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Let a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

We consider stable representations of non-Dynkin quivers with respect to a central charge. These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states. Another motivation is the work of Dimitrov et al. on the phases of stable representations of the generalized Kronecker quiver. One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples. In addition, we study especially the behavior of preinjective, postprojective and regular indecomposable modules. We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges. In this case the phases of the stable representations approach one or two limit points. In particular, the phases are not dense in two arcs. The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship among three n-cluster tilting subcategories of triangulated categories in a recollement. Let (D ′ , D , D ″) be a recollement of triangulated categories. We construct an n-cluster tilting subcategory of D by gluing together n-cluster tilting subcategories of D ′ and D ″ under certain conditions. On the other hand, we prove that an n-cluster tilting subcategory of D can induce n-cluster tilting subcategories of D ′ and D ″ . Using these two results, we give a one-to-one correspondence between some n-cluster tilting subcategories of D and pairs of n-cluster tilting subcategories of D ′ and D ″ . Furthermore, we obtain a recollement of abelian categories from a recollement of triangulated categories. [ABSTRACT FROM AUTHOR]

We study the category Rep (Q , C) of representations of a quiver Q with values in an abelian category C. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver Q and we use them to impose conditions on C that allow us to prove interesting homological properties of Rep (Q , C) that can be constructed from C. For example, we compute the global dimension of Rep (Q , C) in terms of the global one of C. We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on C) every hereditary complete cotorsion pair (A , B) in C induces the hereditary complete cotorsion pairs (Rep (Q , A) , Rep (Q , A) ⊥ 1 ) and (Ψ ⊥ 1 (B) , Ψ (B)) in Rep (Q , C) , and then we obtain a strengthened version of this and other related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of Rep (Q , C) , finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in Rep (Q , C) can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs. [ABSTRACT FROM AUTHOR]

In [16] , a theory of universal extensions in abelian categories is developed; in particular, the notion of Ext 1 -universal object is presented. In the present paper, we show that an Ab3 abelian category which is Ext 1 -small satisfies the Ab4 condition if, and only if, each one of its objects is Ext 1 -universal. We use the dual of this result to construct projective effacements in Grothendieck categories. In particular, we complete the classical result of Roos on Grothendieck categories which are Ab4*, with a new proof, independent of [20]. We also give a characterization of the co- Ext 1 -universal objects of the category of torsion abelian groups. In particular, we show that such groups are the ones admitting a decomposition Q ⊕ R , in which Q is injective and R is a reduced group in which each p -component is bounded. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and I be an ideal of R such that the R -modules H I i (M) are I -cofinite, for all finitely generated R -modules M and all i ∈ ℕ 0 . In this paper, we investigate a question of Hartshorne concerning the Abelianness of the category of all I -cofinite modules. [ABSTRACT FROM AUTHOR]

Given a right exact functor from an abelian category into another abelian category, there is an associated abelian category called the comma category of the functor. In this paper, we characterize when left Frobenius pairs (resp. strong left Frobenius pairs) in abelian categories can induce left Frobenius pairs (resp. strong left Frobenius pairs) in their comma categories. This leads to the construction of recollements of right triangulated categories (resp. triangulated categories) from the stable categories of left Frobenius pairs (resp. strong left Frobenius pairs). Applications are given to complete hereditary cotorsion pairs and Gorenstein projective objects. [ABSTRACT FROM AUTHOR]

It is well known that the Baer criterion for injectivity of R -modules, for a ring R with unit, is not true in a general category, even in a general abelian category. In this paper, we prove some results analogous to the Baer criterion for injectivity in abelian categories and Grothendieck categories. In particular, we generalize the known fact that G -injectivity is the same as injectivity, if G is a generator in a Grothendieck category. Furthermore, some Baer type theorems for general abelian categories are proved. Finally, equivalent conditions to satisfying a classical kind of Baer criterion are found in (locally presentable) abelian categories. [ABSTRACT FROM AUTHOR]

Let C be an abelian monoidal category. It is proved that the nilpotent category Nil (C) of C admits almost monoidal structure except the unit axiom. As an application, it is proved that Hom and Tensor functors exist over Nil (C) and tensor–hom adjunction remains true over the nilpotent category of the category of finite-dimensional vector spaces, which develops some recent results on this topic. [ABSTRACT FROM AUTHOR]

In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to the Calabi-Yau reduction, silting reduction and τ -tilting reduction. More precisely, if an abelian category A admits a recollement relative to abelian categories A ′ and A ′ ′ , which is denoted by (A ′ , A , A ′ ′ , i ∗ , i ∗ , i ! , j ! , j ∗ , j ∗) , then the assignment C ↦ j ∗ (C) defines a bijection between wide subcategories in A containing i ∗ (A ′) and wide subcategories in A ′ ′. Moreover, a wide subcategory C of A containing i ∗ (A ′) admits a new recollement relative to A ′ and j ∗ (C) which is induced from the original recollement. [ABSTRACT FROM AUTHOR]

Let C be an additive category. The nilpotent category Nil (C) of C , consists of object pairs (X, x) with X ∈ C , x ∈ End C (X) such that xn = 0 for some positive integer n, and a morphism f: (X, x) → (Y, y) is f ∈ Hom C (X , Y) satisfying fx = yf. A general theory of Nil (C) is established and it is abelian in the case that C is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a result in [Comm. Algebra, 2018, 46(7): 3062–3070]. As an application, it is proved that all self-adjoint functors are naturally isomorphic to Hom and Tensor functors over the category V of finite-dimensional vector spaces. Both Hom and Tensor can be naturally generalized to HOM and Tensor functor over Nil (V) . They are still self-adjoint, but intrinsically different. [ABSTRACT FROM AUTHOR]

We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories. [ABSTRACT FROM AUTHOR]

Let 풜 be an abelian category and (풳,풴) a pair of classes of objects in 풜. Inspired by Bouchiba’s work on generalized Gorenstein projective modules, we give a new way of measuring (풳,풴)-Gorenstein projective dimension by defining a complete n-(풳,풴) resolution. Then we relate the relative global Gorenstein homological dimension to the invariants silp(풜) and spli(풜) under some conditions. Furthermore, we prove that, in the setting of a left and right coherent ring R, the supremum of Ding projective dimensions of all finitely presented (left or right) R-modules and the (left or right) Gorenstein weak global dimension are identical, generalizing a theorem of Ding, Li and Mao. [ABSTRACT FROM AUTHOR]

Let (C,E,s)\left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}}) be an extriangulated category. Given a composition of two commutative squares in C{\mathcal{C}}, if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [Categories for the Working Mathematician, Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [On good morphisms of exact triangles, J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.

We consider an arbitrary Abelian category A and a subcategory T closed under extensions and direct summands, and characterize those T that are (semi-)special preenveloping in A ; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories T of A closed under extensions and direct summands are precisely those for which ( ⊥ 1 T , T) is a right complete cotorsion pair, where ⊥ 1 T : = Ker (Ext A 1 (− , T)) . Particular cases appear when T = V ⊥ 1 : = Ker (Ext A 1 (V , −)) , for an Ext1-universal object V such that Ext A 1 (V , −) vanishes on all (existing) coproducts of copies of V. For many choices of A , we show that these latter examples exhaust all the possibilities. We then show that, when A has an epi-generator, the (semi-)special preenveloping torsion classes T given by (quasi-)tilting objects are exactly those for which any object T ∈ T is the epimorphic image of some object in ⊥ 1 T (and the subcategory B : = Sub (T) of subobjects of objects in T is reflective) and they are, in turn, the right constituents of complete cotorsion pairs in A (resp., B ). In a final section, we apply the results when A = mod- R is the category of finitely presented modules over a right coherent ring R, something that gives new results and raises new questions even at the level of classical tilting theory in categories of modules. [ABSTRACT FROM AUTHOR]

Let be a specialization closed subset of Spec R and be a Serre subcategory of Mod R. As a generalization of the notion of cofiniteness, we introduce the concept of -coserreness with respect to (see Definition 4.1). First, as a main result, for some special Serre subcategories , we show that an R -module M with dim R M ≤ 1 is -coserre with respect to if and only if Ext R 0 , 1 (R / , M) ∈ for all ideals ∈ F (). Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of -coserre R -modules M with dim R M ≤ 1 is a full Abelian subcategory of Mod R. Also, for every homologically bounded R -complex X whose homology modules belong to we show that the local cohomology modules H i (X) for all i , are -coserre in all the cases where dim R ≤ 1 , dim R X ≤ 2 − sup X and cd (, X) ≤ 1 − sup X. [ABSTRACT FROM AUTHOR]

Let A be a essentially small abelian category and C be a Serre subcategory of A. Consider the quotient functor q : A → A / C. For an object A ∈ A and a non-negative integer k we investigate when the natural map q X , A i : Ext A i (X , A) → Ext A / C i (q (X) , q (A)) is invertible for every X ∈ A and every i ∈ { 0 , 1 , ⋯ , k }. In the end we give an application of the main theorem. [ABSTRACT FROM AUTHOR]

Let (R, m) be a commutative Noetherian complete local ring and let I be a proper ideal of R such that the category of all I-cofinite R-modules is an Abelian subcategory of the category of all R-modules. Then for each I-cofinite R-module M it is shown that dim R/(I + AnnRM) = dim M. As a consequence of this result it is shown that if J is an ideal of R such that for each p ∈ mAssRR, dim R/(J + p) ≤ 1 or cd(J, R/p) ≤ 1, then dim R/(J + AnnRM) = dim M, for each J-cofinite R-module M. [ABSTRACT FROM AUTHOR]

We introduce relative CS-Rickart objects in abelian categories, as common generalizations of relative Rickart objects and extending objects. We study direct summands and (co)products of relative CS-Rickart objects as well as classes all of whose objects are self-CS-Rickart. Corresponding results for dual relative CS-Rickart objects may be automatically obtained by the duality principle. Applications are given to Grothendieck categories and, in particular, to module and comodule categories. [ABSTRACT FROM AUTHOR]

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A → M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B , a functor A → B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M → B. In particular, this provides a description of the left heart of A as a concrete category. We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R -modules, for a given Polish group or Polish ring R ; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field. [ABSTRACT FROM AUTHOR]

We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories. [ABSTRACT FROM AUTHOR]

In this paper, we study the cotorsion dimensions and cotorsion triples in the recollements of abelian categories. The main results are that recollements induce new (resp. complete hereditary) cotorsion triples from the middle category and that the cotorsion dimensions are bounded under certain conditions. As an application, the cotorsion triples in the recollements of module categories with respect to triangular matrix algebras are recovered. [ABSTRACT FROM AUTHOR]

Let be an abelian category and M   ∈ A. Then M is called a (D 4) -object if, whenever A and B are subobjects of M with M   =   A   ⊕   B and f : A → B is an epimorphism, Ker ⁡ f is a direct summand of A. In this paper we give several equivalent conditions of (D 4) -objects in an abelian category. Among other results, we prove that any object M in an abelian category is (D 4) if and only if for every subobject K of M such that K is the intersection K 1 ∩ K 2 of perspective direct summands K 1 and K 2 of M with M   =   K 1   +   K 2 , every morphismr φ :   M   →   M / K can be lifted to an endomorphism θ : M → M in E n d A (M). [ABSTRACT FROM AUTHOR]

Let A be an abelian category with enough projective objects, and let X be a quasi-resolving subcategory of A . In this paper, we investigate the affinity of the Spanier–Whitehead category S W (X) of the stable category of X with the singularity category D s g (A) of A . We construct a fully faithful triangle functor from S W (X) to D s g (A) , and we prove that it is dense if and only if the bounded derived category D b (A) of A is generated by X . Applying this result to commutative rings, we obtain characterizations of the isolated singularities, the Gorenstein rings and the Cohen–Macaulay rings. Moreover, we classify the Spanier–Whitehead categories over complete intersections. Finally, we explore a method to compute the (Rouquier) dimension of the triangulated category S W (X) in terms of generation in X . [ABSTRACT FROM AUTHOR]

Let T be a right exact functor from an abelian category ℬ into another abelian category A . Then there exists a functor p from the product category A × ℬ to the comma category ( (T ↓ A) ). In this paper, we study the property of the extension closure of some classes of objects in ( (T ↓ A) ), the exactness of the functor p and the detailed description of orthogonal classes of a given class P (X , Y) in ( (T ↓ A) ). Moreover, we characterize when special precovering classes in abelian categories A and ℬ can induce special precovering classes in ( (T ↓ A) ). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring Λ = ( R M 0 S ) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them. [ABSTRACT FROM AUTHOR]

We introduce (dual) strongly relative CS-Rickart objects in abelian categories, as common generalizations of (dual) strongly relative Rickart objects and strongly extending (lifting) objects. We give general properties, and we study direct summands, (co)products of (dual) strongly relative CS-Rickart objects and classes all of whose objects are (dual) strongly self-CS-Rickart. [ABSTRACT FROM AUTHOR]

Let k be a field. We show that locally presentable, k-linear categories C dualizable in the sense that the identity functor can be recovered as ∐ i x i ⊗ f i for objects x i ∈ C and left adjoints f i from C to Vect k are products of copies of Vect k . This partially confirms a conjecture by Brandenburg, the author and T. Johnson-Freyd. Motivated by this, we also characterize the Grothendieck categories containing an object x with the property that every object is a copower of x: they are precisely the categories of non-singular injective right modules over simple, regular, right self-injective rings of type I or III. [ABSTRACT FROM AUTHOR]

Let (R ,) be a commutative Noetherian complete local ring and I be a proper ideal of R. Suppose that M is a nonzero I -cofinite R -module of Krull dimension n. In this paper, it shown that dim R / (I + Ann R H n (M)) = n. As an application of this result, it is shown that dim R / (I +) = n , for each ∈ Att R H n (M). Also it shown that for each j ≥ 0 the submodule Σ j (M) : = ∪ { K : K ≤ M and dim K ≤ j } of M is I -cofinite, dim R / (I + Ann R H j (M / Σ j (M))) < j and dim R / (I + Ann R H j (M)) ≤ j , whenever the category of all I -cofinite R -modules is an Abelian subcategory of the category of all R -modules. Also some applications of these results will be included. [ABSTRACT FROM AUTHOR]

We study the transfer of D4 and dually C4 objects via functors between abelian categories. We also give applications to Grothendieck categories, comodule categories, (graded) module categories. [ABSTRACT FROM AUTHOR]

We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to coproducts and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module categories, and comodule categories. [ABSTRACT FROM AUTHOR]

We introduce and study (strongly) π -Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) π -Rickart. We derive consequences for module and comodule categories. [ABSTRACT FROM AUTHOR]

We introduce and study (dual) relative F-Baer objects as specializations of (dual) relative split objects with respect to a fully invariant short exact sequence in A B 3 * (AB3) abelian categories. We analyze their relationship with (dual) relative Baer objects, and we study direct summands and direct sums of (dual) relative F-Baer objects. We give applications to module and comodule categories. [ABSTRACT FROM AUTHOR]

Let \mathscr{C} be a triangulated category and \mathscr{X} be a cluster tilting subcategory of \mathscr{C} . Koenig and Zhu showed that the quotient category \mathscr{C} / \mathscr{X} is Gorenstein of Gorenstein dimension at most one. But this is not always true when \mathscr{C} becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let \mathscr{C} be an extriangulated category with enough projectives and enough injectives, and \mathscr{X} a cluster tilting subcategory of \mathscr{C} . We show that under certain conditions, the quotient category \mathscr{C} / \mathscr{X} is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu. [ABSTRACT FROM AUTHOR]

noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(mathit{Q}_j)$ of representations of a quiver $mathit{Q}_j$ as a full subcategory of the category $Rep_k(mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}})$ of representations of $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$ for every $j in {1,2,ldots , n}$.

We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n -abelian categories. More precisely, for a positive integer n and a small n -abelian category M , we show that M is equivalent to a full subcategory of an abelian category L 2 (M , G) , where L 2 (M , G) is the category of absolutely pure group valued functors over M. We also show that n -kernels and n -cokernels in M are precisely exact sequences of L 2 (M , G) with terms in M. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category ℬ, a functorially finite, extension closed subcategory of ℬ is cluster tilting if and only if ℬ / is an abelian category. [ABSTRACT FROM AUTHOR]

In this paper we introduce a generalization of the Nakayama functor for finite-dimensional algebras. This is obtained by abstracting its interaction with the forgetful functor to vector spaces. In particular, we characterize the Nakayama functor in terms of an ambidextrous adjunction of monads and comonads. In the second part we develop a theory of Gorenstein homological algebra for such Nakayama functor. We obtain analogues of several classical results for Iwanaga-Gorenstein algebras. One of our main examples is the module category Λ-Mod of a k-algebra Λ, where k is a commutative ring and Λ is finitely generated projective as a k-module. [ABSTRACT FROM AUTHOR]