Canonical Module Research Articles

$C$-CANONICAL MODULES

Let $C$ be a semidualizing module over a commutative Noetherian local ring $R$. In this paper we introduce a new class of modules, namely $C$-canonical modules which are a generalization of canonical modules. It is shown that if the canonical module exists then the $C$-canonical module exists and the converse holds under special conditions. Also, a new characterization of Gorenstein local rings is given via $C$-canonical modules.

Seminormality, canonical modules, and regularity of cut polytopes

Motivated by a conjecture of Sturmfels and Sullivant we study normal cut polytopes . After a brief survey of known results for normal cut polytopes it is in particular observed that for simplicial and simple cut polytopes their cut algebras are normal and hence Cohen–Macaulay. Moreover, seminormality is considered. It is shown that the cut algebra of K 5 is not seminormal which implies again the known fact that it is not normal. For normal Gorenstein cut algebras and other cases of interest we determine their canonical modules. The Castelnuovo–Mumford regularity of a cut algebra is computed for various types of graphs and bounds for it are provided if normality is assumed. As an application we classify all graphs for which the cut algebra has regularity less than or equal to 4.

Criteria for prescribed bound on projective dimension

We establish a prescribed upper bound for the projective dimension of a finitely generated module over a Cohen-Macaulay local ring (with canonical module) satisfying certain cohomological conditions. The central hypothesis is the vanishing of finitely many suitable Ext modules. We then derive corollaries which provide freeness criteria and a characterization of regular local rings. Finally, we raise two main questions related to our results; one generalizes a problem due to D. Jorgensen, and the other retrieves the long-standing Auslander-Reiten conjecture in the Gorenstein case.

Ulrich elements in normal simplicial affine semigroups

Let $H\subseteq \mathbb{N}^d$ be a normal affine semigroup, $R=K[H]$ its semigroup ring over the field $K$ and $\omega_R$ its canonical module. The Ulrich elements for $H$ are those $h$ in $H$ such that for the multiplication map by $\mathbf{x}^h$ from $R$ into $\omega_R$, the cokernel is an Ulrich module. We say that the ring $R$ is almost Gorenstein if Ulrich elements exist in $H$. For the class of slim semigroups that we introduce, we provide an algebraic criterion for testing the Ulrich propery. When $d=2$, all normal affine semigroups are slim. Here we have a simpler combinatorial description of the Ulrich property. We improve this result for testing the elements in $H$ which are closest to zero. In particular, we give a simple arithmetic criterion for when is $(1,1)$ an Ulrich element in $H$.

Trace ideals of canonical modules, annihilators of Ext modules, and classes of rings close to being Gorenstein

In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of near Gorensteinness, introduced by Herzog, Hibi and Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer is given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and show that they are almost Gorenstein rings in dimension one, and are closely related to Teter rings in the artinian case. We explain their basic properties and give some examples.

Local rings with self-dual maximal ideal

Let R be a Cohen–Macaulay local ring possessing a canonical module. In this paper, we consider when the maximal ideal of R is self-dual—i.e., it is isomorphic to its canonical dual as an R -module. Local rings satisfying this condition are called Teter rings, studied by Teter, Huneke–Vraciu, Ananthnarayan–Avramov–Moore, and others. In the one-dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness.

Reducing invariants and total reflexivity

Motivated by a recent result of Yoshino and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over local rings. Our main result considers modules which have finite reducing Gorenstein dimension and determines a criterion for such modules to be totally reflexive in terms of the vanishing of Ext. Along the way, we give examples and applications, and in particular, prove that a Cohen–Macaulay local ring with canonical module is Gorenstein if and only if the canonical module has finite reducing Gorenstein dimension.

Cotilting sheaves on Noetherian schemes

We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks.

Singular equivalences of functor categories via Auslander-Buchweitz approximations

The aim of this paper is to construct singular equivalences between functor categories. As a special case, we show that there exists a singular equivalence arising from a cotilting module T, namely, the singularity category of (T⊥)/[T] and that of (modΛ)/[T] are triangle equivalent. In particular, the canonical module ω over a commutative Noetherian ring R induces a singular equivalence between (CMR)/[ω] and (modR)/[ω], which generalizes Matsui-Takahashi's theorem. Our result is based on a sufficient condition for an additive category A and its subcategory X so that the canonical inclusion X↪A induces a singular equivalence Dsg(A)≃Dsg(X), which is a functor category version of Xiao-Wu Chen's theorem.

Functional and Structural Analyses of the Split-Dehydratase Domain in the Biosynthesis of Macrolactam Polyketide Cremimycin

In the biosynthesis of the macrolactam antibiotic cremimycin, the 3-aminononanoic acid starter unit is formed via a non-2-enoyl acyl carrier protein thioester intermediate, which is presumed to be constructed by cis-acyltransferase (AT) polyketide synthases (PKSs) CmiP2, CmiP3, and CmiP4. While canonical cis-AT PKS modules are comprised of a single polypeptide, the PKS module formed by CmiP2 and CmiP3 is split within the dehydratase (DH) domain. Here, we report the enzymatic function and the structural features of this split-DH domain. In vitro analysis showed that the split-DH domain catalyzes the dehydration reaction of (R)-3-hydroxynonanoyl N-acetylcysteamine thioester (SNAC) to form (E)-non-2-enoyl-SNAC, suggesting that the split-DH domain is catalytically active in cremimycin biosynthesis. In addition, structural analysis revealed that the CmiP2 and CmiP3 subunits of the split-DH domain form a tightly associated heterodimer through several hydrogen bonding and hydrophobic interactions, which are similar to those of canonical DH domains of other cis-AT PKSs. These results indicate that the split-DH domain has the same function and structure as common cis-AT PKS DH domains.

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.

The trace of the canonical module

The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen--Macaulay $K$-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.

Cohen–Macaulayness and canonical module of residual intersections

We show the Cohen–Macaulayness and describe the canonical module of residual intersections J = a : R I J={\mathfrak {a}}\colon _R I in a Cohen–Macaulay local ring R R , under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second author, a family of complexes that contains important information on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo–Mumford regularity and the type. Finally, whenever I I is strongly Cohen–Macaulay, we show duality results for residual intersections that are closely connected to results by Eisenbud and Ulrich. It establishes some tight relations between the Hilbert series of some symmetric powers of I / a I/{\mathfrak {a}} . We also provide closed formulas for the types and for the Bass numbers of some symmetric powers of I / a . I/{\mathfrak {a}}.

Relative Canonical Modules and Some Duality Results

Let (R, m) be a relative Cohen–Macaulay local ring with respect to an ideal a of R and set c to be ht a. We investigate some properties of the Matlis dual of the R-module [Formula: see text], and we show that such modules behave like canonical modules over Cohen–Macaulay local rings. Moreover, we provide some duality and equivalence results with respect to the module [Formula: see text], and these results lead us to achieve generalizations of some known results, such as the local duality theorem, which have been provided over a Cohen–Macaulay local ring admiting a canonical R-module.

Some criteria for regular and Gorenstein local rings via syzygy modules

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

On delta invariants and indices of ideals

Let $R$ be a Cohen–Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of $R$. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invariants. As an application of this result, we give upper bounds for orders of Ulrich ideals of $R$ when $R$ has Gorenstein punctured spectrum. Secondly, we extend the definition of indices to the ideal case, and generalize the result of Avramov–Buchweitz–Iyengar–Miller on the relationship between the index and regularity.

Resolutions of co-letterplace ideals and generalizations of Bier spheres

We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicial spheres comprehensively generalizing Bier spheres.

Non Cohen–Macaulay locus of canonical modules

Let (R,m) be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let M be a finitely generated R-module and KM the canonical module of M. Denote by nCM(M) the non Cohen–Macaulay locus of M. In this paper, the relationship between the two loci nCM(KM) and nCM(M) is given. The structure of M is clarified in case where dim⁡nCM(KM)≤0 (i.e. KM is generalized Cohen–Macaulay).

Tangent cones of monomial curves obtained by numerical duplication

Given a numerical semigroup ring $$R=k\llbracket S\rrbracket $$ , an ideal E of S and an odd element $$b \in S$$ , the numerical duplication $$S \bowtie ^b E$$ is a numerical semigroup, whose associated ring $$k\llbracket S \bowtie ^b E\rrbracket $$ shares many properties with the Nagata’s idealization and the amalgamated duplication of R along the monomial ideal $$I=(t^e \mid e\in E)$$ . In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen–Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when $$\mathrm{gr}_{\mathfrak {m}}(I)$$ is Cohen–Macaulay and when $$\mathrm{gr}_{\mathfrak {m}}(\omega _R)$$ is a canonical module of $$\mathrm{gr}_{\mathfrak {m}}(R)$$ in terms of numerical semigroup’s properties, where $$\omega _R$$ is a canonical module of R.

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.