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Publication Date: 2023
Kyoto Journal of Mathematics (21543321) 63(4)pp. 829-849
Let C be a locally bounded k-category, where k is a field.We prove that C is pure-semisimple, that is, every object of Mod-C is pure-projective if and only if every family of morphisms between indecomposable finitely generated C-modules is Noetherian. Our formalism establishes the pure-semisimplicity of Galois coverings, that is, if C is a G-category with a free G-action on ind-C, then C is pure-semisimple if and only if C/G is so. © 2023 by Kyoto University.
Publication Date: 2023
Journal of Algebra and its Applications (17936829) 22(7)
Let Λ be a Z-graded artin algebra. It is proved that the category of graded Λ-modules is pure-semisimple if and only if there are only finitely many nonisomorphic indecomposable finitely generated graded Λ-modules. As a consequence of this result together with a known result of Gordon and Green (which states that Λ is of finite representation type if and only if there are only finitely many non-isomorphic indecomposable finitely generated graded Λ-modules), we see that the category of all Λ-modules is pure-semisimple if and only if the category of all graded Λ-modules is so. © World Scientific Publishing Company.
Publication Date: 2021
Journal of Algebra and its Applications (17936829) 20(4)
In this paper, ideal balanced pairs in an abelian category will be introduced and studied. It is proved that every ideal balanced pair gives rise to a triangle equivalence of relative derived categories. We define complete ideal cotorsion triplets and investigate their relation with ideal balanced pairs. © 2021 World Scientific Publishing Company.
Publication Date: 2020
Kyoto Journal of Mathematics (21543321) 60(1)pp. 61-91
According to Auslander's formula, one way of studying an abelian category C is to study mod-C, which has nicer homological properties than C, and then translate the results back to C. Recently, Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander's formula, including the versions for all modules and for unbounded derived categories. We also include some results concerning recollements of triangulated categories. © 2020 by Kyoto University.
Publication Date: 2020
Communications in Algebra (00927872) 48(5)pp. 2167-2182
In this paper, (higher) Gorenstein flat phantom morphisms over rings will be introduced and studied. To study their relationship, a characterization of Gorenstein flat objects in morphism category is given. Communicated by Alberto Facchini. © 2020, © 2020 Taylor & Francis Group, LLC.
Publication Date: 2020
Communications in Algebra (00927872) 48(7)pp. 3133-3156
A sufficient condition for the existence of recollements of functor categories is provided. Using this criterion, we show that a recollement of rings induces a recollement of their path rings (resp. incidence rings, monomial rings) over a locally finite quiver. Also, we present a covering technique for recollement of derived categories of functor categories. © 2020, © 2020 Taylor & Francis Group, LLC.
Publication Date: 2019
Journal of Pure and Applied Algebra (00224049) 223(3)pp. 1073-1096
Let Mod-S denote the category of S-modules, where S is a small pre-additive category. Using the notion of relative derived categories of functor categories, we generalize Rickard's theorem on derived equivalences of module categories over rings to Mod-S. Several interesting applications will be provided. In particular, it will be shown that derived equivalence of two coherent rings not only implies the equivalence of their homotopy categories of projective modules, but also implies that they are Gorenstein derived equivalent. As another application, it is shown that a good tilting module produces an equivalence between the unbounded derived category of the module category of the ring and the relative derived category of the module category of the endomorphism ring of it. © 2018 Elsevier B.V.
Publication Date: 2018
Communications in Algebra (00927872) 46(10)pp. 4377-4391
For a (right and left) coherent ring A, we show that there exists a duality between homotopy categories (Formula presented.) and 𝕂b(mod-A). If A = Λ is an artin algebra of finite global dimension, this duality induces a duality between their subcategories of acyclic complexes, (Formula presented.) and (Formula presented.) As a result, it will be shown that, in this case, (Formula presented.) admits a Serre functor and hence has Auslander–Reiten triangles. © 2018, © 2018 Taylor & Francis.
Publication Date: 2018
Journal of Algebra (00218693) 507pp. 320-361
Let C be a G-category for some group G. We show that a G-covering functor p:C⟶C/G induces G-precovering of their bounded derived categories, singularity categories and Gorenstein defect categories. Then we provide a Gorenstein version of Gabriel's theorem. Using this result we investigate the number of summands in a decomposition of the middle term of almost split sequences over monomial algebras. © 2018
Publication Date: 2017
Journal of Algebra and its Applications (17936829) 16(2)
In this paper, we study the category of sheaves over an infinite partially ordered set with its natural topological structure. Totally acyclic complexes in this category will be characterized in terms of their stalks. This leads us to describe Gorenstein projective, injective and flat sheaves. As an application, we get an analogue of a formula due to Mitchell, giving an upper bound on the Gorenstein global dimension of such categories. Based on these results, we present situations in which the class of Gorenstein projective sheaves is precovering as well as situations in which the class of Gorenstein injective sheaves is preenveloping. © 2017 World Scientific Publishing Company.
Publication Date: 2016
Communications in Algebra (00927872) 44(12)pp. 5454-5477
The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalize two results of Happel by proving the existence of AR-triangles in Gorenstein-derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent, and finally study relations between the Gorenstein-derived category of a quiver and its image under a reflection functor. Some interesting applications are provided. © 2016, Copyright © Taylor & Francis Group, LLC.
Publication Date: 2016
Applied Categorical Structures (09272852) 24(4)pp. 331-371
This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category S and a maximal object s of S and construct a recollement of Mod- S in terms of Mod-End S(s) and Mod- (S\{s}) in four different levels. In case S is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N-complexes. © 2015, Springer Science+Business Media Dordrecht.
Publication Date: 2016
Forum Mathematicum (09337741) 28(2)pp. 377-389
We describe explicitly the Auslander-Reiten translation in the category of bounded complexes of finitely generated maximal Cohen-Macaulay modules, Cb(CM R), over a commutative local Cohen-Macaulay ring R with a canonical module ω. Then the Auslander-Reiten formula is generalized for complexes in Cb(CM R) and we prove the existence theorem of Auslander-Reiten sequences. As an application of our results, we investigate the existence of Auslander-Reiten triangles in the category of perfect complexes as a full triangulated subcategory of Db(mod R). © 2016 by De Gruyter 2016.
Publication Date: 2015
Journal of Algebra and its Applications (17936829) 14(3)
Let λ be an artin algebra. By letting the Nakayama functor act degree-wise, we define a translation ? in the category of complexes of finitely generated λ-modules, C(mod λ). Then we investigate the existence of almost split sequences in the category C(mod λ). As an application of our results, we see that the full subcategory of D(mod λ) consisting of complexes isomorphic to perfect complexes admits almost split sequences. © World Scientific Publishing Company.
Publication Date: 2015
Canadian Journal of Mathematics (0008414X) 67(1)pp. 28-54
We study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.
Publication Date: 2014
Journal of Pure and Applied Algebra (00224049) 218(5)pp. 888-903
The main objective of this paper is to study the relative derived categories from various points of view. Let A be an abelian category and C be a contravariantly finite subcategory of A. One can define C-relative derived category of A, denoted by DC*(A). The interesting case for us is when A has enough projective objects and C=GP-A is the class of Gorenstein projective objects, where DC*(A) is known as the Gorenstein derived category of A. We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickard's theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories. © 2013 Elsevier B.V.
Publication Date: 2009
EXCLI Journal (16112156) 30pp. 211-217
In this paper, a new approach for prediction of protein solvent accessibility is presented. The prediction of relative solvent accessibility gives helpful information for the prediction of na-tive structure of a protein. Recent years several RSA prediction methods including those that generate real values and those that predict discrete states (buried vs. exposed) have been de-veloped. We propose a novel method for real value prediction that aims at minimizing the prediction error when compared with existing methods. The proposed method is based on Pace Regression (PR) predictor. The improved prediction quality is a result of features of PSI-BLAST profile and the PR method because pace regression is optimal when the number of coefficients tends to infinity. The experiment results on Manesh dataset show that the pro-posed method is an improvement in average prediction accuracy and training time.
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