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...
Value semigroups of non-irreducible singular algebraic curves and their fractional ideals are submonoids of [Formula: see text] that are closed under infimums, have a conductor and fulfill a special compatibility property on their elements. Monoids of [Formula: see text] fulfilling these three conditions are known in the literature as good semigroups and there are examples of good semigroups that are not realizable as the value semigroup of an algebraic curve. In this paper, we consider good semigroups independently from their algebraic counterpart, in a purely combinatorial setting. We define the concept of good system of generators, and we show that minimal good systems of generators are unique. Moreover, we give a constructive way to compute the canonical ideal and the Arf closure of a good subsemigroup when [Formula: see text].
Macaulay's inverse system is an effective method to construct Artinian K-algebras with additional properties like, Gorenstein, level, more generally with any socle type. Recently, Elias and Rossi [ER17] gave the structure of the inverse system of d-dimensional Gorenstein K-algebras for any d > 0. In this paper we extend their result by establishing a one-to-one correspondence between d-dimensional level K-algebras and certain submodules of the divided power ring. We give several examples to illustrate our result.
Kyoji Saito's notion of a free divisor was generalized by the first author to reduced Gorenstein spaces and by Delphine Pol to reduced Cohen-Macaulay spaces. Starting point is the Aleksandrov-Terao theorem: A hypersurface is free if and only if its Jacobian ideal is maximal Cohen-Macaulay. Pol obtains a generalized Jacobian ideal as a cokernel by dualizing Aleksandrov's multi-logarithmic residue sequence. Notably it is essentially a suitably chosen complete intersection ideal that is used for dualizing. Pol shows that this generalized Jacobian ideal is maximal Cohen-Macaulay if and only if the module of Aleksandrov's multi-logarithmic differential k-forms has (minimal) projective dimension k − 1, where k is the codimension in a smooth ambient space. This equivalent characterization reduces to Saito's definition of freeness in case k = 1. In this article we translate Pol's duality result in terms of general commutative algebra. It yields a more conceptual proof of Pol's result and a generalization involving higher multi-logarithmic forms and generalized Jacobian modules.(1.7)]), unitary reflection arrangements and their discriminants (see [Ter83, Thm. C]) and discriminants of versal deformations of isolated complete intersection singularities and space curves (see [Loo84, (6.13)] and [vS95]). Free divisors also occur as discriminants in prehomogeneous vector spaces (see [GMS11]). In case of hyperplane arrangements the study of freeness attracted a lot of attention (see [Yos14]).Let D be a germ of reduced hypersurface in Y ∼ = (C n , 0) defined by h ∈ O Y . The O Y -modules Ω q (log D) of logarithmic differential q-forms along D and the O D -modules ω p D of regular meromorphic differential p-forms on D fit into a short exact logarithmic residue sequence (see [Sai80, §2] and [Ale88, §4])In particular I is a regular ideal of R and hence any R-module is R-torsion.We assume further that R admits a canonical module ω R . Then also R admits a canonical module ω R (see [BH93, Thm. 3.3.7]). Notation 2.1. Abbreviating ω R := R ⊗ R ω R we deal with the following functors − * := Hom R (−, R), − ∨ := Hom R (−, ω R ), − I := Hom R (−, Iω R ), − ∨ := Hom R (−, ω R ). In general ω R ∼ = ω R and − ∨ is not the duality of R-modules. For an R-module N, N * = Hom R (N, R) but N ∨ means either Hom R (N, ω R ) or Hom R (N, ω R ), depending on the context. For R-modules M and N, we denote the canonical evaluation map by δ M,N : M → Hom R (Hom R (M, N), N), m → (ϕ → ϕ(m)).4
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