Duality on value semigroups

Abstract

We establish a combinatorial counterpart of the Cohen-Macaulay duality on a class of curve singularities which includes algebroid curves. For such singularities the value semigroup and the value semigroup ideals of all fractional ideals satisfy axioms that define so-called good semigroups and good semigroup ideals. We prove that each good semigroup admits a canonical good semigroup ideal which gives rise to a duality on good semigroup ideals. We show that the Cohen-Macaulay duality and our good semigroup duality are compatible under taking values.2010 Mathematics Subject Classification. Primary 14H20; Secondary 13C14, 20M12. Key words and phrases. curve singularity, value semigroup, canonical module, duality. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007(FP7/ -2013 under REA grant agreement n o PCIG12-GA-2012-334355. 1 (suitably normalized) canonical ideals K of R are characterized by having value semigroup ideal Γ K = K 0 .Waldi [Wal72] was the first to describe a symmetry property of the value semigroup for plane algebroid curves with two branches. Delgado [DdlM87, DdlM88] then made the step to general algebroid curves proving an analog of Kunz's result. Later Campillo, Delgado, and Kiyek [CDK94] relaxed the hypotheses to include analytically reduced and residually rational local rings R with infinite residue field. D'Anna [D'A97] extended Jäger's approach under the preceding hypotheses. He turned Delgado's symmetry definition into an explicit formula for a semigroup ideal K 0 (see Definition 5.2.1) such that any (suitably normalized) fractional ideal K of R is canonical if and only if Γ K = K 0 . In the process he studied axioms satisfied by value semigroup ideals which lead to the notion of a good semigroup ideal (see Definition 4.1.1).Barucci, D'Anna, and Fröberg [BDF00] studied some more special classes of rings such as almost Gorenstein rings, Arf rings, and rings of small multiplicity in relation with their value semigroups. Their setup includes the case of semilocal rings. Notably they found an example of a good semigroup which is not the value semigroup of any ring.Recently Pol [Pol16, Thm. 5.2.1] gave an explicit formula for the value semigroup ideal of the dual of a fractional ideal for Gorenstein algebroid curves.In this paper, we extend and unify D'Anna's and Pol's results for a general class of rings R that we call admissible (see Definition 3.1.5). We show that any good semigroup admits a canonical semigroup ideal K that is defined by a simple maximality property (see Definition 5.2.3). Equivalently, such a K induces a duality E → K − E on good semigroup ideals (see Theorem 5.2.6). This means thatfor all good semigroup ideals. It turns out that our canonical semigroup ideals are exactly the translations of D'Anna's K 0 . In particular, D'Anna's characterization of canonical ideals in terms of their value semigroup ideals persists for admissible rings (see Theorem 5.3.2). For a...

On the Colength of Fractional Ideals

“…The central results in this section will be several generalizations of results in [5], [2], [10], [11] and [8], which establish some symmetry among E(J : I) and E(I) mediated by E(J ) and among their maximal points.…”

“…, v r (h)) (cf. [8,Definition 3.2]). We will consider on Z r the natural partial order ≤ induced by the order of Z.…”

“…In our context, canonical ideals exist (cf. [3] or [8]); two canonical ideals differ up to a multiplication by a unit in Q and a multiple by such a unit of a canonical ideal is also a canonical ideal. The ring R will be called Gorenstein if R itself is a canonical ideal.…”

“…which contradicts the fact that q(E(J : I), f(J ) − α) = r − n + 1. Therefore, we must have p(E(I), α) = n. Now, assume that we have equality in (8). Let I be a fractional ideal of R and let α ∈ Z r .…”

“…Also, recently, Pol, in the work [10], extended Delgado's result by showing that the Gorensteiness of the ring of a singularity is equivalent to some symmetry relation among sets of values of any dual pair of regular fractional ideal, duality taken with respect to the ring itself, and also showed that in this case one has a pairing between absolute and relative maximal points of the value set of an ideal and that of its dual. In the work [8] (see also [11]), the authors show that this symmetry relation among value sets of dual pairs of ideals holds without any extra assumption on the ring, if one takes duality with respect to a canonical ideal.…”