2 Preliminaries

Let R denote the power series ring $ {\mathbf {k}}[\![x_1, \dots , x_n]\!] $ over an algebraically closed characteristic zero field ${\mathbf {k}}$ and we denote by $\max =(x_1,\dots , x_n)$ its maximal ideal.

Let A be a one-dimensional local ring with maximal ideal $\max $ . We denote by ${\operatorname {H\!F}}_A$ the Hilbert function of A, i.e., ${\operatorname {H\!F}}_A(i)=\operatorname {Length}_A(\max ^i/\max ^{i+1})$ , $i\ge 0$ . It is well-known that ${\operatorname {H\!F}}^0_A(i)=e_0(A)$ , $i\gg 0$ , where $e_0(A)$ is the multiplicity of A. The first integral of ${\operatorname {H\!F}}_A$ is defined by, $i\ge 0$ ,

$$ \begin{align*}{\operatorname{H\!F}}^1_A(i)=\sum_{j=0}^i {\operatorname{H\!F}}_A(j)= \operatorname{Length}_A(A/{\max}^{i+1}).\end{align*} $$

We write ${\operatorname {H\!F}}_A^0={\operatorname {H\!F}}_A$ . There exists an integer $e_1(A)$ such that ${\operatorname {H\!F}}^1_A(i)=e_0(A)( i+1) -e_1(A)$ for $i\gg 0$ ; the (first) Hilbert polynomial is $\operatorname {H\!P}^1_A(T)=e_0(A)( T+1) -e_1(A)$ . See [Reference Matlis22, Chapter XII] for the basic properties of the Hilbert functions of one-dimensional Cohen–Macaulay local rings.

A branch X is an irreducible curve singularity of $({\mathbf {k}}^n,0)=\operatorname {Spec}(R)$ , i.e., X is a one-dimensional, integral scheme $X=\operatorname {Spec}(R/I)$ ; we write ${\mathcal O}_{X}=R/I$ and $I(X)=I$ .

Let $\nu : \overline {X}=\operatorname {Spec} (\overline {{\mathcal O}_{X}})\longrightarrow (X,0)$ be the normalization of $(X,0)$ , where $\overline {{\mathcal O}_{X}}\cong {\mathbf {k}}[\![t]\!]$ is the integral closure of ${\mathcal O}_{X}$ on its full field of fractions ${\mathrm { tot}}({\mathcal O}_{X})$ . The singularity order of X is $ \delta (X)=\operatorname {dim}_{{\mathbf {k}}}\left ({\mathcal O}_{\overline {X}}/{\mathcal O}_{X}\right ). $ We denote by $\mathcal C$ the conductor of the finite extension $\nu ^*: {\mathcal O}_{X} \hookrightarrow \overline {{\mathcal O}_{X}}$ and by $c(X)$ the dimension of $\overline {{\mathcal O}_{X}}/\mathcal C$ .

Given a set of nonnegative integers $1\le a_1<\cdots <a_n$ , we consider the monomial curve singularity $X(a_1,\dots , a_n)$ defined by the parameterization

$$ \begin{align*}\begin{array}{cccc} \gamma:&R & \longrightarrow &{\mathbf{k}}[\![t]\!]\\ &x_i & \mapsto & t^{a_i}, \end{array} \end{align*} $$

i.e., $I(X(a_1,\dots , a_n))=\ker (\gamma )$ . If $gcd(a_1,\dots , a_n)=1,$ then the induced map

$$ \begin{align*}\gamma:R/I(X(a_1,\dots, a_n))\longrightarrow {\mathbf{k}}[\![t]\!]\end{align*} $$

is the normalization map of $\mathcal O_{X(a_1,\dots , a_n)}=R/I(X(a_1,\dots , a_n))={\mathbf {k}}[\![t^{a_1},\dots , t^{a_n}]\!]$ .

We denote by $D_X$ the semigroup of values of X: the set of integers $v_t(f)=ord_t(t)$ where $f\in {\mathcal O}_{X}\setminus \{0\}$ . It is easy to see that $\delta (X)=\#(\mathbb N\setminus D_X)$ . If B is a finite codimension sub- ${\mathbf {k}}$ -algebra of $\Gamma $ then $X=\operatorname {Spec}(B)$ is branch. We write $D_B=D_X$ .

Let $\omega _{X}$ be the dualizing module of X; we can consider the composition of ${\mathcal O}_{X}$ -module morphisms

$$ \begin{align*}\gamma_X: \Omega_{X} \longrightarrow \nu_* \Omega_{\overline{X}}\cong \nu_* \omega_{\overline{X}} \longrightarrow \omega_{X}. \end{align*} $$

Let $d: {\mathcal O}_{X} \longrightarrow \Omega _{X}$ the universal derivation, then we have a ${\mathbf {k}}$ -linear map $\gamma _X d$ that we also denote by $d: {\mathcal O}_{X} \longrightarrow \omega _{X}.$ Recall that the Milnor number of X is $\mu (X)=\operatorname {dim}_{{\mathbf {k}}}(\omega _{X}/d {\mathcal O}_{X})$ , [Reference Buchweitz and Greuel5]. Since we only consider branches we have that $\mu (X)=2\delta (X)$ (see [Reference Buchweitz and Greuel5, Proposition 1.2.1]). Notice that X is non-singular iff $\mu (X)=0$ iff $\delta (X)=0$ iff $c(X)=0$ .

We denote by $\pi :Bl(X)\longrightarrow X$ the blowing-up of X on its closed point. The fiber of the closed point of X has a finite number of closed points: the so-called points of the first neighborhood of X. We can iterate the process of blowing-up until we get the normalization of X (see [Reference Cutkosky7, Reference Northcott24]). We denote by $\operatorname {Inf}(X)$ the set of infinitely near points of X. The curve singularity defined by an infinitely point p of X will be denote by $(X,p)$ ; we set $(X,0)=X$ .

Proposition 2.1 Let X be a branch. Then

$\mathrm{(i)}$

$$ \begin{align*}\delta(X)=\sum_{p \in \operatorname{Inf}(X)} e_i(X,p). \end{align*} $$

$\mathrm{(ii)}$ It holds

$$ \begin{align*}e_0(X)-1\le e_1(X) \le \delta(X)\le \mu(X) \end{align*} $$

and $e_1(X)\le \binom {e_0(X)}{2} - \binom {n-1}{2}$ .

$\mathrm{(iii)}$ If X is singular, then $\delta (X)+1 \le c(X)\le 2 \delta (X)$ , and $c(X) = 2 \delta (X)$ if and only if ${\mathcal O}_{X}$ is a Gorenstein ring.

3 Macaulay-like duality

In this section, we establish a Macaulay-like duality for the family of sub- ${\mathbf {k}}$ -algebras B of $\Gamma ={\mathbf {k}}[\![t]\!]$ of finite codimension. For the classical Macaulay’s duality, see [Reference Iarrobino and Kanev20], [Reference Elias, Cuong, Hoa and Trung14], and for the generalization to higher dimension of Macaulay’s duality, see [Reference Elias and Rossi15]. Recall that Macaulay’s duality is a particular case of Matlis’ duality (see [Reference Bruns and Herzog4]).

We write $\Delta = {\mathbf {k}}[u]$ ; $\Gamma $ is a $\Delta $ -module with $\Delta $ acting on $\Gamma $ by derivation. This action denoted by $\circ $ is defined by

$$ \begin{align*}\begin{array}{ clc} \circ: \Delta \times \Gamma &\longrightarrow & \Gamma \\ (g,f) & \to & g \circ f = g (\partial_{t})(f), \end{array} \end{align*} $$

where $ \partial _{t} $ denotes the derivative with respect to t. This action induces a non-singular ${\mathbf {k}}$ -bilinear perfect pairing:

(1) $$ \begin{align} \begin{array}{ lclc} \perp: & \Delta \times \Gamma &\longrightarrow & {\mathbf{k}} \\ & (g,f)&\mapsto & g\perp f=(g\circ f)(0). \end{array} \end{align} $$

Let B be a finite codimension sub- ${\mathbf {k}}$ -algebra of $\Gamma $ . Then we have a non-singular ${\mathbf {k}}$ -bilinear perfect pairing:

(2) $$ \begin{align} \begin{array}{ cclc} \perp: &B^{\perp} \times \frac{\Gamma}{B}&\longrightarrow & {\mathbf{k}} \\ & (g,\overline{f})&\mapsto & g\perp f. \end{array} \end{align} $$

We denote by $Perp(B)$ , the ${\mathbf {k}}$ -vector space of maps

$$ \begin{align*}\begin{array}{cclc} g^{\perp}: &B &\longrightarrow & {\mathbf{k}} \\ &f &\mapsto & g\perp f \end{array} \end{align*} $$

for all $g\in \Delta $ . These maps are the elements of the dual space of B with finite support: $g^{\perp }(\max _B^d)=0$ for $d>\operatorname {deg}(g)$ . We denote by $Der_{{\mathbf {k}}}(B)$ the ${\mathbf {k}}$ -vector space of ${\mathbf {k}}$ -derivations of B. Since $Der_{{\mathbf {k}}}(B)\cong (\max _B/\max _B^2)^*$ , we can identify $Der_{{\mathbf {k}}}(B)$ with the ${\mathbf {k}}$ -vector space of elements $\sigma $ of the dual space of B such that $\sigma (\max _B^2)=0$ .

We have $Der_{{\mathbf {k}}}(B)\subset Perp(B)$ , this inclusion is strict. Let us consider the codimension $8$ algebra $B={\mathbf {k}}[\![t^4,t^7,t^{17}]\!]$ . The linear map $(u^{11})^{\perp }:B\longrightarrow {\mathbf {k}}$ is not a derivation since $t^{11}\in \max _B^2$ and $(u^{11}){\perp }(t^{11})=11!\neq 0$ .

Next step is to characterize the vector ${\mathbf {k}}$ -vector subspaces $B^{\perp }$ of $\Delta ,$ where B ranges the family of finite codimension sub- ${\mathbf {k}}$ -algebras of $\Gamma $ . First, we give some properties of $B^{\perp }$ that we will use along the paper.

Given a polynomial $g=\sum _{i=0}^d a_i u^i\in \Delta $ we denote by $\mathrm {{Supp}}(g)$ the support of g: the finite set of integers i such that $a_i\neq 0$ .

Proof $(1)$ Since $\perp $ is a ${\mathbf {k}}$ -bilinear perfect pairing, we get $\operatorname {dim}_{{\mathbf {k}}}(B^{\perp })=\delta $ , see the equation (2).

$(2)$ Since B is a ${\mathbf {k}}$ -algebra, we have $1\in B$ , so if $g=\sum _{j\ge 0} a_iu^i\in B^{\perp }$ , then $0=g\perp 1= a_0$ . Hence $B^{\perp }\subset \langle u, u^2, \dots \rangle $ . We know that $(t^c)\subset B$ so for all $g=\sum _{j\ge 0} a_iu^i\in B^{\perp }$ , we have

$$ \begin{align*}0=g\perp t^{c+i}=(c+i)! a_{c+i} \end{align*} $$

$i\ge 0$ . Hence, if $g\in B^{\perp }$ , then $\operatorname {deg}(g)\le c-1$ . From this, we deduce that $B^{\perp }\subset \langle u, u^2, \dots , u^{c-1}\rangle $ .

Notice that $v_t(f)\ge e_0(B)$ for all $f\in B\setminus \{1\}$ , so given $i\in [1, e_0(B)-1]$ we have $u^i\perp f=0$ . Hence $u^i\in B^{\perp }$ and then $u^{[1, e_0(B)-1]}\subset B^{\perp }$ .

$(3)$ The condition of $(i)$ is equivalent to $(ii)$ . $(ii)$ trivially implies $(iii)$ and this implies $(iv)$ . If $B^{\perp }\subset \langle u^2,u^3,\dots \rangle $ , then $t\in B$ , since B is a ${\mathbf {k}}$ -algebra, we get $(ii)$ .

For all power series $f=\sum _{i\ge 0} b_i t^i\in \Gamma $ and given a nonnegative integer $s\in \mathbb N$ , we denote by $[f]_{\le s}$ the truncated polynomial $[f]_{\le s}=\sum _{i\ge 0}^s b_i t^i$ .

Let B be a finite codimension sub- ${\mathbf {k}}$ -algebra of $\Gamma $ with conductor c. Then B is a finitely generated ${\mathbf {k}}$ -algebra; let $f_1,\dots ,f_r$ be a system of generators of B as ${\mathbf {k}}$ -algebra. We denote by $\natural _{B,d}$ , $d\ge c-1$ , the finite set of polynomials $[f_1^{l_1}\dots f_r^{l_r}]_{\le d}$ with $l_i\ge 0$ , $i=1,\dots ,r$ , and $l_1+\cdots +l_r\le d$ . We denote by $W(\{f_1,\dots , f_r\},d)\subset \Delta $ the ${\mathbf {k}}$ -vector space generated by the polynomials of $\natural _{B,d}$ . Notice that $W(\{f_1,\dots , f_r\},d)+\langle t^{d+1}\rangle =W(\{f_1,\dots , f_r\},d+1)$ .

Proof Let $f_1,\dots ,f_r$ be a system of generators of B as ${\mathbf {k}}$ -algebra, and let $\natural _{B,c-1}$ be the associated set of polynomials.

If $g\in B^{\perp }$ , then $\operatorname {deg}(g)\le c-1$ , Proposition 3.2(2), so

$$ \begin{align*}0=g\perp (f_1^{l_1}\dots f_r^{l_r})= g\perp [f_1^{l_1}\dots f_r^{l_r}]_{\le c-1}. \end{align*} $$

Hence, $g\perp h=0$ for all $h\in \natural _{B,c-1}$ .

Let $g \in \Delta $ be a polynomial with $\operatorname {deg}(g)\le c-1$ and such that $g\perp h=0$ for all $h\in \natural _{B,c-1}$ . Any $f\in B$ can be written as

$$ \begin{align*}f=\sum_{l_1,\dots,l_r\in \mathbb N} c_{l_1,\dots,l_r}f_1^{l_1}\dots f_r^{l_r} \end{align*} $$

with $c_{l_1,\dots ,l_r}\in {\mathbf {k}}$ . Since $\operatorname {deg}(g)\le c-1,$ we have

$$ \begin{align*}g\perp f= \sum_{l_1,\dots,l_r\in \mathbb N} c_{l_1,\dots,l_r} (g\perp f_1^{l_1}\dots f_r^{l_r})= \sum_{l_1,\dots,l_r\in \mathbb N} c_{l_1,\dots,l_r} (g\perp [f_1^{l_1}\dots f_r^{l_r}]_{\le c-1})=0, \end{align*} $$

so $g\in B^{\perp }$ .

Example 3.7 Let us consider the codimension $\delta =4$ algebra $B={\mathbf {k}}[\![t^3+t^4,t^5]\!]$ of $\Gamma $ . The conductor of B is $c=8$ . Then $B^{\perp }$ is the set of polynomials $g\in \Delta $ of degree at most $7$ such that $g\perp f=0$ for $f\in \natural _{B,c-1}=\{t^3+t^4, t^5, t^6+2t^7\}$ . A simple computation shows that $B^{\perp }$ is the ${\mathbf {k}}$ -vector space generated by the four linear independent polynomials $u, u^2, u^3-\frac {1}{4}u^4, u^6-\frac {1}{2.7}u^7$ . Let us consider

$$ \begin{align*}B_2={\mathbf{k}}[\![t^3,t^4,t^5]\!]\subset B_3={\mathbf{k}}[\![t^2,t^3]\!], \end{align*} $$

then we have $B_2=\operatorname {Ann}\langle u^2\rangle \cap B_3$ , i.e., $u^2$ is an algebra-forming element with respect to $B_2$ .

In the following result, we prove that, in fact, if $V\subset \Delta $ is algebra-forming with respect to a ${\mathbf {k}}$ algebra $B\subset \Gamma ,$ then $\operatorname {Ann}(V)\cap B$ is a sub- ${\mathbf {k}}$ -algebra of $\Gamma $ .

Proof Clearly $C=\operatorname {Ann}(V)\cap B$ is a ${\mathbf {k}}$ -vector subspace of $\Gamma $ . Given $f_1, f_2\in C$ we have that $f_1+f_2\in C$ and from

$$ \begin{align*}f_1 f_2=\frac{1}{2}((f_1+f_2)^2-f_1^2-f_2^2), \end{align*} $$

we deduce that $g\perp (f_1 f_2)=0$ , i.e., $f_1f_2\in C$ . Since $g(0)=0$ for all $g\in V$ we get $1\in C$ , so C is a sub- ${\mathbf {k}}$ -algebra of $\Gamma $ .

The following result is an extension of Macaulay’s duality to finite codimension sub- ${\mathbf {k}}$ -algebras $B\subset \Gamma $ .

Proof Let B be a sub- ${\mathbf {k}}$ -algebra B of $\Gamma $ . Since we have a non-singular ${\mathbf {k}}$ -bilinear pairing:

$$ \begin{align*}\begin{array}{ cclc} \perp: &B^{\perp} \times \frac{\Gamma}{B}&\longrightarrow & {\mathbf{k}} \\ & (g,\overline{f})&\mapsto & g\perp f, \end{array} \end{align*} $$

we get that $B^{\perp }$ is a ${\mathbf {k}}$ -vector subspace of dimension $\delta $ of $\Delta $ . By definition $B^{\perp }$ is algebra-forming with respect to $\Gamma $ . Being c the conductor we have $(t^c)\subset B$ , so $\operatorname {deg}(g)\le c-1$ for all $g\in B^{\perp }$ and there exist $g\in B^{\perp }$ of degree $c-1$ .

Let V be an algebra forming, with respect to $\Gamma $ , ${\mathbf {k}}$ -vector subspace satisfying the conditions of (2). Let us consider the ${\mathbf {k}}$ -algebra $B=\operatorname {Ann}(V)$ . From the perfect pairing (1), we get that the codimension of B in $\Gamma $ is $\delta $ . Since V is generated by polynomials of degree at most $c-1$ we have that $(t^c)\subset B$ , so the conductor of B is at most c. Furthermore, since there is $g\in V$ with $\operatorname {deg}(g)=c-1$ we deduce that c is the conductor of B.

It is straightforward to prove the inclusion reversing from the definition of the inverse system $B^{\perp }$ .

We end this section by describing the ${\mathbf {k}}$ -linear maps $B_2^{\perp }\longrightarrow B_1^{\perp }$ induced by ${\mathbf {k}}$ -algebra isomorphisms $B_1\longrightarrow B_2$ between two finite codimension ${\mathbf {k}}$ -algebras $B_1$ and $B_2$ of $\Gamma $ . Let c be an integer bigger than the conductors of $B_1$ and $B_2$ .

The perfect pairing (1) induce a perfect pairing

$$ \begin{align*}\begin{array}{ lclc} \perp: & \Delta_{\le c-1} \times \frac{\Gamma}{(t^c)} &\longrightarrow & {\mathbf{k}} \\ & (g,\overline{f})&\mapsto & g\perp f=(g\circ f)(0), \end{array} \end{align*} $$

where $\Delta _{\le c-1}$ is the ${\mathbf {k}}$ -vector space of polynomials of degree at most $c-1$ . We consider the usual ${\mathbf {k}}$ -vector basis of $\Gamma /(t^c)$ of the cosets of $t^i$ , $i=0,\dots , c-1$ . Its dual basis is $\frac {1}{i!} u^i$ , $i=0,\dots , c-1$ , since

$$ \begin{align*}\left(\frac{1}{i!} u^i\right)\perp t^j=\delta_{i,j} \end{align*} $$

$1\le i, j \le c-1$ .

The ${\mathbf {k}}$ -algebra $B_i$ has conductor at most c so we can consider that $B_i\subset \Gamma /(t^{c})$ , $i=1,2$ . On the other hand, from Proposition 3.2, we have that $B_i^{\perp }\subset \Delta _{\le c-1}$ , $i=1,2$ .

If $B_1$ is isomorphic to $B_2$ by $\phi $ , then their normalizations are isomorphic:

$$ \begin{align*}\Gamma =\overline{B_1}\overset{\overline{\phi}}{\cong}\overline{B_2}=\Gamma. \end{align*} $$

This automorphism is determined by a power series $h(t)\in (t)$ such that $u\perp h\neq 0$ and

$$ \begin{align*}\begin{array}{ lclc} \overline{\phi}: & \Gamma &\longrightarrow & \Gamma \\ & f &\mapsto & f(h). \end{array} \end{align*} $$

Then we have an isomorphism of ${\mathbf {k}}$ -vector spaces

$$ \begin{align*}\frac{\Gamma}{B_1}\overset{\overline{\phi}}{\longrightarrow} \frac{\Gamma}{B_2} \end{align*} $$

and the perfect pairing induces a ${\mathbf {k}}$ -vector isomorphism

$$ \begin{align*}\phi^*:B_2^{\perp}\longrightarrow B_1^{\perp}. \end{align*} $$

The matrix $M_{\phi }$ associated with $\phi $ in the basis ${t^i}$ , $i=0,\dots , c-1$ , is the $c\times c$ matrix whose columns are the coefficients of $\phi ({t}^i)={h}^i$ , $i=0,\dots , c-1$ , with respect to this basis. Hence, the matrix of $\phi ^*:B_2^*=B_2^{\perp }\longrightarrow B_1^*=B_1^{\perp }$ with respect to the basis $\frac {1}{i!} u^i$ , $i=0,\dots , c-1$ , is the transpose matrix $^\tau M_{\phi }$ of $M_{\phi }$ .

Example 3.10 Let $B_2\subset \Gamma $ be a ${\mathbf {k}}$ -algebra generated by two elements $f_1$ , $f_2$ with $v_t(f_1)=2$ and $v_t(f_2)=7$ . We may assume that $f_1=t^2+$ monomials of higher degree. Then $B_2$ is of finite codimension $\delta =3$ and conductor $c=6$ .

Since $\Gamma $ is complete there exist a power series $h\in (t)$ such that $h^2=f_1$ ; we write $h=t+h_2 t^2+\dots +h_5 t^5+\dots $ . Notice that $\Gamma ={\mathbf {k}}[\![h]\!]$ .

Let ${\phi }$ the automorphism of $\Gamma $ defined by h, i.e., ${\phi }(f)=f(h)$ . Then $\phi ^{-1}(B_2)$ is a ${\mathbf {k}}$ -algebra $B_1$ generated by $f_1'=t^2$ and $f_2'(h)$ such that $v_h(f_2' )=7$ . After a change of generators $B_1$ is generated by $f_1'=t^2$ and $f_2'=t^7$ .

The induced isomorphism $\phi :B_1\longrightarrow B_2$ has the following $6 \times 6$ associated matrix with respect the basis ${t^i}$ , $i=0,\dots , 5$ ,

$$ \begin{align*}M_{\phi}=\left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & h_2 & 1 & 0 & 0 & 0 \\ 0 & h_3 & 2h_2 & 1 & 0 & 0 \\ 0 & h_4 & 2h_3+h_2^2 & 3h_2 & 1 & 0 \\ 0 & h_5 & 2b_4+2h_2h_3 & 3h_3+3h_2^2 & 4h_2 & 1 \\ \end{array} \right). \end{align*} $$

Then the matrix of the isomorphism $\phi ^*:B_2^{\perp }\longrightarrow B_1^{\perp }$ with respect to $\frac {1}{i!} u^i$ , $i=0,\dots , 5$ , is $M_{\phi }^\tau $ . Since $B_1$ is the monomial ${\mathbf {k}}$ -algebra ${\mathbf {k}}[\![t^2,t^7]\!]$ , the ${\mathbf {k}}$ -vector space $B_1^{\perp }$ is generated by $u, u^3, u^5$ . From this, we can compute $B_2^{\perp }$ by considering $(^\tau M_{\phi })^{-1}$ .

4 Algebra-forming vector spaces

The first goal of this section is to characterize the algebra-forming ${\mathbf {k}}$ -vector spaces.

Proposition 4.1 Let B be a ${\mathbf {k}}$ -sub-algebra of finite codimension of $\Gamma $ with conductor c, and let $f_1,\dots , f_s$ be a system of generators of B. Given an integer $d\ge c-1$ , let $h_1,\dots , h_m$ be a system of generators of $W(\{f_1,\dots , f_s\},d)$ .

Let V be a dimension $\delta $ ${\mathbf {k}}$ -vector subspace of $(u)\subset \Delta $ generated by polynomials of degree at most $d-1$ . Let $g_1,\dots ,g_{\delta }\in V$ be a basis of V.

Then V is algebra-forming with respect to B iff for all r-upla $(\lambda _1,\dots ,\lambda _m)\in {\mathbf {k}}^m$ such that

(3) $$ \begin{align} \sum_{j=1}^m \lambda_j (g_i\perp h_j)=0 \end{align} $$

for all $i=1,\dots ,\delta $ , then

(4) $$ \begin{align} \sum_{j=1}^m \lambda_j^2 (g_i\perp h_j^2)+ 2 \sum_{j=1,l=1,j\neq l}^m \lambda_j \lambda_j (g_i\perp h_j h_l)=0 \end{align} $$

for all $i=1,\dots ,\delta $ .

Proof From Proposition 3.2, we have to prove that for all $f\in B$ such that $g\perp f=0$ for all $g\in V$ we have that $g\perp f^2=0$ for all $g\in V$ . Since the polynomials of V are of degree at most $d-1$ we only have to prove that for all $f\in W=W(\{f_1,\dots , f_s\},d)$ such that $g\perp f=0$ for all $g\in V$ , we have that $g\perp f^2=0$ for all $g\in V$ .

A general element of W can be written as $f=\sum _{j=1}^m \lambda _j h_j$ . Hence the condition $g_i\perp f=0$ is equivalent to

$$ \begin{align*}\sum_{j=1}^m \lambda_j (g_i\perp h_j)=0 \end{align*} $$

for all $i=1,\dots ,\delta $ . Similarly, the condition $g_i\perp f^2=0$ is equivalent to

$$ \begin{align*}\sum_{j=1}^m \lambda_j^2 (g_i\perp h_j^2)+ 2 \sum_{j=1,l=1,j\neq l}^m \lambda_j \lambda_j (g_i\perp h_j h_l)=0 \end{align*} $$

for all $i=1,\dots ,\delta $ .

Definition 4.3 Let B be a sub- ${\mathbf {k}}$ -algebra of finite codimension $\delta $ of $\Gamma $ and conductor c. Let D be the semigroup of B; we write the set $t^{\mathbb N\setminus D_B}=\{t^i; i\in \mathbb N\setminus D_B \}$ as $g_1=t^{c-1},\dots , g_{\delta }=t$ . Then we define the so-called standard filtration of B as follows: $B_i$ is the ${\mathbf {k}}$ -algebra generated by B and $g_1,\dots , g_i$ for $i=1, \dots , \delta $ ; we set $B_0=B$ . Notice that $B_{\delta }=\Gamma $ and that we have

$$ \begin{align*}B=B_0\subset B_1\subset \dots \subset B_{\delta}=\Gamma \end{align*} $$

and $\operatorname {dim}_{{\mathbf {k}}}(B_{i+1}/B_i)=1$ , $i=0,\dots , \delta -1$ .

After the definition of standard filtration, we only have to consider algebra-forming elements $g\in \Delta $ , with respect a suitable sub- ${\mathbf {k}}$ -algebras of $\Gamma $ , in order to define a ${\mathbf {k}}$ -algebra recursively. The algebra-forming elements are not unique as the following example shows.

Example 4.4 Let us consider the Example 3.7. The standard filtration of B is

$$ \begin{align*}B={\mathbf{k}}[\![t^3+t^4,t^5]\!]\subset B_1={\mathbf{k}}[\![t^3+t^4,t^5,t^7]\!]\subset B_2={\mathbf{k}}[\![t^3,t^4,t^5]\!]\subset B_3={\mathbf{k}}[\![t^2,t^3]\!]\subset \Gamma. \end{align*} $$

The chain of ${\mathbf {k}}$ -algebras is defined as follows. The cosets of $t,t^2, t^4, t^7$ in $\Gamma /B$ form a basis of $\Gamma /B$ as ${\mathbf {k}}$ -vector space. Then $B_1$ is the ${\mathbf {k}}$ -algebra generated by B and $t^7$ , $B_2$ is the ${\mathbf {k}}$ -algebra generated by $B_1$ and $t^4$ , $B_3$ is the ${\mathbf {k}}$ -algebra generated by B and $t^2$ , and finally $\Gamma $ is the ${\mathbf {k}}$ -algebra generated by B and t.

We know that $B^{\perp }$ is a four-dimensional ${\mathbf {k}}$ -vector space generated by $u, u^2, u^3-\frac {1}{4}u^4, u^6-\frac {1}{2.7}u^7$ ; we have $B_3=\operatorname {Ann}\langle u\rangle $ , $B_2=\operatorname {Ann}\langle u^2\rangle \cap B_3$ , $B_1=\operatorname {Ann}\langle u^3-\frac {1}{4}u^4\rangle \cap B_2$ , $B=\operatorname {Ann}\langle u^6-\frac {1}{2.7}u^7\rangle \cap B_1$ . On the other hand, the ${\mathbf {k}}$ -algebra $C_1={\mathbf {k}}[\![t^3+t^5,t^4]\!]\subset B_1$ can be obtained as

$$ \begin{align*}C_1=\operatorname{Ann}\langle u^3-\frac{1}{4.5}u^5\rangle\cap B_2, \end{align*} $$

i.e., $u^3-\frac {1}{4.5}u^5$ is an algebra-forming element with respect to $B_2$ . Notice that $B_1$ and $C_1$ are non analytically isomorphic codimension one ${\mathbf {k}}$ -algebras of $B_2$ .

Next, we show how to build the standard filtration by using derivations.

Proof If we denote by $\max _B$ , the maximal ideal of B then $\max _C\subset \max _B$ , $\operatorname {dim}_{{\mathbf {k}}}(\max _B/\max _C)=1$ and $\max _B^2\subset \max _C$ . Since we have

$$ \begin{align*}\frac{\max_C}{\max_B^2} \subset \frac{\max_B}{\max_B^2}, \end{align*} $$

we deduce that there exists a linear form $\alpha : \frac {\max _B}{\max _B^2}\longrightarrow {\mathbf {k}}$ such that $\ker (\alpha )=\frac {\max _C}{\max _B^2}$ . From this, we get the claim.

Corollary 4.6 Let B be a sub- ${\mathbf {k}}$ -algebra of finite codimension $\delta $ of $\Gamma $ . Let us consider the standard filtration of B:

$$ \begin{align*}B=B_0\subset B_1\subset \dots \subset B_{\delta}=\Gamma. \end{align*} $$

For all $i=1,\dots ,\delta ,$ there exists a derivation $\partial _{l_i}\in Der_{{\mathbf {k}}}(B_i)$ , $l_i\in \max _{B_i}$ , such that $\ker (\partial _{l_i})=B_i$ .

5 Monomial algebras

In this section, we first compute the inverse system of a monomial ${\mathbf {k}}$ -algebra. After this, we characterize monomial Gorenstein curve singularities in terms of its inverse system. We end the section relating the inverse system of a curve singularity with its generic plane projection and its saturation.

The following result it is easy to deduce from the proof of the second part of Proposition 3.2(2).

The inverse system of a monomial Gorenstein ${\mathbf {k}}$ -algebra case can be handled. Let us recall the definition of symmetric semi-group and the celebrate result of Kunz.

Kunz proved that the ring ${\mathbf {k}}[\![D]\!]$ is Gorenstein ring if and only if D is a symmetric semigroup,[Reference Kunz21]. This symmetry is inherited by $B^{\perp }$ .

Given a finite codimension subalgebra B of $\Gamma $ , we consider the curve singularity $X=\operatorname {Spec}(B)$ defined by B. Let $X'$ be the generic plane projection of X, [Reference Briançon, Galligo and Granger3], and let $\widetilde {X}$ be the saturation of X, [Reference Zariski28] and the references therein. We have

$$ \begin{align*}{\mathcal O}_{X'}\subset {\mathcal O}_{X}=B\subset {\mathcal O}_{\widetilde{X}}\subset \Gamma, \end{align*} $$

and then

$$ \begin{align*}{\mathcal O}_{\widetilde{X}}^{\perp}\subset B^{\perp} \subset {\mathcal O}_{X'}^{\perp}. \end{align*} $$

We have, [Reference Elias9],

$$ \begin{align*}\delta(\widetilde{X}) \le \delta(X) \le \delta(X')\le (e_0(X)-1) \delta(\widetilde{X})- \binom{e_0(X)-1}{2}. \end{align*} $$

From [Reference Zariski27, Proposition 1.6, page 971], we know that $\widetilde {X}$ is also the saturation of $X'$ .

On the other hand, $\widetilde {X}$ is a monomial curve singularity. Assume that the coset of $x_1$ in B is $t^{e_0}$ with $e_0$ the multiplicity of B. Since the rings are complete and the ground field is algebraically closed, we can assumed it after a suitable election of the uniformization parameter of $\Gamma $ . Let $\{e_0; \beta _1,\dots , \beta _g\}$ be the characteristic of $X'$ , [Reference Zariski28, Section 3, page 993], then ${\mathcal O}_{\widetilde {X}}$ is the monomial subalgebra with generators:

$$ \begin{align*}\left\{ \begin{array}{ll} t^{e_0}, & \mbox{} \\ t^{s_\nu n_{\nu+1}\dots n_g}, & \mbox{} m_{\nu}\le s_{\nu}\le [m_{\nu+1}/n_{\nu+1}], \nu=1,\dots, g-1, \\ t^{m_g+i}, & \mbox{} 0\le i\le e_0-1, \end{array} \right. \end{align*} $$

where $\beta _{\nu }/e_0=m_{\nu }/n_1\dots n_{\nu }$ is the $\nu $ th characteristic exponent of $X'$ , $\nu =1,\dots , g-1$ , and $\gcd (m_i, n_i)=1$ for all $i=1,\dots ,g$ (see [Reference Zariski28, Section 3, page 995]).

The facts ${\mathcal O}_{\widetilde {X}}^{\perp }\subset B^{\perp }$ and Proposition 5.2 can be useful in order to simplify the computation of $B^{\perp }$ as the next example shows.

6 The canonical module

As in the Artin case, we can relate the canonical module with the inverse system. In that case, we have that if I is an Artinian ideal, then $I^{\perp }\cong E_{R/I}({\mathbf {k}})\cong \omega _{R/I}$ (see [Reference Bruns and Herzog4, Reference Elias, Cuong, Hoa and Trung14]). In the case of branches, we can determine the “negative” part of the canonical module.

Let X be a branch of $({\mathbf {k}}^n,0)$ and $\overline {X}$ its normalization. We first describe the canonical module $\omega _X$ by using Rosenlicht’s regular differential forms (see [Reference Serre26, Chapter IV 9], [Reference Buchweitz and Greuel5, Section 1], see also [Reference Elias13]). We denote by $\Omega _{\overline {X}}(p),$ the set of meromorphic forms in $\overline {X}$ with a pole at most in $p=\nu ^{-1}(0)$ . Then Rosenlicht’s differential forms are defined as follows: $\omega ^R_{X}$ is the set of $\nu _*(\alpha )$ , $\alpha \in \Omega _{\overline {X}}(p)$ , such that for all $F\in {\mathcal O}_{X},$

$$ \begin{align*}\mathrm{res}_{p}(F \alpha)=0. \end{align*} $$

Notice that we have a mapping that we also denote by

$$ \begin{align*}d_R : {\mathcal O}_{X} \longrightarrow \Omega_{X} \longrightarrow \nu_* \Omega_{\overline{X}} \hookrightarrow \omega^R_{X}. \end{align*} $$

In [Reference Altman and Kleiman1, Chapter VIII], it is proved that $\omega _{X} \stackrel {\phi }{\cong } \omega _{X}^R$ and $d_R=\phi d$ , where $d: {\mathcal O}_{X} \longrightarrow \omega _{X}$ is the map defined in the Section 1. Since ${\mathcal O}_{X}$ is a one-dimensional reduced ring, we know that $\omega _{(X,0)}$ is a sub- ${\mathcal O}_{X}$ -module of $\mathrm {tot}({\mathcal O}_{X})$ (see [Reference Bruns and Herzog4, Proposition 3.3.18]). There is a perfect pairing, [Reference Serre26, Chapter IV],

$$ \begin{align*}\begin{array}{ccccc} \frac{\nu_* {\mathcal O}_{\overline{X}}}{{\mathcal O}_{X}} & \times& \frac{\omega_{(X,0)}}{\nu_* \Omega_{\overline{X}}} & \stackrel{\eta}{\longrightarrow} & \mathbb C \\ F & \times & \alpha & \longrightarrow & {\mathrm{res}}_{p}(F \alpha) \end{array} \end{align*} $$

notice that for all $\lambda \in R$ it holds $ \eta (\lambda F, \alpha )= {\mathrm {res}}_{p}(\lambda F \alpha )=\eta ( F, \lambda \alpha ). $

Proposition 6.1 Let X be a branch of $({\mathbf {k}} ^n,0)$ and $\overline {X}$ its normalization. Then we have an isomorphism of the $\delta (X)$ dimensional ${\mathbf {k}}$ -vector spaces:

$$ \begin{align*}B^{\perp}\overset{\epsilon}{\cong} \frac{\omega_{X}}{\nu_* \Omega_{\overline{X}}} \end{align*} $$

such that $\epsilon (g)$ is the coset defined by $\alpha = \sum _{i=0}^{c-1} i! c_{i} t^{-i-1}$ , for all $g=\sum _{i=0}^{c-1} c_i u^i\in B^{\perp }$ .

Proof We write $B={\mathcal O}_{X}$ , $\Gamma =\nu _* {\mathcal O}_{\overline {X}}$ , and $\Omega _{\overline {X}}= \Gamma dt$ . Then $\epsilon $ is the composition of the isomorphisms induced by the above two perfect pairings

$$ \begin{align*}B^{\perp} \overset{\epsilon_1}{\cong} \left(\frac{\Gamma}{B}\right)^* \overset{\epsilon_2}{\cong} \frac{\omega_{X}}{\nu_* \Omega_{\overline{X}}}. \end{align*} $$

Next, we describe both morphisms $\epsilon _1, \epsilon _2$ . Given $g\in B^{\perp }$ , we can write it as

$$ \begin{align*}g= c_0+ c_1 u+\dots, c_{c-1} u^{c-1}, \end{align*} $$

so $\epsilon _1(g)$ is the linear form induced by $\xi :\Gamma ^*\longrightarrow {\mathbf {k}}$ defined by: if $f=\sum _{i\ge 0} a_i t^i\in \Gamma ,$ then

$$ \begin{align*}\xi(f)=\sum_{i=0}^{c-1} i! a_i c_i. \end{align*} $$

On the other hand, every $\alpha \in \omega _X$ can be written as $\alpha =t^n h(t) dt$ with $n\in \mathbb Z$ and $h(t)\in \Gamma $ an invertible series. From [Reference Elias13, Proposition 2.6], we get that $\alpha = \sum _{i\ge -c} e_i t^i$ such that $\mathrm {{res}}_0(\alpha F)=0$ for all $f\in B$ . Given $f=\sum _{i\ge 0} a_i t^i\in \Gamma $ , we have

$$ \begin{align*}\mathrm{{res}}_0(f \alpha)= \sum_{i=0}^{c-1} a_i e_{-i-1} \end{align*} $$

so $\epsilon _2^{-1}(\alpha )$ is the linear form induced by $\xi ':\Gamma ^*\longrightarrow {\mathbf {k}}$ defined by

$$ \begin{align*}\xi'(f)=\sum_{i=0}^{c-1} a_i e_{-i-1}. \end{align*} $$

From this, we deduce that $e_{-i-1}=i! c_i$ for $i=0,\dots , c-1$ .