma $ induces a morphism of k-algebras $\gamma ^{\ast }\colon \Gamma (V)\to \kappa (\gamma ){[[ {t} ]]}$ . Composing with $\operatorname {\mathrm {ord}}_t$ , one defines, following [Reference IshiiIsh05], a k-semivaluation $\operatorname {\mathrm {ord}}_{\gamma }$ on V. Note that $\operatorname {\mathrm {ord}}_{\gamma }$ is a valuation if and only if $\gamma ^{\ast }$ is injective, that is, if and only if the generic point of $\gamma $ is the generic point of V. Such arcs are called fat arcs in the literature. Note that any nonfat arc on a curve is necessarily a constant arc.

An elementary yet crucial fact is that the semivaluation associated with an arc “increases by specialization,” as expressed by the following proposition (see [Reference IshiiIsh05, Proposition 2.7]).

Proposition 3.3. We keep the preceding notation. Let $\gamma _1,\gamma _2$ be elements of ${\mathscr {L}_{\infty }}(V)$ , such that $\gamma _2$ is a specialization of $\gamma _1$ , that is, it lies in the closure of $\gamma _1$ . Then

$$\begin{align*}\forall f\in \Gamma(V),\quad \operatorname{\mathrm{ord}}_{\gamma_1}(f)\le \operatorname{\mathrm{ord}}_{\gamma_2}(f). \end{align*}$$

3.4 Contact loci in arc spaces

Let $F\subset V$ be a closed subscheme of V, defined by the ideal ${\mathfrak {f}}\subset \Gamma (V)$ . Then the contact order $\operatorname {\mathrm {ord}}_F(\gamma )$ of $\gamma $ with F is the integer N, such that $t^N$ generates the ideal ${\langle {\gamma ^{\ast }({\mathfrak {f}})} \rangle }$ in $\kappa (\gamma ){[[ {t} ]]}$ , or $+\infty $ in case $\gamma ^{\ast }({\mathfrak {f}})=\{0\}$ , that is, in case $\gamma $ is entirely contained in F. Note that $\operatorname {\mathrm {ord}}_F(\gamma )=\operatorname {\mathrm {ord}}_{F^{\operatorname {\mathrm {red}}}}(\gamma )$ only depends on the underlying closed set supporting F and not on the schematic structure of F. One sets

$$\begin{align*}\operatorname{\mathrm{Cont}}^{N}(V,F):=\left\{\gamma\in {\mathscr{L}_{\infty}}(V)\;:\; \operatorname{\mathrm{ord}}_{F}(\gamma)=N\right\}; \end{align*}$$

$$\begin{align*}\operatorname{\mathrm{Cont}}^{\ge N}(V,F):=\left\{\gamma\in {\mathscr{L}_{\infty}}(V)\;:\; \operatorname{\mathrm{ord}}_{F}(\gamma)\ge N\right\}. \end{align*}$$

Note that $\operatorname {\mathrm {Cont}}^{\ge N}(V,F)$ is the closed subset of ${\mathscr {L}_{\infty }}(V)$ defined by the ideal

$$\begin{align*}{\langle {\Delta_i(f)} \rangle}_{\substack{0\le i\le N-1\\f\in {\mathfrak{f}}}}. \end{align*}$$

Also, $\operatorname {\mathrm {Cont}}^{N}(V,F)$ is the intersection of $\operatorname {\mathrm {Cont}}^{\ge N}(V,F)$ with the open set

$$\begin{align*}\bigcup_{f\in {\mathfrak{f}}}\{\Delta_N(f)\neq 0\} \end{align*}$$

and is thus locally closed.

The following fact, firstly observed in [Reference Ein, Lazarsfeld and MustaţǎELM04], follows from a computation in local (étale) coordinates.

3.7 Maximal divisorial sets in arc spaces

Let $ V $ be an affine $ k $ -variety. Being given $\nu $ a semivaluation on V trivial on k, one defines the closed set ${\mathcal {N}}_V(\nu )$ of ${\mathscr {L}_{\infty }}(V)$ as the Zariski closure of the set $\left \{\gamma \in {\mathscr {L}_{\infty }}(V)\;:\;\operatorname {\mathrm {ord}}_{\gamma }=\nu \right \}$ (which is nonempty by [Reference IshiiIsh05, Proposition 2.11] in case $\nu $ is a divisorial valuation, and by [Reference MoreMor09, Proposition 3.12] in general).

In case $\nu $ is a divisorial valuation, and following [Reference IshiiIsh08], ${\mathcal {N}}_V(\nu )$ is called the maximal divisorial set associated with $\nu $ . By op.cit., it may be described as follows: take a resolution $\pi \colon W\to V$ of the singularities of V, such that the center of $\nu $ on W is a divisor E, and let N be the positive integer, such that $\nu =N\cdot \operatorname {\mathrm {ord}}_E$ . Then ${\mathcal {N}}_V(\nu )$ coincides with the closure of the image by ${\mathscr {L}_{\infty }}(\pi )$ of the set $ \operatorname {\mathrm {Cont}}^{N}(W,E) $ , and hence, also of the set $ \operatorname {\mathrm {Cont}}^{\ge N}(W,E) $ . In particular, ${\mathcal {N}}_V(\nu )$ is irreducible by Lemma 3.5. We denote by $\eta _{V,\nu }$ or $\eta _{\nu }$ its generic point. It follows from the definitions and Proposition 3.3 that $\operatorname {\mathrm {ord}}_{\eta _{\nu }}=\nu $ .

3.8

Let $ \nu $ be a semivaluation on V. Let us consider the subset $ {\mathcal D}_{V}(\nu ) $ of ${\mathscr {L}_{\infty }}(V)$ given by

$$\begin{align*}{\mathcal D}_{V}(\nu):=\left\{\gamma\in {\mathscr{L}_{\infty}}(V)\,:\, \forall f\in \Gamma(V),\,\operatorname{\mathrm{ord}}_{\gamma}(f)\ge \nu(f)\right\}. \end{align*}$$

The following lemma collects some basic properties connecting the different subsets of the arc scheme associated to valuations that we have defined.

Proof. Part (i) directly follows from the definition of $\operatorname {\mathrm {ord}}_{\gamma }$ .

As for part (ii), by its very definition, ${\mathcal D}_{V}(\nu )$ contains $\left \{\gamma \in {\mathscr {L}_{\infty }}(V)\;:\;\operatorname {\mathrm {ord}}_{\gamma }=\nu \right \} $ , thus also its closure ${\mathcal {N}}_{V}(\nu )$ . Since $ \nu $ is a divisorial valuation, $ {\mathcal {N}}_{V}(\nu ) $ is irreducible. Let $\eta $ be the generic point of an irreducible component of ${\mathcal D}_{V}(\nu )$ containing ${\mathcal {N}}_{V}(\nu )$ . Let $\gamma $ be an arc with $\operatorname {\mathrm {ord}}_{\gamma }=\nu $ . Then $\gamma $ is a specialization of $\eta $ . Thus, by Proposition 3.3

$$\begin{align*}\forall f\in \Gamma(V),\quad \operatorname{\mathrm{ord}}_{\eta}(f)\le \nu(f). \end{align*}$$

Since $\eta \in {\mathcal D}_{V}(\nu )$ , and by the very definition of ${\mathcal D}_{V}(\nu )$ , one infers that $\operatorname {\mathrm {ord}}_{\eta }=\nu $ . Hence, $\eta \in {\mathcal {N}}_{V}(\nu )$ , thus, $\eta $ is the generic point of ${\mathcal {N}}_{V}(\nu )$ , which shows part (ii).

Let us prove part (iii). We argue by contradiction and assume that there exists $ f\in \Gamma (V) $ with $ f\ne 0 $ , such that $\Delta _{\nu (f)}(f)\in {\mathfrak {p}}_{\nu }$ . In particular, for any $\gamma \in {\mathcal {N}}_{V}(\nu )$ , the $t^{\nu (f)}$ -coefficient of $\gamma ^{\ast }(f)$ vanishes. Since $\gamma \in {\mathcal D}_{V}(\nu )$ , one infers that $\operatorname {\mathrm {ord}}_{\gamma }(f)>\nu (f)$ . Taking $\gamma =\eta _{\nu }$ gives a contradiction since $\operatorname {\mathrm {ord}}_{\eta _{\nu }}=\nu $ .

The case of curve singularities

3.11

Let us keep the assumptions and notation from Section 2.6; in particular, $ {\mathcal C} $ is an affine $ k $ -curve and $ c\in {\mathcal C}(k)$ is the only singular point of ${\mathcal C}$ , and is analytically irreducible. For any divisorial valuation $ \nu $ on $ {\mathcal C} $ centered at $ c $ , our aim in this article is to compare the formal neighborhood of the generic point of the maximal divisorial set $ {\mathcal {N}}_{{\mathcal C}}(\nu ) $ with the formal neighborhood of a generic element of the set of $ k $ -rational points of $ {\mathcal {N}}_{{\mathcal C}}(\nu ) $ .

Let p be a primitive parametrization of $\widehat {({\mathcal C},c)}$ . Then it induces a valuation

$$\begin{align*}\nu\colon f\in \Gamma({\mathcal C})\mapsto \operatorname{\mathrm{ord}}_t(p(f))\end{align*}$$

on $ {\mathcal C} $ , centered at $ c $ , which is a divisorial valuation. The following result is well-known.

Note that the constant arc $\{c\}$ is the only element $\gamma \in {\mathscr {L}_{\infty }}({\mathcal C})$ , such that $\operatorname {\mathrm {ord}}_{\gamma }=\nu _c$ .

3.15

The following corollary provides a presentation (up to radical) of the prime ideal $ {\mathfrak {p}}_{N\cdot \nu } $ of $ \Gamma ({\mathcal C})_{\infty } $ defining the generic point $ \eta _{N\cdot \nu } $ of the maximal divisorial set $ {\mathcal {N}}_{{\mathcal C}}(N\cdot \nu ) $ .

Corollary 3.16. For any nonempty family $\{f_i\}_{i\in I}$ of nonzero elements $f_i\in \Gamma ({\mathcal C})$ , such that $\nu (f_i)>0$ , the prime ideal ${\mathfrak {p}}_{N\cdot \nu }$ of $\Gamma ({\mathcal C})_{\infty }$ corresponding to the generic point of ${\mathcal {N}}_{{\mathcal C}}(N\cdot \nu )$ is the radical of the ideal

$$\begin{align*}{\langle {\Delta_0(f_i),\dots,\Delta_{N\cdot \nu(f_i)-1}(f_i)} \rangle}_{i\in I}. \end{align*}$$

Moreover, if $f\in \Gamma ({\mathcal C})$ is a nonzero element, such that $\nu (f)>0$ , then $\Delta _{N\cdot \nu (f)}(f)\notin {\mathfrak {p}}_{N\cdot \nu }$ .

3.18

As a direct consequence of Lemmas 2.9 and 3.13, we can affirmatively answer the first part of Question 1.3.

4.1

We first introduce some generic notation and a definition. Let R be a k-algebra, ${\mathfrak {I}}$ be an ideal of R, and ${\mathfrak {P}}$ be a prime ideal of R containing ${\mathfrak {I}}$ . Let $(A,{\mathfrak {M}}_A)$ be the localization of $ R/{\mathfrak {I}}$ with respect to $ {\mathfrak {P}}$ , and let $(\widehat {A},{\mathfrak {M}}_{\widehat {A}}) $ be its completion, with residue field $\kappa _A=\kappa _{\widehat {A}}$ . Let $ \iota \colon R\to \widehat {A} $ be the composition of the natural morphism $ R\to A $ with the completion morphism $A\to \widehat {A} $ . Recall that a coefficient field in $ \widehat {A} $ is a subfield K of $\widehat {A}$ , such that the quotient morphism $p_{\widehat {A}}\colon \widehat {A} \to \kappa _{\widehat {A}}$ induces an isomorphism $K{\overset {\sim }{\to }} \kappa _{\widehat {A}}$ . Since R is a k-algebra, such a coefficient field exists by Cohen’s theorem (see, e.g., [Sta21, Tag 032A]).

Let $\pi \colon R\to \operatorname {\mathrm {Frac}}(R/{\mathfrak {P}}) $ be the composition of the quotient morphism $R\to R/{\mathfrak {P}}$ with the canonical injection $R/{\mathfrak {P}}\to \operatorname {\mathrm {Frac}}(R/{\mathfrak {P}})$ . Then $\pi $ factors through $R\to A\to \kappa _A$ , giving a natural isomorphism from $\kappa _A=\kappa _{\widehat {A}}$ with $ \operatorname {\mathrm {Frac}}(R/{\mathfrak {P}}) $ . The composition of the quotient morphism $p_{\widehat {A}}$ with the latter isomorphism is denoted by $\tau $ .

In order to unpack the above definitions a little bit, note that, identifying $\rho $ with the induced morphism $R/{\mathfrak {P}}\to \widehat {A}$ , one has the following commutative diagram:

where the leftmost vertical arrow is the quotient morphism and the lower horizontal arrow is the natural inclusion. In our situation, we will have a rather explicit description of $\operatorname {\mathrm {Frac}}(R/{\mathfrak {P}})$ and the morphism $\tau $ is to be understood as a technical device to consider the reduction of an element of $\widehat {A}$ in an explicit field rather than in the a priori abstract residue field $\kappa _{\widehat {A}}$ . Let us now give a basic example: take $R=k[(X_i)_{i\in {\mathbf {N}}}]$ , ${\mathfrak {I}}=\{0\}$ and ${\mathfrak {P}}={\langle {X_0,\dots ,X_{N-1}} \rangle }$ . With the notation of the previous section, it corresponds to the generic point of ${\mathcal {N}}_{{\mathcal C}}(N\cdot \nu )$ , where $({\mathcal C},c)=({\mathbf {A}}^1_k,\text {origin})$ . Then for the most natural choice $\rho _0$ of $\rho $ , the above diagram reads as follows (all the injective arrows are the natural inclusions):

Of course, there are other possible “less natural” choices for $\rho $ , for example, choose any family $(Y_i)_{i\geq N}$ of elements of the maximal ideal of $k((X_i)_{i\geq N}){[[ {X_0,\dots ,X_{N-1}} ]]}$ and set $\rho (X_i)=X_i+Y_i$ for $i\geq N$ . Roughly speaking, in the case of a general curve singularity, our aim will be to construct an algebraic presentation of a coefficient field of the formal neighborhood of the generic primitive arc that is “as natural” as $\rho _0$ in the above situation.

4.5

We now come back to our algebraic pointed k-curve $({\mathcal C},c)$ . We work with the notation and under the hypotheses of Sections 2.6 and 2.10. We denote by $ p $ a primitive parametrization of $\widehat {({\mathcal C},c)}$ . Let $ \nu $ be the divisorial valuation induced by $ p $ as defined in Section 3.11. In particular, $ {\mathfrak {p}}_{\nu }$ is the prime ideal of the ring $ \Gamma ({\mathscr {L}_{\infty }}({\mathcal C}))\cong k[(X_j)_{j\in {\mathbf {N}}},(Y_{i,j})_{\substack {1\le i\le \ell \\ j\in {\mathbf {N}}}}]/{\mathfrak {i}}_{\infty }$ corresponding to the generic point $\eta _{{\mathcal C},\nu }$ of the maximal divisorial set $ {\mathcal {N}}_{{\mathcal C}}(\nu ) $ . In a slight abuse of notation, we also denote by $ {\mathfrak {p}}_{\nu }$ its unique preimage in $ k[(X_j)_{j\in {\mathbf {N}}},(Y_{i,j})_{\substack {1\le i\le \ell \\ j\in {\mathbf {N}}}}] $ containing $ {\mathfrak {i}}_{\infty } $ .

Recall that $n = \operatorname {\mathrm {ord}}_t(X(t))$ is the multiplicity of the germ (see at the end of Section 2.10). We consider the $ k $ -algebras

$$\begin{align*}R:=k[(X_j)_{j\in {\mathbf{N}}},(Y_{i,j})_{\substack{1\le i\le \ell\\ j\in{\mathbf{N}}}},\frac{1}{X_{{n}}}] \end{align*}$$

$$\begin{align*}\text{and}\quad R^{\ge \nu}:=k[(X_j)_{j\ge {n}},(Y_{i,j})_{\substack{1\le i\le \ell\\ j\ge{m}_{i}}},\frac{1}{X_{{n}}}]. \end{align*}$$

For any $F\in k[X,Y_1,\dots ,Y_{\ell }]$ , let $ (F_{j})_{j\in {\mathbf {N}}} $ (respectively, $ (F^{\ge \nu }_{j})_{j\in {\mathbf {N}}} $ , respectively, $(F_{j}^{\ge \nu ,\circ })_{j\in {\mathbf {N}}}$ ) be the family of elements of R (respectively, $R^{\ge \nu }$ , respectively, $R^{\ge \nu }$ ) defined by the relation

(4.1) $$ \begin{align} \sum_{j\ge 0} F_{j}\cdot t^j:=F\left(\sum_{j\ge 0} X_j\cdot t^j,\left(\sum_{j\ge 0} Y_{i,j}\cdot t^j\right)_{1\le i\le \ell}\right), \end{align} $$

respectively

$$\begin{align*}\sum_{j\ge 0} F^{\ge\nu}_{j}\cdot t^j:=F\left(\sum_{j\ge {n}} X_j\cdot t^j,\left(\sum_{j\ge {m}_{i}} Y_{i,j}\cdot t^j\right)_{1\le i\le \ell}\right), \end{align*}$$

respectively

(4.2) $$ \begin{align} \sum_{j\ge 0} F_{j}^{\ge\nu,\circ}\cdot t^j:=F\left(X_{{n}}\cdot t^{{n}},\left(\sum_{j\ge {m}_{i}} Y_{i,j}\cdot t^j\right)_{1\le i\le \ell}\right). \end{align} $$

In particular, (the extension of) ${\mathfrak {i}}_{\infty }$ (in R) is the ideal of R generated by the elements $\left \{F_{j}; F\in {\mathfrak {i}}, j\in {\mathbf {N}}\right \}$ . We call $ {\mathfrak {i}}_{\infty }^{\ge \nu }$ the ideal of $R^{\ge \nu }$ generated by the elements $\left \{F^{\ge \nu }_{j}; F\in {\mathfrak {i}}, j\in {\mathbf {N}}\right \}$ , and $ {\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }$ the ideal of $R^{\ge \nu }$ generated by the elements $\left \{F^{\ge \nu ,\circ }_{j}; F\in {\mathfrak {i}}, j\in {\mathbf {N}}\right \}$ .

4.7

Let ${\mathfrak {q}}_{\nu }$ be the ideal of R given by ${\langle {(X_j)_{0\le j<{n}}, (Y_{i,j})_{\substack {1\le i\le \ell \\ 0\le j< {m}_i}}}\rangle }$ . By Corollary 3.16, $ {\mathfrak {p}}_{\nu } $ is a prime ideal of $ R/{\mathfrak {i}}_{\infty } $ and

(4.3) $$ \begin{align} {\mathfrak{p}}_{\nu}=\operatorname{\mathrm{rad}}({\mathfrak{q}}_{\nu}+{\mathfrak{i}}_{\infty})=\operatorname{\mathrm{rad}}({\mathfrak{q}}_{\nu}+{\mathfrak{i}}_{\infty}^{\ge \nu}). \end{align} $$

In order to study the formal neighborhood of $ \eta _{{\mathcal C},\nu } $ in $ {\mathscr {L}_{\infty }}({\mathcal C}) $ , we shall work on $R/{\mathfrak {i}}_{\infty } $ .

From now on, and until the end of the section, we use the notation of Section 4.1, with ${\mathfrak {I}}:={\mathfrak {i}}_{\infty }$ and ${\mathfrak {P}}:={\mathfrak {p}}_{\nu }$ . The following proposition is the main result in this section.

Remark 4.10. Assume that $\rho $ is a morphism as in the statement of Proposition 4.8. Then $\operatorname {\mathrm {Ker}}(\rho )={\mathfrak {p}}_{\nu }$ (Remark 4.3). In particular, $\operatorname {\mathrm {Ker}}(\rho )$ contains ${\mathfrak {q}}_{\nu }$ and $\sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu }}$ , and for every $F\in {\mathfrak {i}}$ , one has by (4.1):

(4.4) $$ \begin{align} F\left(\sum_{j\ge {n}}\rho(X_{j})\cdot t^{j},\left(\sum_{j\ge {m}_{i}}\rho(Y_{i,j})\cdot t^j\right)_{1\le i\le \ell}\right)=0. \end{align} $$

4.11

In the rest of this section, we will state and prove some technical results leading to the proof of Proposition 4.8. First, observe that the k-algebras $ R/{\mathfrak {p}}_{\nu }$ and $ R^{\ge \nu }/\sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu }} $ are isomorphic. The next two results will provide a better understanding of the latter quotient ring.

Proof. There exist elements $\{H_j\}_{j\ge 1}$ in $k[(X_{j})_{j\ge {n}}][\frac {1}{X_{{n}}}]$ , such that

$$\begin{align*}\sum\limits_{j\ge {n}}X_{j}\cdot t^j = X_{{n}}\cdot t^{{n}}\cdot \left(1+\sum\limits_{j\ge 1}\frac{X_{{n}+j}}{X_{{n}}}\cdot t^j\right)=X_{{n}}\cdot t^{{n}}\cdot \left(1+\sum_{j\ge 1}H_j\cdot t^j\right)^{{n}}. \end{align*}$$

Let $u\theta (u)$ be the image of t by the inverse of the isomorphism $k[(X_{j})_{j\ge {n}}][\frac {1}{X_{{n}}}]{[[ {u} ]]}\to k[(X_{j})_{j\ge {n}}][\frac {1}{X_{{n}}}]{[[ {t} ]]}$ mapping u to $t\cdot \left (1+\sum _{j\ge 1}H_j\cdot t^j\right )$ .

Let $\{\widetilde {Y}_{i,j}\}_{\substack {1\le i\le \ell \\j\ge {m}_{i}}}$ be the family of elements of $R^{\ge \nu }$ defined, for $i\in \{1,\dots ,\ell \}$ , by the equality

$$\begin{align*}\sum_{j\ge {m}_{i}}\widetilde{Y}_{i,j}\cdot u^j=\sum_{j\ge {m}_{i}} Y_{i,j}\cdot (u\theta(u))^j. \end{align*}$$

The sought-for automorphism maps $X_{j}$ to $X_{j}$ for $j\ge {n}$ and $Y_{i,j}$ to $\widetilde {Y}_{i,j}$ for $1\le i\le \ell $ and $j\ge {m}_{i}$ .

For the next proposition, recall that the $Y_{p,i} (t)$ ’s are the explicit formal series induced by our choice of a primitive parametrization p and of an embedding of our germ (see Section 2.10 for our assumptions in that regard).

In the proof of the proposition, we will use the following well-known result, the proof of which, we include for the convenience of the reader.

Proof of proposition 4.13

Since $ (t^{{n}},Y_{p,1}(t),\dots ,Y_{p,\ell }(t)) $ is a Puiseux parametrization of $({\mathcal C},c)$ , one has, for every $ F\in {\mathfrak {i}} $ ,

(4.5) $$ \begin{align} F\left(\phi(X_{{n}})\cdot t^{{n}},\left(\sum_{j\ge {m}_{i}}\phi(Y_{i,j})\cdot t^j\right)_{1\le i\le \ell}\right) =F((ut)^{{n}},Y_{p,1}(ut),\dots,Y_{p,\ell}(ut))=0. \end{align} $$

Thus, by the definition of the family $(F^{\ge \nu ,\circ }_{j})_{j\in {\mathbf {N}}}$ , one has $\sum _{j\ge 0}\phi (F^{\ge \nu ,\circ }_{j})\cdot t^j=0$ , and we deduce that the kernel of $\phi $ contains ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }$ , thus also $\sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }}$ since $k[u,u^{-1}]$ is a domain. Therefore, $\phi $ factors through

$$\begin{align*}\psi:k[X_{{n}},(Y_{i,j})_{\substack{1\le i\le \ell\\j\ge {m}_{i}}},\frac{1}{X_{{n}}}]/\sqrt{{\mathfrak{i}}_{\infty}^{\ge\nu,\circ}}\to k[u,u^{-1}]. \end{align*}$$

We now show that $\psi $ is injective by applying Lemma 4.14.

Let K be an algebraically closed k-extension. A morphism

$$\begin{align*}k[X_{{n}},(Y_{i,j})_{\substack{1\le i\le \ell\\j\ge {m}_{i}}},\frac{1}{X_{{n}}}]/\sqrt{{\mathfrak{i}}_{\infty}^{\ge\nu,\circ}}\to K\end{align*}$$

is entirely and uniquely determined by a collection of elements $\beta _{{n}}\in K^{{\times }}$ , $(\beta _{i,j})_{\substack {0\le i\le \ell \\j\ge {m}_{i}}}\in K $ , such that one has

$$\begin{align*}\forall F\in {\mathfrak{i}},\quad F\left(\beta_{{n}}\cdot t^{{n}}, \left(\sum_{j\ge {m}_{i}}\beta_{i,j}\cdot t^j\right)_{1\le i\le \ell}\right)=0.\end{align*}$$

Let us fix such a collection. We have to prove that there exists $\omega \in K^{{\times }}$ , such that $ \beta _{{n}}=\omega ^{{n}} $ and $Y_{p,i}(\omega \cdot t)=\sum _{j\ge {m}_i}\beta _{i,j}\cdot t^j$ for $1\le i\le \ell $ .

Since K is algebraically closed, one may find $\mu \in K^{{\times }}$ , such that $\beta _{{n}}=\mu ^{{n}}$ . Thus, one has

(4.6) $$ \begin{align} \forall F\in {\mathfrak{i}},\quad F\left((\mu t)^{{n}},\left(\sum_{j\ge {m}_{i}}\beta_{i,j}\mu^{-j}\cdot (\mu t)^j\right)_{1\le i\le \ell}\right)=0. \end{align} $$

By Proposition 2.11, the $ \ell +1 $ -tuple $\left (t^{{n}},\left (\sum _{j\ge {m}_{i}}\beta _{i,j}\mu ^{-j}\cdot t^j\right )_{1\le i\le \ell }\right )$ can be obtained from $\left (t^{{n}},\left (Y_{p,i}(t)\right )_{1\le i\le \ell }\right )$ by composition with an automorphism of $K{[[ {t} ]]}$ . Since such an automorphism has to fix $t^{{n}} $ , it is of the form $t\mapsto \zeta t $ , with $ \zeta $ a ${n}$ -th root of unity. Hence, for any $1\le i\le \ell $ , we have

(4.7) $$ \begin{align} \sum_{j\ge {m}_i}\beta_{i,j}\cdot \mu^{-j}t^j=Y_p(\zeta\cdot t). \end{align} $$

Now we take $\omega =\zeta \cdot \mu $ .

It remains to show that the fraction field of $\operatorname {\mathrm {Im}}(\psi )$ is $k(u)$ . For $1\le i\le \ell $ , write $ Y_{p,i}(t)=\sum _{j\ge {m}_{i}}p_{i,j}\cdot t^j $ , where $p_{i,j}\in k$ and $p_{i,{m}_i}\neq 0$ .

Since $\left (t^{{n}},\left (\sum _{j\ge {m}_{i}}p_{i,j}\cdot t^j\right )_{1\le i\le \ell }\right )$ is a primitive Puiseux k-parametrization of $({\mathcal C},c)$ , one has

$$\begin{align*}\gcd({n},\{j\ge {m}_{i}: 1\le i\le \ell, p_{i,j}\neq 0\})=1.\end{align*}$$

Thus, one may find, for $1\le i \le \ell $ , integers $j_{i_1},\dots ,j_{i_r}\ge {m}_{i}$ with $p_{i,j_{i_q}}\neq 0$ for $ 1\le q\le r $ and $ a_0\in {\mathbf {Z}} $ , $a_{i_1}\dots ,a_{i_r},b_{i_1}\dots ,b_{i_r}\in {\mathbf {N}}$ , such that

(4.8) $$ \begin{align} a_0\cdot {n}+\sum_{q=1}^r a_{i_q}\cdot j_{i_{q}}=1+\sum_{q=1}^r b_{i_{q}}\cdot j_{i_q}. \end{align} $$

Thus, since $\phi (Y_{i,j} ) = u^j p_{i,j}$ , one has

$$\begin{align*}\phi\left(X_{{n}}^{a_0}\prod_{i=1}^{\ell}\prod_{q=1}^r Y_{i,j_{i_{q}}}^{a_{i_q}}\right)=\delta\cdot u\cdot \phi\left(\prod_{i=1}^{\ell}\prod_{q=1}^r Y_{i,j_{i_{q}}}^{b_{i_q}}\right)\end{align*}$$

with $\delta \in k^{{\times }}$ , and we conclude that $u\in \operatorname {\mathrm {Frac}}(\operatorname {\mathrm {Im}}(\phi ))$ , hence, also $u\in \operatorname {\mathrm {Frac}}(\operatorname {\mathrm {Im}}(\psi ))$ .

Proof. The situation is described by the following diagram.

By Proposition 4.13, the morphism of k-algebras $ \phi \colon R^{\ge \nu }\to k[(X_j)_{j> {n}}, u, u^{-1}] $ sending $ X_j$ to $X_j $ for $ j>{n} $ , $ X_{{n}}$ to $u^{{n}} $ and such that $\sum _{j\ge {m}_i}\phi (Y_{i,j})\cdot t^j=Y_{p,i}(ut)$ for $ 1\le i\le \ell $ induces an isomorphism between $ R^{\ge \nu }/\sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }}$ and a subring of $ k[(X_j)_{j> {n}}, u, u^{-1}] $ , with fraction field $ k((X_j)_{j> {n}}, u)$ . By Lemma 4.16, the previous isomorphism induces an isomorphism $\theta $ of $ \operatorname {\mathrm {Frac}}\left (R/{\mathfrak {p}}_{\nu }\right )$ with $k((X_j)_{j> {n}}, u)$ which maps $ \pi '(X_j)$ to $X_j $ for $ j>{n} $ , $ \pi '(X_{{n}})$ to $u^{{n}} $ and such that $\sum _{j\ge {m}_i}\theta (\pi '(Y_{i,j}))\cdot t^j=Y_{p,i}(ut)$ for $ 1\le i\le \ell $ . Let $v\in \operatorname {\mathrm {Frac}}\left (R/{\mathfrak {p}}_{\nu }\right )$ be the preimage of u by $\theta $ . Since $v^{{n}}=\pi '(X_{{n}})=\tau (\iota (\sigma (X_{{n}})))$ and $\sigma (X_{{n}})=X_{{n}}$ (Lemma 4.12), by Hensel’s lemma, there exists a unique element $ {\mathcal U}\in \widehat {A} $ , such that ${\mathcal U}^{{n}}=\iota (X_{{n}})$ and $ \tau ({\mathcal U})=v $ . Now we define the morphism of $ k $ -algebras $\rho '\colon R^{\ge \nu }{\longrightarrow } \widehat {A}$ by setting

$$\begin{align*}\begin{array}{crcll} & \rho'(X_j) &=& \iota(\sigma(X_j))=\iota(X_j)&\text{ for } j\ge {n},\\ \text{and} &\sum_{j\ge {m}_i}\rho'(Y_{i,j})\cdot t^j&=&Y_{p,i}({\mathcal U}\cdot t)&\text{ for } 1\le i\le \ell. \end{array}\end{align*}$$

Since $\tau ({\mathcal U})=v$ , one has $ \tau \circ \rho '=\pi '$ (by the the above description of the isomorphism $\theta $ ), as well as $ \rho '(X_{j})=\iota (X_{j}) $ for every $ j\ge {n} $ .

Let us finally prove that $\sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }}\subset \ker (\rho ')$ . It suffices to show that $\rho '(R^{\ge \nu })$ is a domain and $\ker (\rho ')$ contains ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }$ .

Let us show that $\rho '(R^{\ge \nu })$ is a domain. By construction, it is clear that $\rho '(R^{\ge \nu })$ is a subring of $k[(\iota (X_j))_{j> {n}},{\mathcal U}]$ . If the elements $(\iota (X_j))_{j> {n}},{\mathcal U} $ were algebraically dependent over k, then their images $ (\tau \circ \iota (X_j)=\pi '(X_j))_{j> {n}},v $ by $\tau $ in $ \operatorname {\mathrm {Frac}}\left (R/{\mathfrak {p}}_{\nu }\right )$ would also be. By the above description of the isomorphism $\theta \colon \operatorname {\mathrm {Frac}}\left (R/{\mathfrak {p}}_{\nu }\right )\cong k((X_j)_{j> {n}},u)$ , this is not the case. Therefore, $\rho '(R^{\ge \nu })$ is a domain.

Let us show that $\ker (\rho ')$ contains ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }$ .

For every $F\in {\mathfrak {i}}$ , one has

$$ \begin{align*} \rho'\left(F\left(X_{{n}}\cdot t^{{n}},\left(\sum_{j\ge {m}_{i}}Y_{i,j}\cdot t^j\right)_{1\le i\le \ell}\right)\right) & =F\left(\rho'(X_{{n}})\cdot t^{{n}},\left(\sum_{j\ge {m}_{i}}\rho'(Y_{i,j})\cdot t^j\right)_{1\le i\le \ell}\right)\\ &=F\left( ({\mathcal U} t)^{{n}},\left(Y_{p,i}({\mathcal U}\cdot t)\right)_{1\le i\le \ell}\right)\\ &=0. \end{align*} $$

Thus, by the definition of $(F^{\ge \nu ,\circ }_{j})_{j\in {\mathbf {N}}}$ (see (4.2)), we have $\sum _{j\ge 0} \rho '(F^{\ge \nu ,\circ }_{j})\cdot t^{j}=0 $ , and we deduce that ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }\subset \ker (\rho ') $ .

Proof of Proposition 4.8

For the ease of the reader, the diagram below shows the morphisms involved in the proof:

Let $\widetilde {\pi }:=\pi \circ (R^{\ge \nu }{\hookrightarrow } R)$ and $\widetilde {\rho }:=\rho '\circ \sigma ^{-1}$ . Since $\tau \circ \rho '=\pi '$ and $\widetilde {\pi }=\pi '\circ \sigma ^{-1}$ , one has $\tau \circ \widetilde {\rho }=\widetilde {\pi }$ . Moreover, since ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }\subset \operatorname {\mathrm {Ker}}(\rho ')$ and $\sigma ^{-1}$ maps ${\mathfrak {i}}_{\infty }^{\ge \nu ,\circ }$ to ${\mathfrak {i}}_{\infty }^{\ge \nu }$ , one has ${\mathfrak {i}}_{\infty }^{\ge \nu } \subset \operatorname {\mathrm {Ker}}(\widetilde {\rho })$ .

Let $\rho \colon R\to \widehat {A}$ be the morphism inducing $\widetilde {\rho }$ on $R^{\ge \nu }$ and such that $\rho (X_{j})=0$ for $j<{n}$ , $\rho (Y_{i,j})=0$ for $ 1\le i\le \ell $ and $j<{m}_i$ . Since $\tau \circ \widetilde {\rho }=\widetilde {\pi }$ , $\tau (X_{j})=0$ for $0\le j<{n}$ , $\tau (Y_{i,j})=0$ for $ 1\le i\le \ell $ , and $0\le j<{m}_i$ and the same holds for $\pi $ , one has $\tau \circ \rho =\pi $ . Moreover, by its very construction, $\operatorname {\mathrm {Ker}}(\rho )$ contains ${\mathfrak {q}}_{\nu }+{\mathfrak {i}}_{\infty }^{\ge \nu }$ and $\rho (R)=\widetilde {\rho }(R^{\ge \nu })=\rho '(R^{\ge \nu })$ is a domain, thus, $\operatorname {\mathrm {Ker}}(\rho )$ contains $\sqrt {{\mathfrak {q}}_{\nu }+{\mathfrak {i}}_{\infty }^{\ge \nu }}={\mathfrak {p}}_{\nu }$ .

The remaining assertion of Proposition 4.8 is a straightforward consequence of Proposition 4.17, recalling that, for any $j\ge {n}$ , one has $\sigma (X_j)=X_j$ .

5.1

The first result may be interpreted as a statement about the existence and the uniqueness of some specific “infinitesimal quasi-deformations” of a nondegenerate arc. It is an application of a version of Hensel’s lemma for an arbitrary set of variables stated in [Reference Bourqui, Morán Cañón and SebagBMCS, Proposition 4.2], which we first recall.

Proposition 5.3. Let K be a field, m and $\ell $ be positive integers, $F_1,\dots ,F_{\ell }$ be $\ell $ elements of $K[X_1,\dots ,X_m,Y_1,\dots ,Y_{\ell }]$ , and J be the Jacobian matrix $\left (\partial _{Y_i} F_r\right )_{\substack {1\le r\le \ell \\1\le i\le \ell }}$ . Let ${\boldsymbol {x}}(t)=(x_i(t))_{1\le i\le m}$ and ${\boldsymbol {y}}(t)=(y_i(t))_{1\le i\le \ell }$ be elements of $K{[[ {t} ]]}$ , such that:

$$\begin{align*}\forall 1\le r\le \ell,\quad F_r\left({\boldsymbol{x}}(t), {\boldsymbol{y}}(t)\right)=0\quad \text{and}\quad \det(J)\left({\boldsymbol{x}}(t), {\boldsymbol{y}}(t)\right)\ne 0. \end{align*}$$

Then there exist $\ell $ nonnegative integers $N_1,\dots ,N_{\ell }$ and $\ell $ elements $\widetilde {F_1},\dots ,\widetilde {F_{\ell }}$ of the ring $K{[[ {t} ]]}[X_1,\dots ,X_m,Y_1,\dots ,Y_{\ell }]$ , such that the following holds:

1. 1. For any K-algebra ${\mathcal A}$ and any elements $({\mathbf {X}}(t),{\mathbf {Y}}(t))\in ({\mathcal A}{[[ {t} ]]})^{m+\ell }$ , one has

   $$\begin{align*}\forall 1\le r\le \ell,\,F_r({\mathbf{X}}(t),{\mathbf{Y}}(t))=0 {\Longleftrightarrow} \forall 1\le r\le \ell,\,\widetilde{F_r}({\mathbf{X}}(t),{\mathbf{Y}}(t))=0. \end{align*}$$

2. 2. Let $(C,{\mathfrak {M}}_C)$ be a complete local K-algebra with residue field K, and let ${\mathbf {X}}(t)=(X_i(t))_{1\le i\le m}$ (respectively, ${\mathbf {Y}}(t)=(Y_i(t))_{1\le i\le \ell }$ ) be an m-tuple (respectively, an $\ell $ -tuple) of elements of $C{[[ {t} ]]}$ whose image in $(C/{\mathfrak {M}}_C){[[ {t} ]]}=K{[[ {t} ]]}$ is ${\boldsymbol {x}}(t)$ (respectively, ${\boldsymbol {y}}(t)$ ). Then there exists a unique family $({\mathcal Z}_{i,j})_{\substack {1\le i\le \ell \\j\ge \operatorname {\mathrm {ord}}_ty_i(t)}}$ of elements of $\;{\mathfrak {M}}_C $ , such that, setting , we have

   

Remark 5.4. The integers $N_r$ and the elements $\widetilde {F_r}$ will be explicitly constructed in the proof. In case $\ell =1$ , by a slight modification of the proof, one sees that one can take $\widetilde {F_1}=F_1$ and $N_1=\operatorname {\mathrm {ord}}_t(\partial _{Y_1}F_1({\boldsymbol {x}}(t),{\boldsymbol {y}}(t)))$ .

We also note that assertion (1) expresses the fact that the arc scheme associated to the K-scheme defined by the $F_i$ ’s on the one hand, and the arc scheme (or Greenberg scheme) associated to the $K{[[ {t} ]]}$ -scheme defined by the $\widetilde {F_i}$ ’s on the other hand (see [Reference Chambert-Loir, Nicaise and SebagCLNS18, Chapter 4, Proposition 3.17 & (2.1.4)]), are isomorphic.

An infinitesimal deformation of the arc $({\boldsymbol {x}}(t),{\boldsymbol {y}}(t))$ in the aforementioned arc scheme with value in an object $(C,{\mathfrak {M}}_C)$ of $\operatorname {\mathrm {\mathbf {CplLoc}}}_K$ is the datum of two families $({\mathcal W}_{i,j})_{1\le i\le m}$ and $({\mathcal Z}_{i,j})_{1\le i\le \ell }$ of elements of ${\mathfrak {M}}_C $ , such that

(5.1)

Thus, assertion (2) may be interpreted as follows: starting from the datum of two families of elements of ${\mathfrak {M}}_C$ as above (not necessarily satisfying (5.1)) there is at most one way of perturbing the $({\mathcal Z}_{i,j})_{\substack {1\le i\le \ell \\j\ge \operatorname {\mathrm {ord}}_ty_i(t)}}$ in order to obtain a deformation of the arc $({\boldsymbol {x}}(t),{\boldsymbol {y}}(t))$ ; and, in general, it is only possible to obtain after this perturbation a “quasi-deformation”, that is, a deformation up to a finite number of terms, as expressed by the condition on the degrees in assertion (2). Note that this allows us to recover a weak form of the Drinfeld-Grinberg-Kazhdan theorem [Reference Grinberg and KazhdanGK00, Reference DrinfeldDri02], more precisely, the fact that the formal neighborhood of a rational nondegenerate arc is the quotient of a power series in countably many variables by a finitely generated ideal. This is due to the fact that the condition on the degrees of the $\widetilde {F_i}$ provide finitely many equations, that are a priori power series in the involved variables, and not necessarily polynomials (which would recover the full Drinfeld-Grinberg-Kazhdan theorem).

Proof. Consider the matrix

$$\begin{align*}M(t):=\left(t^{\operatorname{\mathrm{ord}}_t y_i(t)}\cdot \partial_{Y_i} F_r({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))\right)_{\substack{1\le r\le \ell\\1\le i\le \ell}}. \end{align*}$$

Since $\det (J)\left ({\boldsymbol {x}}(t), {\boldsymbol {y}}(t)\right )\neq 0$ , one also has $\det (M(t))\neq 0$ . Thus, there exist $\ell \times \ell $ matrices $A(t)$ and $B(t)$ with entries in $K{[[ {t} ]]}$ and determinant in $K{[[ {t} ]]}^{{\times }}$ , and nonnegative integers $N_1,\dots ,N_{\ell }$ , such that $A(t)\cdot M(t)\cdot B(t)^{-1}=\operatorname {\mathrm {Diag}}(t^{N_1},\dots ,t^{N_{\ell }})$ .

Let $[\widetilde {F_r}]_{1\le r\le \ell }$ be the $\ell $ -tuple of elements of $K{[[ {t} ]]}[X_1,\dots ,X_m,Y_1,\dots ,Y_{\ell }]$ defined by the relation $[\widetilde {F_r}]_{1\le r\le \ell }=A(t)\cdot [F_r]_{1\le r\le \ell }$ . It easily follows that assertion (1) holds.

Let us consider the set of variables $(Z_{i,j})_{\substack {1\le i\le \ell \\ j\ge \operatorname {\mathrm {ord}}_t(y_i(t))}}$ and

$$\begin{align*}{\mathbf{Z}}(t):=\left(\sum_{j\ge \operatorname{\mathrm{ord}}_t(y_i(t))}Z_{i,j}\cdot t^{j}\right)_{1\le i\le \ell}=:\left(Z_i(t)\right)_{1\le i\le \ell}. \end{align*}$$

For $1\le r\le \ell $ and $j\ge 0$ , let $\widetilde {F_{r,j}}\in C[(Z_{i,j})]$ be defined by the relation

$$\begin{align*}\sum_{j\ge 0} \widetilde{F_{r,j}}\cdot t^j=\widetilde{F_r}({\mathbf{X}}(t),{\mathbf{Y}}(t)+{\mathbf{Z}}(t)).\end{align*}$$

We also consider the following $\ell $ -tuple of elements of $C[(Z_{i,j})]{[[ {t} ]]}$ :

$$\begin{align*}\widetilde{{\mathbf{F}}}_{\ge \bullet}:= \left[ \sum\limits_{j\ge N_r} \widetilde{F_{r,j}}\cdot t^j \right]_{1\le r\le \ell}.\end{align*}$$

Thus, we aim to prove the existence of a unique solution with values in ${\mathfrak {M}}_C$ of the system of polynomial equations $\widetilde {{\mathbf {F}}}_{\ge \bullet }=0 $ with unknowns $Z_{i,j}$ (i.e., the equations obtained by requiring that all the t-coefficients of any component of the $\ell $ -tuple $\widetilde {{\mathbf {F}}}_{\ge \bullet }$ be zero). We seek to apply Proposition 5.2. By hypotheses and (1), for $1\le r\le \ell $ , one has $\widetilde {F_r}\left ({\boldsymbol {x}}(t), {\boldsymbol {y}}(t)\right )=0 $ . Thus, $ Z_{i,j}=0 $ for $ 1\le i\le \ell $ and $ j\ge \operatorname {\mathrm {ord}}_t(y_i) $ is a solution of the reduction modulo $ {\mathfrak {M}}_C $ of the system. Now we have to study the Jacobian matrix of the system at this solution. Let $ \widetilde {{\mathbf {F}}}_{\ge \bullet }^{\operatorname {\mathrm {red}}}=[\widetilde {F_r}({\boldsymbol {x}}(t),{\boldsymbol {y}}(t)+{\mathbf {Z}}(t))]_{1\le r\le \ell }= 0 $ be the reduction of the system $\widetilde {{\mathbf {F}}}_{\ge \bullet }=0 $ modulo $ {\mathfrak {M}}_{C}$ . By the Taylor formula, we have

(5.2) $$ \begin{align} \widetilde{{\mathbf{F}}}_{\ge \bullet}^{\operatorname{\mathrm{red}}}=\widetilde{J}({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))\cdot {\mathbf{Z}}(t)+H(t), \end{align} $$

where $\widetilde {J}:=\left (\partial \widetilde {F_r}/\partial Y_i\right )_{\substack {1\le r\le \ell \\1\le i\le \ell }}$ and the t-coefficients of the components of $H(t)$ contain only terms of degree at least 2 in the $Z_{i,j}$ ’s. By the definition of the $\widetilde {F_r}'s$ , one has

$$\begin{align*}\widetilde{J}({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))=A(t)\cdot J({\boldsymbol{x}}(t),{\boldsymbol{y}}(t)).\end{align*}$$

Now, setting ${\mathbf {Z}}^{\circ }(t):=\left (\sum _{j\ge 0}Z_{i,j-\operatorname {\mathrm {ord}}_t y_i(t)}\cdot t^{j}\right )_{1\le i\le \ell }$ , one has

$$\begin{align*}J({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))\cdot {\mathbf{Z}}(t)=M(t)\cdot {\mathbf{Z}}^{\circ}(t); \end{align*}$$

therefore

$$\begin{align*}\widetilde{J}({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))\cdot {\mathbf{Z}}(t)=\operatorname{\mathrm{Diag}}(t^{N_1},\dots,t^{N_{\ell}})\cdot B(t)\cdot {\mathbf{Z}}^{\circ}(t). \end{align*}$$

Since $\det (B(t))\in K{[[ {t} ]]}^{{\times }}$ , we deduce from the former relation that the K-linear map

$$\begin{align*}\prod_{i=1}^{\ell} K^{ {\mathbf{Z}}_{\ge \operatorname{\mathrm{ord}}_t(y_i(t))}}\to \prod_{i=1}^{\ell} K^{{\mathbf{Z}}_{\ge N_i}},\quad (z_{i,j})\mapsto \phi_{i,j}({\boldsymbol{z}}) \end{align*}$$

defined by

$$\begin{align*}[{\underset{{j\ge N_j}}{\sum}}\phi_{i,j}({\boldsymbol{z}})\cdot t^j]_{1\le i\le \ell} =\widetilde{J}({\boldsymbol{x}}(t),{\boldsymbol{y}}(t))\cdot [{\underset{{j\ge \operatorname{\mathrm{ord}}_ty_i(t)}}{\sum}}z_{i,j}\cdot t^j]_{1\le i\le \ell} \end{align*}$$

is invertible. Thus, Proposition 5.2 guarantees the existence and the uniqueness of a family $({\mathcal Z}_{i,j})_{\substack {1\le i\le \ell \\j\ge \operatorname {\mathrm {ord}}_t(y_i(t))}} $ of elements of $ {\mathfrak {M}}_C $ which is a solution of the system $ \widetilde {{\mathbf {F}}}_{\ge \bullet }=0 $ . This concludes the proof of (2).

5.5

We recall, here, the following known result about completions of local rings (see, e.g., [Reference de Fernex and DocampodFD20, Lemma 10.12] and [Reference HaiechHai21, Theorem A and Corollary 3.9]).

The next technical proposition, whose output is a presentation of the involved completion $\widehat {A}$ , will be applied in the next section to obtain a presentation of the formal neighborhood of the generic primitive arc of a curve singularity that is suited to our needs.

6.1

In this section, we recover the setting and notation from Section 4. As already explained, our aim is to provide a presentation of the formal neighborhood of $ {\mathscr {L}_{\infty }}({\mathcal C}) $ at its generic primitive arc $\eta _{{\mathcal C},\nu }$ which can be naturally interpreted in terms of infinitesimal deformations of the corresponding $\kappa (\eta _{{\mathcal C},\nu })$ -arc. This will be a consequence of the existence of a coefficient field with specific properties (Proposition 4.8) and of the technical results in Section 5, whose assumptions hold in our setting, as we will check. As in Section 4, all the results and arguments hold, with the corresponding modifications, if we consider a valuation $ N\cdot \nu $ of $ {\mathcal C} $ , for $ N\ge 1 $ (see Remark 4.6).

Let $\rho $ be a morphism as in the statement of Proposition 4.8. We set $K:=\operatorname {\mathrm {Frac}}(\rho (R))\subset \widehat {A}$ which is a coefficient field of $\widehat {A}$ (Remark 4.3). For simplicity, we set

$$\begin{align*}x_{\rho}(t):=\sum_{j\ge {n}}\rho(X_j)\cdot t^j,\quad y_{\rho,i}(t):=\sum_{j\ge {m}_i}\rho(Y_{i,j})\cdot t^j\quad \text{and} \quad {\boldsymbol{y}}_{\rho}(t):=(y_{\rho,i}(t))_{1\le i\le \ell}. \end{align*}$$

Recall that, by Remark 4.10, one has

(6.1) $$ \begin{align} \forall F\in {\mathfrak{i}}, \quad F(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))=0. \end{align} $$

In other words, $(x_{\rho }(t),{\boldsymbol {y}}_{\rho }(t))$ defines a K-arc on ${\mathcal C}$ , which is nondegenerate (see Section 2.4), since the image of this arc by $p_{\widehat {A}}$ is the nonconstant $\kappa _{\eta _{{\mathcal C},\nu }}$ -arc on ${\mathcal C}$ induced by $\eta _{{\mathcal C},\nu }$ .

For the sake of convenience, set $Y_0:=X$ . By a standard application of “Elkik’s trick” (see, e.g., [Reference Bourqui and SebagBS17a, Section 4.2]), there exist elements $F_1,\dots ,F_{\ell }\in {\mathfrak {i}}$ , an $\ell $ -minor $\Delta $ of the Jacobian matrix $[\partial _{Y_{i}}F_j]_{\substack {0\le i\le \ell \\1\le j\le \ell }}$ , and an element H of the quotient ideal ${\langle {F_1,\dots ,F_\ell } \rangle } :{\mathfrak {i}}$ , such that $H(x_{\rho }(t),{\boldsymbol {y}}_{\rho }(t))\neq 0$ and $\Delta (x_{\rho }(t),{\boldsymbol {y}}_{\rho }(t))\neq 0$ . In fact, by Lemma 6.7, one has

(6.2) $$ \begin{align} \det([\partial_{Y_j}{F_i}(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))]_{1\le i,j\le \ell}\neq 0. \end{align} $$

Remark 6.2. Let $({\mathcal A},{\mathfrak {M}}_{\mathcal A})$ be any object of $\operatorname {\mathrm {\mathbf {CplLoc}}}_K$ . Then for any $(x_A(t),{\boldsymbol {y}}_{A}(t))\in {\mathfrak {M}}_{{\mathcal A}}{[[ {t} ]]}^{\ell +1}$ , the conditions

$$\begin{align*}\forall F\in {\mathfrak{i}},\quad F(x_{A}(t) +x_{\rho}(t),{\boldsymbol{y}}_{A}(t)+{\boldsymbol{y}}_{\rho}(t))=0 \end{align*}$$

$$\begin{align*}\text{and}\quad \forall 1\le r\le \ell, \quad F_r(x_{A}(t) +x_{\rho}(t),{\boldsymbol{y}}_{A}(t)+{\boldsymbol{y}}_{\rho}(t))=0 \end{align*}$$

are equivalent. Indeed, $H(x_{A}(t)+x_{\rho }(t),{\boldsymbol {y}}_{A}(t)+{\boldsymbol {y}}_{\rho }(t))\in {\mathcal A}{[[ {t} ]]}$ is not a zero divisor, since it reduces modulo ${\mathfrak {M}}_{{\mathcal A}}$ to $H(x_{\rho }(t),{\boldsymbol {y}}_{\rho }(t))\neq 0$ .

6.3

We will first prove that the complete ring $ \widehat {A} $ is Noetherian and obtain a suitable system of generators of its cotangent space (which is, in fact, a basis, though we do not need this fact for our present purposes; see Remark 6.15 below). The result is a particular case of Reguera and Reguera-Mourtada’s general study of stable points and their cotangent space (see, in particular, [Reference Mourtada and RegueraMR18, Theorem 3.4] combined with [Reference RegueraReg09, Theorem 3.13]; see also [Reference RegueraReg18, Reference MourtadaMou17]); as pointed out by the referee, this may also be seen as a consequence of [Reference Chiu, de Fernex and DocampoCdFD22, Theorem 8.1] by considering the projection associated to $k[X]\to k[X,Y_1,\dots ,Y_{\ell } ]$ . We provide a direct proof in our setting for the convenience of the reader, and since our viewpoint is more deformation-theoretic than in the above references and in some sense in the same vein as the arguments that are to be used later in the section.

Proof. Recall that $ A $ is defined to be the localization of $ R/{\mathfrak {i}}_{\infty } $ with respect to $ {\mathfrak {p}}_{\nu } $ . By (4.3), $ {\mathfrak {M}}_{A} $ is generated by the image of ${\mathfrak {p}}_{\nu }:=\operatorname {\mathrm {rad}}\left ({\mathfrak {i}}_{\infty }+{\langle {(X_j)_{0\le j<{n}}, (Y_{i,j})_{\substack {1\le i\le \ell \\ 0\le j< {m}_i}}} \rangle }\right )\subset R$ in $ A $ .

Let $ {\mathfrak {h}} $ be the ideal of $\widehat {A}$ generated by the set

(6.3) $$ \begin{align} \{\iota(X_j)\}_{0\le j<{n}} \cup \{\iota(Y_{i,j})\}_{\substack{1\le i\le \ell\\ 0\le j< {m}_i}} \cup \{\iota(Y_{i,j})-\rho(Y_{i,j})\}_{\substack{1\le i\le \ell\\ j\ge {m}_i}}. \end{align} $$

We first show that the ideal generated by $\iota ({\mathfrak {p}}_{\nu })$ is contained in ${\mathfrak {h}}$ . Let $P\in {\mathfrak {p}}_{\nu }$ , seen as a polynomial in the indeterminates $(X_j)_{0\le j<{n}},(Y_{i,j})_{\substack {1\le i\le \ell \\ 0\le j< {m}_i}}$ with coefficients in $R^{\ge \nu }$ . Let us set $ Q:=P(0)\in R^{\ge \nu } $ , then it suffices to show that $ \iota (Q) $ lies in $ {\mathfrak {h}} $ . Since $P\in {\mathfrak {p}}_{\nu }$ , and by the definition of the latter ideal, we deduce that $Q\in \sqrt {{\mathfrak {i}}_{\infty }^{\ge \nu }}$ , in particular, $ \rho (Q)=0$ by Remark 4.10. Let us write $Q=\widetilde {Q}(Y_{i,j})$ , where $\widetilde {Q}$ is seen as a polynomial in the indeterminates $\{Y_{i,j}\}_{\substack {1\le i\le \ell \\ j\ge {m}_i}}$ with coefficients in $T:=k[(X_j)_{j\ge {n}}][\frac {1}{X_{{n}}}]$ . By Proposition 4.8, $\rho $ and $\iota $ coincide on T. Thus, the relation $\rho (Q)=0$ may be rewritten as $\iota (\widetilde {Q})(\rho (Y_{i,j}))=0$ , which shows that $\iota (\widetilde {Q})(Y_{i,j})$ lies in the ideal of $\iota (T)[(Y_{i,j})]$ generated by $Y_{i,j}-\rho (Y_{i,j})$ . Thus, $\iota (Q)=\iota (\widetilde {Q})(\iota (Y_{i,j}))$ lies in the ideal generated by the $\iota (Y_{i,j})-\rho (Y_{i,j})$ , in particular, $\iota (Q)$ lies in ${\mathfrak {h}}$ .

Since the image by the projection $\widehat {A}\to A/{\mathfrak {M}}_A^2$ of the ideal generated by $\iota ({\mathfrak {p}}_{\nu })$ is ${\mathfrak {M}}_A/{\mathfrak {M}}_A^2$ , we deduce that ${\mathfrak {M}}_A/{\mathfrak {M}}_A^2$ is generated by the image of the set (6.3). In the remainder of the proof, we still denote by $\rho $ and $\iota $ the composition of $\rho $ and $\iota $ with the projection morphism $\widehat {A}\to A/{\mathfrak {M}}_A^2$ . For every $1\le i\le \ell $ , we set $Z_{i,j}:=\iota (Y_{i,j})-\rho (Y_{i,j})\in {\mathfrak {M}}_{A}/{\mathfrak {M}}_A^2$ for $ j\ge {m}_i$ and $Z_{i,j}:=\iota (Y_{i,j})\in {\mathfrak {M}}_{A}/{\mathfrak {M}}_A^2$ for $0\le j <{m}_i $ .

For every $ 1\le r\le \ell $ , we have

$$\begin{align*}0=\iota(F_r)=F_r\left(\sum_{0\le j<{n}}\iota(X_j)\cdot t^{j} +x_{\rho}(t),\left(\sum_{j\ge 0}Z_{i,j}\cdot t^j+y_{\rho,i}(t) \right)_{1\le i\le \ell}\right).\end{align*}$$

Applying the Taylor expansion formula, we obtain, for $1\le r\le \ell $ , the relation

$$ \begin{align*} 0&=F_r(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))+\partial_{X} F_r(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))\cdot\sum_{0\le j<{n}}\iota(X_j)\cdot t^{j}\\ & \quad +\sum_{1\le i\le \ell}\partial_{Y_i}F_r(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))\cdot\sum_{j\ge 0}Z_{i,j}\cdot t^j. \end{align*} $$

For $0\le r \le \ell $ (and setting $Y_0:=X$ for the sake of convenience), set $J_{r}(t):=[\partial _{Y_i}{F_j}({\boldsymbol {y}}_{\rho }(t))]_{\substack {1\le j \le \ell \\ 0 \le i\neq r \le \ell }}$ . By (6.1), one obtains

$$\begin{align*}[\partial_{X} {\mathbf{F}}(x_{\rho}(t),{\boldsymbol{y}}_{\rho}(t))]\cdot \sum_{0\le j<{n}}\iota(X_j)\cdot t^{j} = -J_0(t)\cdot \left[\sum_{j\ge 0}Z_{i,j}\cdot t^j\right]_{1\le i\le \ell}.\end{align*}$$

Multiplying this equality by the adjugate matrix of $ J_0(t) $ and using Lemma 6.8 below, we obtain

(6.4) $$ \begin{align} \left(\sum_{0\le j<{n}}\iota(X_j)\cdot t^{j} \right)[\det(J_{i}(t))]_{1\le i\le \ell} =-\det(J_0(t))\cdot \left[\sum_{j\ge 0}Z_{i,j}\cdot t^j\right]_{1\le i\le \ell}. \end{align} $$

By Lemma 6.7 below (recall that for any $1\le i\le \ell $ , one has $\operatorname {\mathrm {ord}}_t(y_0(t))\le \operatorname {\mathrm {ord}}_t(y_i(t))<+\infty $ , see Section 2.10), we know that

(6.5) $$ \begin{align} \operatorname{\mathrm{ord}}_t(\det(J_0(t))\le \operatorname{\mathrm{ord}}_t \det(J_i(t)),\; 1\le i\le \ell. \end{align} $$

Thus, we deduce from (6.4) that in ${\mathfrak {M}}_{A}/{\mathfrak {M}}_{A}^2 $ the elements $ Z_{i,j} $ lie in the $\kappa _{A}$ -vector space generated by the $\{\iota (X_j)\}_{0\le j<{n}}$ .

The last assertion of Proposition 6.4 is a consequence of Lemma 5.6.

6.6

We state and prove two elementary lemmas used before.

Proof. Differentiating the relations $F_r({\boldsymbol {y}}(t))=0,\,1\le r\le \ell $ , one obtains

$$\begin{align*}J_0(t)\cdot [y^{\prime}_i(t)]_{1\le i\le \ell}+y^{\prime}_0(t)\cdot [\partial_{Y_0}F_i({\boldsymbol{y}}(t))]_{1\le i\le \ell}=0. \end{align*}$$

Multiplying by the adjugate matrix of $J_0(t)$ and using Lemma 6.8 below, one finds the relation

(6.6) $$ \begin{align} \det(J_0(t))\cdot [y^{\prime}_i(t)]_{1\le i\le \ell}+y^{\prime}_0(t)\cdot [\det(J_i(t))]_{1\le i\le \ell}=0. \end{align} $$

If $r_0=0$ , there is nothing to do. Otherwise, one has $r_0\in \{1,\dots ,\ell \}$ , since $y^{\prime }_0(t)\neq 0$ and $k{[[ {t} ]]}$ is a domain, (6.6) for $i=r_0$ shows that $\det (J_0(t))\neq 0$ . The last assertion is also a straightforward consequence of (6.6).