Canonical Module Research Articles

English

Let $$\mathfrak{a}$$ , I, J be ideals of a Noetherian local ring $$(R,\mathfrak{m},k)$$ . Let M and N be finitely generated R-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $$H^t_{I,J}$$ (M) and $$H^t_{I,J}$$ (M)), where t is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and D(−):= HomR(−, Er(k)) is the Matlis dual functor. We show that if R is a d-dimensional complete Cohen-Macaulay ring and $$H^i_{I,J}$$ (R) = 0 for all i ≠ t, the natural homomorphism R → Homr( $$H^t_{I,J}$$ (KR), $$H^t_{I,J}$$ (KR)) is an isomorphism, where KR denotes the canonical module of R. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.

Totally reflexive modules over rings that are close to Gorenstein

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor+R(T,N)=0, for some finitely generated R-module N, then N is free.

On the vanishing of self extensions over Cohen-Macaulay local rings

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously.

Identifiability beyond Kruskal’s bound for symmetric tensors of degree 4

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in \mathrm{Sym}^4(\mathbb C^{n+1}) , i.e., quartic hypersurfaces in a projective space \mathbb P^n , that have a decomposition in 2n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley–Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, Cayley–Bacharach schemes and their canonical modules, Trans. Amer. Math. Soc. 339:443–452, 1993.

On the structure of $S_2$-ifications of complete local rings

Motivated by work of Hochster and Huneke, we investigate several constructions related to the $S_2$-ification $T$ of a complete equidimensional local ring $R$: the canonical module, the top local cohomology module, topological spaces of the form $Spec (R)-V(J)$, and the (finite simple) graph $\Gamma _R$ with vertex set $Min (R)$ defined by Hochster and Huneke. We generalize one of their results by showing, e.g., that the number of connected components of $\Gamma _R$ is equal to the maximum number of connected components of $Spec (R)-V(J)$ for all $J$ of height $2$. We further investigate this graph by exhibiting a technique for showing that certain graphs $G$ can be realized in the form $\Gamma _R$.

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.

Embeddings of canonical modules and resolutions of connected sums

For an ideal Im,n generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of R/Im,n to R/Im,n itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by Im,n. Using this embedding, we give a resolution of connected sums of several copies of certain Artin k-algebras where k is a field.

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.

When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

Let A be a Cohen–Macaulay local ring with dimA=d≥3, possessing the canonical module KA. Let a1,a2,…,ar (3≤r≤d) be a subsystem of parameters of A, and set Q=(a1,a2,…,ar). We show that if the Rees algebra R(Q) of Q is an almost Gorenstein graded ring, then A is a regular local ring and a1,a2,…,ar is a part of a regular system of parameters of A.

Relative derived dimensions for cotilting modules

For a Noetherian ring R and a cotilting R-module T of injective dimension at least 1, we prove that the derived dimension of R with respect to the category XT is precisely the injective dimension of T by applying Auslander–Buchweitz theory and Ghost Lemma. In particular, when R is a commutative Noetherian Cohen–Macaulay local ring with a canonical module ωR and dim⁡R≥1, the derived dimension of R with respect to the category of maximal Cohen–Macaulay modules is precisely dim⁡R.

A duality in Buchsbaum rings and triangulated manifolds

Let $\Delta$ be a triangulated homology ball whose boundary complex is $\partial\Delta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $\Delta$, $\mathbb F[\Delta]$, is isomorphic to the Stanley--Reisner module of the pair $(\Delta, \partial\Delta)$, $\mathbb F[\Delta,\partial \Delta]$. This result implies that an Artinian reduction of $\mathbb F[\Delta,\partial \Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $\mathbb F[\Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h"$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h"$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h"$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.

On the generators of the canonical module of a Hibi ring: A criterion of level property and the degrees of generators

In this paper, we study the minimal generating system of the canonical module of a Hibi ring. Using the results, we state a characterization of a Hibi ring to be level. We also give a characterization of a Hibi ring to be of type 2. Further, we show that the degrees of the elements of the minimal generating system of the canonical module of a Hibi ring form a set of consecutive integers.

On canonical modules of idealizations

Let $(R,\mathfrak {m})$ be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let $M$ be a finitely generated $R$-module. In this paper, we study the structure of the canonical module $K(R\ \mathbb {n}\ M)$ of the idealization $R\ \mathbb {n}\ M$ via the polynomial type introduced by Cuong~\cite {C}. In particular, we give a characterization for $K(R\ \mathbb {n}\ M)$ being Cohen-Macaulay and generalized Cohen-Macaulay.

CANONICAL AND -CANONICAL MODULES OF A NOETHERIAN ALGEBRA

We define canonical and $n$-canonical modules of a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$-canonical modules, we generalize a theorem on $(n,C)$-syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a noncommutative version of Aoyama’s theorem which states that a canonical module descends with respect to a flat local homomorphism.

A canonical module characterization of Serre’s (R1)

ABSTRACTIn this short note, we give a characterization of domains satisfying Serre’s condition (R1) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9] where the normality is described in terms of the “shape” of the canonical module.

Residual intersections and the annihilator of Koszul homologies

Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with a small number of generators. Acyclicity of a prime case of these complexes is shown to be equivalent to find a common annihilator for higher Koszul homologies. So that, a tight relation between residual intersections and the uniform annihilator of positive Koszul homologies is unveiled that sheds some light on their structure.

Computing global dimension of endomorphism rings via ladders

This paper deals with computing the global dimension of endomorphism rings of maximal Cohen–Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type An for all n, Dn for n≤13 and E6,7,8 and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [12]. Moreover, we determine the centre of an endomorphism ring of a MCM-module over any curve singularity of finite MCM-type.In general, we describe a method for the computation of the global dimension of an endomorphism ring EndRM, where R is a Henselian local ring, using add(M)-approximations. When M≠0 is a MCM-module over R and R is Henselian local of Krull dimension ≤2 with a canonical module and of finite MCM-type, we use Auslander–Reiten theory and Iyama's ladder method to explicitly construct these approximations.

Canonical modules of complexes

We define the notion of the canonical module of a complex. We then consider Serre's conditions for a complex and study their relationship to the local cohomology of the canonical module and its ring of endomorphisms.

The monomial conjecture and order ideals II

Let I be an ideal of height d in a regular local ring (R,m,k=R/m) of dimension n and let Ω denote the canonical module of R/I. In this paper we first prove the equivalence of the following: the non-vanishing of the edge homomorhpism ηd:ExtRn−d(k,Ω)→ExtRn(k,R), the validity of the order ideal conjecture for regular local rings, and the validity of the monomial conjecture for all local rings. Next we prove several special cases of the order ideal conjecture/monomial conjecture.

Extension Closedness of Syzygies and Local Gorensteinness of Commutative Rings

We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen-Macaulay local ring with canonical module to an arbitrary local ring.