1 Introduction and Motivation

Let R be a complex algebroid curve with s branches and normalization \(R\rightarrow \overline{R}\cong \mathbb {C}[[t_1]]\times \cdots \times \mathbb {C}[[t_s]]\). Then, there is a multivaluation map

$$ \nu =(\nu _1,\dots ,\nu _s):\overline{R}\rightarrow (\mathbb {Z}\cup \{\infty \})^s,\quad x\mapsto ({{\mathrm{ord}}}_{t_1}(x),\dots ,{{\mathrm{ord}}}_{t_s}(x)) $$

which associates to R its value semigroup \(\Gamma _R=\nu (\{x\in R\mid x\,\text {non zero-divisor}\})\)

The value semigroup of a curve singularity is an important combinatorial invariant with a long history. It determines the topological type of plane curves. In case R is an irreducible curve Kunz [1] showed that R is Gorenstein if and only if its value semigroup \(\Gamma _R\) is symmetric.

Example 1

Consider the plane algebroid curve \(R=\mathbb {C}[[x,y]]/{\left\langle x^7-y^4\right\rangle }\cong \mathbb {C}[[t^4,t^7]]\). Then R is Gorenstein and \(\Gamma _R=\langle 4,7\rangle \) is symmetric.

figure a

Later Delgado [2] introduced a notion of symmetry in the reducible case, and extended Kunz’s result. D’Anna used Delgado’s symmetry to define a canonical semigroup ideal. Based on this definition, he characterized canonical ideals of R in terms of their value semigroup ideals. More recently, Pol [3] proved a formula for the value semigroup of the dual of a fractional ideal. Our aim is to generalize both the duality results by D’Anna and by Pol.

2 Good Value Semigroups

Including complex algebroid curves as a special case we consider admissible rings in the following sense: let R be a one-dimensional semilocal Cohen–Macaulay ring that is analytically reduced, residually rational and has large residue fields (i.e. \(|R/{\mathfrak {m}}|\ge |\{\text {branches of}{\widehat{R_{\mathfrak {m}}}}\}|\) for any \(\mathfrak {m}\) maximal ideal of R). Value semigroup (ideals) are then defined as follows.

Definition 2

Let R be an admissible ring, and let \(\mathfrak {V}_R\) be the set of (discrete) valuation rings of \(Q_R\) over R with corresponding valuations \(\nu =(\nu _V)_{V\in \mathfrak {V}_R}:Q_R \rightarrow (\mathbb {Z}\cup \{\infty \})^{\mathfrak {V}_R}\). To each regular fractional ideal \(\mathcal {E}\) of R we associate its value semigroup ideal \(\Gamma _\mathcal {E}:= \nu ( \mathcal {E}^\mathrm {reg}) \subset \mathbb {Z}^{\mathfrak {V}_R}\). If \(\mathcal {E}= R\), then the monoid \(\Gamma _R\) is called the value semigroup of R.

If \(\mathcal {E}\) is a regular fractional ideal of R, then \(\Gamma _\mathcal {E}\) is a semigroup satisfying particular properties, that we consider for any subset \(E \subset \mathbb {Z}^s\):

  1. (E0)

    there exists an \(\alpha \in \mathbb {Z}^s\) such that \(\alpha +\mathbb {N}^s\subset E\);

  2. (E1)

    for any \(\alpha ,\beta \in E\), their component-wise minimum \(\min \{\alpha ,\beta \}\in E\);

  3. (E2)

    for any \(\alpha ,\beta \in E\) with \(\alpha _j=\beta _j\) for some j there exists an \(\epsilon \in E\) such that \(\epsilon _j>\alpha _j=\beta _j\) and \(\epsilon _i \ge \min \{\alpha _i,\beta _i\}\) with equality if \(\alpha _i\not =\beta _i\).

Definition 3

A submonoid S of \(\mathbb {N}^s\) with group of differences \(D_S=\mathbb {Z}^s\) is called a good semigroup if properties (E0), (E1), and (E2) hold for \(E=S\).

A semigroup ideal of S is subset \(E \subset \mathbb {Z}^s\) such that \(E + S \subset E\) and \(\alpha +E\subset S\) for some \(\alpha \in \mathbb {Z}^s\). It is called a good semigroup ideal of the good semigroup S if it satisfies (E1) and (E2).

Proposition 4

Let R be an admissible ring. Then,

  1. (i)

    the value semigroup \(\Gamma _R\) is a good semigroup;

  2. (ii)

    for any regular fractional ideal \(\mathcal {E}\) of R, \(\Gamma _\mathcal {E}\) is a good semigroup ideal of \(\Gamma _R\).      \(\square \)

On value semigroup ideals there is a distance function that mirrors the relative length of fractional ideals.

Definition 5

Let S be a good semigroup, and let \(E\subset D_S\) be a subset. Then \(\alpha ,\beta \in E\) with \(\alpha <\beta \) are called consecutive in E if \(\alpha<\delta <\beta \) implies \(\delta \not \in E\) for any \(\delta \in D_S\). For \(\alpha ,\beta \in E\), a chain of points \(\alpha ^{(i)}\in E\),

$$\begin{aligned} \alpha =\alpha ^{(0)}<\cdots <\alpha ^{(n)}=\beta , \end{aligned}$$
(1)

is said to be saturated of length n if \(\alpha ^{(i)}\) and \(\alpha ^{(i+1)}\) are consecutive in E for all \(i=0,\ldots , n-1\). If E satisfies

  1. (E4)

    for fixed \(\alpha ,\beta \in E\), any two saturated chains (1) in E have the same length n;

then we call \(d_E(\alpha ,\beta ):=n\) the distance of \(\alpha \) and \(\beta \) in E.

D’Anna [4, Prop. 2.3] proved that any good semigroup ideal E satisfies property (E4).

Definition 6

For a good semigroup ideal E, the conductor of E is defined as \({\gamma ^E} := \min \{ \alpha \in E \mid \alpha + \mathbb {N}^s \subset E \}\). We denote \(\gamma :=\gamma ^S\) and \(\tau :=\gamma -\mathbf 1 \).

Definition 7

Let S be a good semigroup, and let \(E \subset F\) be two semigroup ideals of S satisfying property (E4). Then we call

$$ d(F\backslash E):=d_F(\mu ^F,\gamma ^E)-d_E(\mu ^E,\gamma ^E) $$

the distance between E and F.

In the following, we collect the main properties of the distance function \(d(-\backslash -)\). It follows from the definition that it is additive, as proven by D’Anna in [4, Prop. 2.7]:

Lemma 8

Let \(E\subset F\subset G\) be semigroup ideals of a good semigroup S satisfying properties (E1) and  (E4). Then \(d(G\backslash E)=d(G\backslash F)+d(F\backslash E)\).    \(\square \)

Moreover, the distance function detects equality as formulated in [4, Prop. 2.8] and proved in [5, Prop. 4.2.6].

Proposition 9

Let S be a good semigroup, and let E, F be good semigroup ideals of S with \(E\subset F\). Then \(E=F\) if and only if \(d(F\backslash E)=0\).    \(\square \)

The length of a quotient of fractional ideals corresponds to the distance between the corresponding good semigroup ideals; see [4, Prop. 2.2] and [5, Prop. 4.2.7].

Proposition 10

Let R be an admissible ring. If \(\mathcal {E}, \mathcal {F}\) are two regular fractional ideals of R such that \(\mathcal {E}\subset \mathcal {F}\) then, \(\ell _R(\mathcal {F}/\mathcal {E})=d(\Gamma _\mathcal {F}\backslash \Gamma _\mathcal {E})\).    \(\square \)

As a corollary, one can check equality of fractional ideals through their value semigroups:

Corollary 11

Let R be an admissible ring, and let \(\mathcal {E},\mathcal {F}\) be two regular fractional ideals of R such that \(\mathcal {E}\subset \mathcal {F}\). Then \(\mathcal {E}= \mathcal {F}\) if and only if \(\Gamma _\mathcal {E}= \Gamma _\mathcal {F}\).    \(\square \)

3 Canonical Ideals and Main Results

The following is the canonical semigroup ideal as defined by D’Anna in [4].

Definition 12

We call the semigroup ideal

$$ {K^0_S}:=\left\{ \alpha \in \mathbb {Z}^s \mid \Delta ^S ( \tau -\alpha )=\emptyset \right\} . $$

the normalized canonical semigroup ideal of S, where

$$ \Delta ^S (\delta ):=\Delta (\delta )\cap S=(\cup _{i \in I}\{\beta \in \mathbb {Z}^s \mid \delta i=\beta _i,\ \delta _j < \beta _j\ \forall \ j\ne i \})\cap S $$

Definition 13

Let S be a good semigroup. Then S is called symmetric if \(S=K^0_S\).

As mentioned in the introduction, Delgado proved that \(S=\Gamma _R\) is symmetric if and only if R is Gorenstein. D’Anna [4] generalized this result: a regular fractional ideal \(\mathcal {K}\) with \(R\subset \mathcal {K}\subset \overline{R}\) is canonical if and only if \(\Gamma _\mathcal {K}=K^0_S\). Recall that by definition a fractional ideal \(\mathcal {K}\) is canonical if \(\mathcal {K}:(\mathcal {K}:\mathcal {E})=\mathcal {E}\) for any regular fractional ideal \(\mathcal {E}\).

Definition 14

Let K be a good semigroup ideal of a good semigroup S. We call K a canonical semigroup ideal of S if \(K \subset E\) implies \(K=E\) for any good semigroup ideal E with \(\gamma ^K=\gamma ^E\).

In analogy with this definition, we give a characterization of canonical semigroup ideals; see [5, Thm 5.2.7].

Theorem 15

For a good semigroup ideal K of a good semigroup S the following are equivalent:

  1. (a)

    K is a canonical semigroup ideal;

  2. (b)

    there exists an \(\alpha \) such that \(\alpha +K = K^0_S\);

  3. (c)

    for all good semigroup ideals E one has \(K-(K-E)=E\).

Moreover, if K satisfies these equivalent conditions, then \(K-E\) is a good semigroup ideal for any good semigroup ideal E.    \(\square \)

Given this characterization, it is natural to ask if taking the dual commutes with taking the semigroup. In the Gorenstein case, Pol [3] gave a positive answer.

Theorem 16

If R is a Gorenstein admissible ring then,

$$ \Gamma _{R:\mathcal {E}}=\left\{ \alpha \in \mathbb {Z}^s \mid \Delta ^E ( \tau -\alpha )=\emptyset \right\} =\Gamma _R-\Gamma _\mathcal {E}$$

for any regular fractional ideal \(\mathcal {E}\) of R.

Our main result extends Pols result beyond the Gorenstein case.

Theorem 17

Let \(\mathcal {K}\) be a canonical ideal of R and let \(K:=\Gamma _\mathcal {K}\). Then, the following diagram commutes:

figure b

Proof

It is not restrictive to assume R local and \(R\subset \mathcal {K}\subset \overline{R}\). Hence \(K:=\Gamma _\mathcal {K}=K^0_S\) by D’Anna [4].

Let \(\mathcal {E}\subset \mathcal {F}\) be regular fractional ideals of R. Proposition 10 then yields

$$ d(\Gamma _{\mathcal {K}:\mathcal {E}}\backslash \Gamma _{\mathcal {K}:\mathcal {F}})=\ell _R((\mathcal {K}:\mathcal {E})/(\mathcal {K}:\mathcal {F}))=\ell _R(\mathcal {F}/\mathcal {E})=d(\Gamma _\mathcal {F}\backslash \Gamma _\mathcal {E})=:n. $$

Notice that \(\ell _R((\mathcal {K}:\mathcal {E})/(\mathcal {K}:\mathcal {F}))=\ell _R(\mathcal {F}/\mathcal {E})\) as K is canonical. There is a composition series of regular fractional ideals

$$ \mathcal {C}_\mathcal {E}=\mathcal {E}_0\subsetneq \mathcal {E}_1\subsetneq \cdots \subsetneq \mathcal {E}_l=\mathcal {E}\subsetneq \mathcal {E}_{l+1}\subsetneq \cdots \subsetneq \mathcal {E}_{l+n}=\mathcal {F}, $$

where \(\mathcal {C}_\mathcal {E}\) is the conductor of \(\mathcal {E}\). By Corollary 11, applying \(\Gamma \) yields a chain of good semigroup ideals of \(\Gamma _R\)

$$ C_{\Gamma _\mathcal {E}}=\Gamma _{\mathcal {E}_0}\subsetneq \Gamma _{\mathcal {E}_1}\subsetneq \cdots \subsetneq \Gamma _{\mathcal {E}_l}=\Gamma _\mathcal {E}\subsetneq \Gamma _{\mathcal {E}_{l+1}}\subsetneq \cdots \subsetneq \Gamma _{\mathcal {E}_{l+n}}=\Gamma _\mathcal {F}. $$

By Corollary 11 and Theorem 15(c), dualizing with K yields a chain of good semigroup ideals of \(\Gamma _R\)

$$\begin{aligned} \Gamma _{\mathcal {K}:\mathcal {C}_\mathcal {E}}=\Gamma _\mathcal {K}-\Gamma _{\mathcal {C}_\mathcal {E}}=K-C_{\Gamma _\mathcal {E}}=K-C_{\Gamma _{\mathcal {E}_0}}\supsetneq \cdots \supsetneq K-\Gamma _{\mathcal {E}_l}=K-\Gamma _\mathcal {E}\nonumber \\ \supsetneq K-\Gamma _{\mathcal {E}_{l+1}}\supsetneq \cdots \supsetneq K-\Gamma _{\mathcal {E}_{l+n}}=K-\Gamma _\mathcal {F}\supset \Gamma _{\mathcal {K}:\mathcal {F}}. \end{aligned}$$
(2)

By Theorem 15, \(K-\Gamma _{\mathcal {E}_i}\) is a good semigroup ideal of S for all \(i=0,\ldots ,l+n\). Hence, using Proposition 9, we obtain \(d(K-\Gamma _{\mathcal {E}_i}\backslash K-\Gamma _{\mathcal {E}_{i+1}})\ge 1\) for all \(i=0,\ldots ,l+n-1\). On the other hand, by Proposition 10,

$$ d(\Gamma _{\mathcal {K}:\mathcal {C}_\mathcal {E}}\setminus \Gamma _{\mathcal {K}:\mathcal {F}})=\ell _R(\mathcal {K}:\mathcal {C}_\mathcal {E}/\mathcal {K}:\mathcal {F})=\ell _R(\mathcal {F}/\mathcal {C}_\mathcal {E})=l+n. $$

By Lemma 8 and (2), it follows that \(d(K-\Gamma _{\mathcal {E}_i}\backslash K-\Gamma _{\mathcal {E}_{i+1}})=1\) for all \(i=0,\ldots ,l+n-1\) and \(d(K-\Gamma _\mathcal {F}\backslash \Gamma _{\mathcal {K}:\mathcal {F}})=0\). By Proposition 9 the latter is equivalent to the second claim.

In particular, this implies the following

Corollary 18

Let \(\mathcal {K}\) be a fractional ideal of an admissible ring R. Then \(\mathcal {K}\) is canonical if and only if \(K:=\Gamma _\mathcal {K}\) canonical.