Brauer–Thrall for Totally Reflexive Modules over Local Rings of Higher Dimension

Let R be a commutative Noetherian local ring. Assume that R has a pair {x,y} of exact zerodivisors such that dim R/(x,y)?≥?2 and all totally reflexive R/(x)-modules are free. We show that the first and second Brauer–Thrall type theorems hold for the category of totally reflexive R-modules. More precisely, we prove that, for infinitely many integers n, there exists an indecomposable totally reflexive R-module of multiplicity n. Moreover, if the residue field of R is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive R-modules of multiplicity n.     

Connections on modules over singularities of finite CM representation type

Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions is it possible to find a connection on M?We consider the maximal Cohen-Macaulay (MCM) modules over complete CM algebras that are isolated singularities, and usually assume that the singularities have finite CM representation type. It is known that any MCM module over a simple singularity of dimension d≤2 admits an integrable connection. We prove that an MCM module over a simple singularity of dimension d≥3 admits a connection if and only if it is free. Among singularities of finite CM representation type, we find examples of curves with MCM modules that do not admit connections, and threefolds with non-free MCM modules that admit connections.Let A be a singularity not necessarily of finite CM representation type, and consider the condition that A is a Gorenstein curve or a -Gorenstein singularity of dimension d≥2. We show that this condition is sufficient for the canonical module ω_A to admit an integrable connection, and conjecture that it is also necessary. In support of the conjecture, we show that if A is a monomial curve singularity, then the canonical module ω_A admits an integrable connection if and only if A is Gorenstein.     

Gorenstein injective and strongly cotorsion modules

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of _R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).     

k-torsionless modules with finite Gorenstein dimension

Let R be a commutative Noetherian ring. It is shown that the finitely generated R-module M with finite Gorenstein dimension is reflexive if and only if M _p is reflexive for p ∈ Spec(R) with depth(R _p) ? 1, and $G - {\dim _{{R_p}}}$ (M _p) ? depth(R _p) ? 2 for p ∈ Spec(R) with depth(R _p) ? 2. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for n ? 2 we give a characterization of n-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every R-module has a k-torsionless cover provided R is a k-Gorenstein ring.     

Reflexive rings and their extensions

A right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if R[x] is reflexive if and only if R[x; x ^?1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reflexive.     

Trigonal Gorenstein curves

We notice that the Maroni invariant of a trigonal Gorenstein curve of arithmetic genus g larger than four may be equal to zero, and we show that this happens if and only if the g₃¹ admits a non-removable base point, which is necessarily a singularity of the curve. We realize and study trigonal curves on rational scrolls, which in the case, where the g₃¹ admits a base point Q, degenerate to a cone with vertex Q.     

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of _R R. Then we get that the injective envelope of _R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.     

Deficiently extremal Gorenstein algebras

The aim of this article is to study the homological properties of deficiently extremal Gorenstein algebras. We prove that if R/I is an odd deficiently extremal Gorenstein algebra with pure minimal free resolution, then the codimension of R/I must be odd. As an application, the structure of pure minimal free resolution of a nearly extremal Gorenstein algebra is obtained.     

Generalized Divisors and Total Reflexivity

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 _X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 _X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

Complete intersection dimensions and Foxby classes

Let R be a local ring and M a finitely generated R-module. The complete intersection dimension of M-defined by Avramov, Gasharov and Peeva, and denoted -is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities .Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ°φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class A_C(R) for each semidualizing R-complex C.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

Totally reflexive modules with respect to a semidualizing bimodule

Let S and {iaR} be two associative rings, let _S C _R be a semidualizing (S,R)-bimodule. We introduce and investigate properties of the totally reflexive module with respect to _S C _R and we give a characterization of the class of the totally C _R -reflexive modules over any ring R. Moreover, we show that the totally C _R -reflexive module with finite projective dimension is exactly the finitely generated projective right R-module. We then study the relations between the class of totally reflexive modules and the Bass class with respect to a semidualizing bimodule. The paper contains several results which are new in the commutative Noetherian setting.     

Retractions and Gorenstein Homological Properties

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free algebras). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra A, there exists a connected self-injective Nakayama algebra A′ such that there is a sequence of left retractions linking A to A′; in particular, the singularity category of A is triangle equivalent to the stable category of A′. We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.     

Singularity categories and singular equivalences for resolving subcategories

Let \(\mathcal {X}\) be a resolving subcategory of an abelian category. In this paper we investigate the singularity category \(\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))\) of the stable category \(\underline{\mathcal {X}}\) of \(\mathcal {X}\). We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type \((\mathsf {A}_1)\). We also generalize several results of Yoshino on totally reflexive modules.     

The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if Ext_R ⁿ (M, R) ⊗M → Ext_R ⁿ (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.     

Syzygy modules with semidualizing or G-projective summands

Let R be a commutative Noetherian local ring with residue class field k. In this paper, we mainly investigate direct summands of the syzygy modules of k. We prove that R is regular if and only if some syzygy module of k has a semidualizing summand. After that, we consider whether R is Gorenstein if and only if some syzygy module of k has a G-projective summand.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.