Canonical Module Research Articles

A new invariant of lattice polytopes

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $\omega (K[\mathcal{P}])$ of the toric ring $K[\mathcal{P}]$ defined by a lattice polytope $\mathcal{P}$ will be studied. It is shown that if $\mathcal{P}$ possesses an interior lattice point, then the maximal degree is at most $\dim \mathcal{P} - 1$, and that this bound is the best possible in general.

Partial trace ideals, torsion and canonical module

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h(M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h(M) in the case when M is the canonical module ωR. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups.

Deformations of commutative Artinian algebras

The paper is devoted to the study of deformations of Artinian algebras and zero-dimensional germs of varieties. In particular, an approach is developed to solving the open problem about the nonexistence of rigid Artinian algebras; it is based essentially on the use of the canonical duality in the cotangent complex. Thus, it is shown that there are no rigid Gorenstein Artinian algebras and rigid almost complete intersections. The proof of the latter statement is based on the properties of the torsion functors. More precisely, the tensor product of the conormal and canonical modules of the corresponding Artinian algebra is calculated. In this case, the homology and cohomology groups of higher degrees are also found. Among other things, some estimates are obtained for the dimension of the spaces of the first lower and upper cotangent functors of Artinian algebras, and the relationship between them is described. In conclusion, several examples of nonsmoothable Artinian noncomplete intersections are examined, and some unusual properties of such algebras are discussed.

Artinian Gorenstein algebras of embedding dimension four and socle degree three

We prove that in the polynomial ring Q=k[x,y,z,w], with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x,y,z,w)4⊆I⊆(x,y,z,w)2 can be obtained by doubling from a grade three perfect ideal J⊂I such that Q/J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q-module Q/I can be completely described in terms of a graded minimal free resolution of the Q-module Q/J and a homogeneous embedding of a shift of the canonical module ωQ/J into Q/J.

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.

On the behavior of F-signatures, splitting primes, and test modules under finite covers

We give a comprehensive treatment on how F-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. To this end, we expand on the notion of transposability along a section of the relative canonical module as first introduced by K. Schwede and K. Tucker.

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Cotilting with balanced big Cohen-Macaulay modules

Over d-dimensional Cohen-Macaulay rings with a canonical module, d-cotilting classes containing the maximal and balanced big Cohen-Macaulay modules are classified. Particular emphasis is paid to the direct limit closure of the balanced big Cohen-Macaulay modules, and the class of modules of depth d, which are shown to respectively be the smallest and largest such cotilting classes. Considerations are then given to the interplay between local cohomology, canonical duality and cotilting modules for the class of Gorenstein flat modules over Gorenstein local rings.

On minimum number of generators for some derivation modules

In this paper we study the generators of the module of derivations of certain rings and obtain bounds on their minimum number of generators. We will also study minimum number of generators of divisorial ideals of two dimensional rings of invariants, in particular the canonical modules of these rings.

Generalization of bi-canonical degrees

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module $$\omega$$ . Attached to each such ring R, when $$\omega$$ is an ideal, there are integers–the type of R, the reduction number of $$\omega$$ –that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In (Ghezzi et al. in JMS 589:506–528, 2017) and (Ghezzi et al. in JMS 571:55–74, 2021) we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where $$\omega$$ is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.

Upper bound on the colength of the trace of the canonical module in dimension one

We study upper bounds of the colength of the trace of the canonical module in one-dimensional Cohen–Macaulay rings. We answer two questions posed by Herzog–Hibi–Stamate and Kobayashi.

Tutte short exact sequences of graphs

We associate two modules, the \(G\)-parking critical module and the toppling critical module, to an undirected connected graph \(G\). The \(G\)-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied \(G\)-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to \(G\), an edge contraction \(G/e\) and an edge deletion \(G \setminus e\) (\(e\) is a non-bridge). We obtain purely combinatorial consequences of Tutte short exact sequences. For instance, we reprove a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial, and relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of \(G/e\) to the equality of the corresponding invariants of \(G\) and \(G \setminus e\).Mathematics Subject Classifications: 13D02, 05E40Keywords: Tutte polynomials, chip firing games, toppling ideals, \(G\)-parking function ideals, canonical modules

RINGS OF TETER TYPE

AbstractLet R be a Cohen–Macaulay local K-algebra or a standard graded K-algebra over a field K with a canonical module $\omega _R$ . The trace of $\omega _R$ is the ideal $\operatorname {tr}(\omega _R)$ of R which is the sum of those ideals $\varphi (\omega _R)$ with ${\varphi \in \operatorname {Hom}_R(\omega _R,R)}$ . The smallest number s for which there exist $\varphi _1, \ldots , \varphi _s \in \operatorname {Hom}_R(\omega _R,R)$ with $\operatorname {tr}(\omega _R)=\varphi _1(\omega _R) + \cdots + \varphi _s(\omega _R)$ is called the Teter number of R. We say that R is of Teter type if $s = 1$ . It is shown that R is not of Teter type if R is generically Gorenstein. In the present paper, we focus especially on zero-dimensional graded and monomial K-algebras and present various classes of such algebras which are of Teter type.

The Canonical Module of GT-Varieties and the Normal Bundle of RL-Varieties

In this paper, we study the geometry of GT-varieties X_{d} with group a finite cyclic group Gamma subset {{,mathrm{GL},}}(n+1,mathbb {K}) of order d. We prove that the homogeneous ideal {{,mathrm{I},}}(X_{d}) of X_{d} is generated by binomials of degree at most 3 and we provide examples reaching this bound. We give a combinatorial description of the canonical module of the homogeneous coordinate ring of X_{d} and we show that it is generated by monomial invariants of Gamma of degree d and 2d. This allows us to characterize the Castelnuovo–Mumford regularity of the homogeneous coordinate ring of X_d. Finally, we compute the cohomology table of the normal bundle of the so-called RL-varieties. They are projections of the Veronese variety nu _{d}(mathbb {P}^{n}) subset mathbb {P}^{left( {begin{array}{c}n+d nend{array}}right) -1} which naturally arise from level GT-varieties.

Canonical Modules and Class Groups of Rees-Like Algebras

Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.

When is $${\mathfrak {m}}:{\mathfrak {m}}$$ an almost Gorenstein ring?

Given a one-dimensional Cohen-Macaulay local ring $$(R,{\mathfrak {m}},k)$$ , we prove that it is almost Gorenstein if and only if $${\mathfrak {m}}$$ is a canonical module of the ring $${\mathfrak {m}}:{\mathfrak {m}}$$ . Then, we generalize this result by introducing the notions of almost canonical ideal and gAGL ring and by proving that R is gAGL if and only if $${\mathfrak {m}}$$ is an almost canonical ideal of $${\mathfrak {m}}:{\mathfrak {m}}$$ . We use this fact to characterize when the ring $${\mathfrak {m}}:{\mathfrak {m}}$$ is almost Gorenstein, provided that R has minimal multiplicity. This is a generalization of a result proved by Chau, Goto, Kumashiro, and Matsuoka in the case in which $${\mathfrak {m}}:{\mathfrak {m}}$$ is local and its residue field is isomorphic to k.

“Infinite” Properties of Certain Local Cohomology Modules of Determinantal Rings

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle dimensions of $H^{m+n-2}_I(\omega_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(\omega_R))$ are not finite. In the case of $m = n$, i.e. a Gorenstein ring, the socle dimensions provide further information about the $\tau$-numbers as studied in \cite{MS}. Moreover, the endomorphism ring of $H^{n-1}_I(\omega_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generated as $R$-module generalizing an example of \cite{Sp6}.

Cohen–Macaulay property and linearity of pinched Veronese rings

In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen–Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

When does the canonical module of a module have finite injective dimension?

When does the canonical module of a module have finite injective dimension?

Symmetry on rings of differential operators

If k is a field and R is a commutative k-algebra, we explore the question of when the ring DR|k of k-linear differential operators on R is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever R Gorenstein local or when R is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right D-module structures.