Artin Algebra Research Articles

Resolving dualities and applications to homological invariants

Abstract Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.

Torsion-simple objects in abelian categories

We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.

A binary tree of complete intersections with the strong Lefschetz property

In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of Artinian algebras defined by ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.

A functorial approach to monomorphism categories II: Indecomposables

AbstractWe investigate the (separated) monomorphism category of a quiver over an Artin algebra . We show that there exists an epivalence (called representation equivalence in the terminology of Auslander) from to , where is the category of finitely generated ‐modules and and denote the respective injectively stable categories. Furthermore, if has at least one arrow, then we show that this is an equivalence if and only if is hereditary. In general, the epivalence induces a bijection between indecomposable objects in and noninjective indecomposable objects in , and we show that the generalized Mimo‐construction, an explicit minimal right approximation into , gives an inverse to this bijection. We apply these results to describe the indecomposables in the monomorphism category of radical square zero Nakayama algebras, and to give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length 3 with the same residue field. The main tool to prove these results is the language of a free monad of an exact endofunctor on an arbitrary abelian category. This allows us to avoid the technical combinatorics arising from quiver representations. The setup also specializes to more general settings, such as representations of modulations. In particular, we obtain new results on the singularity category of the algebras that were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the algebras that were introduced by Lu and Wang to realize groups via semiderived Hall algebras.

The derived dimensions and representation distances of Artin algebras

The derived dimensions and representation distances of Artin algebras

Invariants and constructions of separable equivalences

In this paper, we first establish relationships between Gorenstein projective modules linked by the separable equivalence of rings, and prove that Gorenstein, CM-finite and CM-free algebras are invariant under separable equivalences. Secondly, we provide a new method to produce separable equivalences. As applications, the following results are obtained. Let Λ and Γ be Artin algebras such that Λ is separably equivalent to Γ. (1) For representation-finite algebras Λ and Γ, their Auslander algebras are separably equivalent; (2) For CM-finite algebras Λ and Γ, the endomorphism algebras of their representative generators are separably equivalent. Finally, we discuss when tilted algebras are invariant under separable equivalences, and give an example to illustrate it.

Self-injective algebras under derived equivalences

The Nakayama permutations of two derived equivalent, self-injective Artin algebras are conjugate. A different but elementary approach is given to showing that the weak symmetry and self-injectivity of finite-dimensional algebras over an arbitrary field are preserved under derived equivalences.

Auslander‐type conditions and weakly Gorenstein algebras

AbstractLet be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles

The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles

Cotorsion pairs and model structures on Morita rings

We study cotorsion pairs and abelian model structures on Morita rings Λ=(ANBAMABB) which are Artin algebras. Given cotorsion pairs (U,X) and (V,Y) in A-Mod and B-Mod, respectively, one can construct four cotorsion pairs in Λ-Mod:((XY)⊥,(XY)),(Δ(U,V),Δ(U,V)⊥),((UV),(UV)⊥),(∇⊥(X,Y),∇(X,Y)). These cotorsion pairs have relations:Δ(U,V)⊥⊆(XY),∇⊥(X,Y)⊆(UV). An important feature is that they are not equal, in general. In fact, there even exists a Morita algebra Λ, such that the four cotorsion pairs are pairwise different. The problem of identifications, i.e., when these inclusions are the same, are studied; the heredity and completeness of these cotorsion pairs are investigated; and finally, various model structures on Λ-Mod are obtained, by explicitly giving the corresponding Hovey triples and Quillen's homotopy categories. In particular, cofibrantly generated Hovey triples, and the Gillespie-Hovey triples induced by compatible generalized projective (respectively, injective) cotorsion pairs, are explicitly constructed. All these Hovey triples obtained are pairwise different and “new” in some sense. Some results are even new when M=0 or N=0.

A non-vanishing result on the singularity category

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras.

The radical of functorially finite subcategories

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if radC∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

G-semisimple algebras

Let Λ be an Artin algebra and mod-(Gprj_-Λ) the category of finitely presented functors over the stable category Gprj_-Λ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which mod-(Gprj_-Λ) is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations Gprj_(Q,Λ) of a finite acyclic quiver Q to the category of representations rep(Q,Gprj_-Λ) over Gprj_-Λ, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra ΛQ of the G-semisimple algebra Λ is CM-finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj(An,Λ) of the linear quiver An over a G-semisimple algebra Λ. We also determine almost split sequences in Gprj(An,Λ) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj(An,Λ).

On the Self-injectivity and CM-free of an Artin Algebra with Radical Cubic Zero

In this paper, we succeed in proving that a connected Artin algebra whose Jacobson cubic radical is zero with each simple module and each Gorenstein projective module having the square radical of the cover projective of its first syzygy to be zero and verifying the coincidence covers, is either self-injective or CM-free.

Extension dimensions of derived and stable equivalent algebras

The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study the behavior of the extensions dimensions of algebras under different equivalences. We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the length of the tilting complex associated with the derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences.

Relative injective envelopes and relative projective covers on ring extensions

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definitions of relative injective envelopes and relative projective covers of modules on ring extensions, and study their basic properties. In particular, we give their equivalent characterizations in terms of relative essential monomorphisms and relative superfluous epimorphisms, and prove that relative injective envelopes and relative projective covers on ring extensions are unique up to isomorphism whenever they exist. Moreover, for an extension of Artin algebras, we show that every finitely generated module has both a relative injective envelope and a relative projective cover. In addition, we compare relative injective envelopes and relative projective covers on two ring extensions linked by surjective homomorphisms of rings respectively.

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.

Gorenstein projective modules over Milnor squares of rings

<p>We construct a class of Gorenstein-projective modules over Milnor squares of rings. As an application, we obtain Gorenstein-projective modules over Morita context rings with two bimodule homomorphisms zero in the general setting instead of Artin algebras or Noetherian rings.</p>

Sincere silting modules and vanishing conditions

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 ( T , − ) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 ( T , − ) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 ( T , − ) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i ( T , T ( J ) ) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang’s paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations.

Applications of resolving subcategories to singularity categories and monomorphism categories

In this paper, we study resolving subcategories and singularity categories. First, if the left perpendicular category of a module [Formula: see text] over an Artin algebra [Formula: see text] is the additive closure of another module [Formula: see text], then the singularity category of [Formula: see text] and that of the endomorphism algebra [Formula: see text] of [Formula: see text] are closed related. This gives a categorical version of a recent result of Zhang ([31], Theorem 2]). Second, we apply the resolution theorem for derived categories to elliptic curves, the monomorphism subcategory of a Gorenstein algebra and of a kind of Eilenberg–Moore category. As consequences, their singularity categories are equivalent, which explains why monomorphism categories are closely related to singularity categories.