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Canadian Journal of Mathematics (0008414X) 77(3)pp. 975-1012
The main theme of this paper is to study τ-tilting subcategories in an abelian category A with enough projective objects. We introduce the notion of τ-cotorsion torsion triples and investigate a bijection between the collection of τ-cotorsion torsion triples in A and the collection of support τ-tilting subcategories of A , generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of A . General definitions and results are exemplified using persistent modules. If A = Mod-R, where R is a unitary associative ring, we characterize all support τ-tilting (resp. all support τ−-tilting) subcategories of Mod-R in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support τ-tilting (resp. support τ−-tilting) subcategory of Mod-R. We also study the theory in Rep(Q, A ), where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support τ-tilting subcategories in Rep(Q, A ) from certain support τ-tilting subcategories of A . © The Author(s), 2024.
Journal of Pure and Applied Algebra (00224049) 229(5)
Higher homological algebra, basically done in the framework of an n-cluster tilting subcategory M of an abelian category A, has been the topic of several recent researches. In this paper, we study a relative version, in the sense of Auslander-Solberg, of the higher homological algebra. To this end, we consider an additive sub-bifunctor F of ExtMn(−,−) as the basis of our relative theory. This, in turn, specifies a collection of n-exact sequences in M, which allows us to delve into the relative higher homological algebra. Our results include a proof of the relative n-Auslander-Reiten duality formula, as well as an exploration of relative Grothendieck groups, among other results. As an application, we provide necessary and sufficient conditions for M to be of finite type. © 2025 Elsevier B.V.
Communications in Algebra (00927872) 52(5)pp. 2148-2166
Assume that B is a finite dimensional algebra, and (Formula presented.) is the one-point extension algebra of B using a finitely generated projective B-module P 0. The categories of B-modules and A-modules are connected via two adjoint functors known as the restriction and extension functors, denoted by (Formula presented.) and (Formula presented.), respectively. These functors have nice homological properties and have been studied in the category (Formula presented.) of finitely presented modules that we extend to the category (Formula presented.) of all A-modules. Our main focus is to investigate the behavior of important subcategories (tilting and τ-tilting subcategories) and objects (finendo quasi-tilting modules, silting modules, and cosilting modules) under these functors. © 2023 Taylor & Francis Group, LLC.
Progress of Theoretical and Experimental Physics (20503911) (1)
The effects of hybridization and impurity (magnetic and nonmagnetic) potentials on the pairing symmetries and magnetic response of a two-band superconductor with an equal-time s-wave interband pairing order parameter in the framework of Green’s function technique are investigated theoretically. First, the effects of spin-independent and spin-dependent hybridization on the generation of even- or odd-frequency Cooper pairs which determines the symmetry classification and the response of the superconductor are studied. Next, the impurity effect on creating different symmetry classes and the kernel response function of a two-band superconductor are discussed. By separating the contributions of even- and odd-frequency pairing to the Meissner kernel, it is shown that the competition between these two terms determines the total Meissner effect of the superconductor. For a two-band spin-singlet superconductor, nonmagnetic impurity scatterings do not change transition temperature according to Anderson’s theorem, while both intra- and interband magnetic impurity scattering cause superconducting transition temperature suppression with the rate following the Abrikosov–Gor’kov theory. For spin-triplet pairing, interband magnetic scattering has no impact on pair breaking, whereas intraband magnetic scattering acts as a pair breaker and suppresses the transition temperature in the Born limit. In this case, the odd-frequency superconducting pairs can be induced in the simultaneous presence of both intra- and interband magnetic impurities. Thus, by controlling the concentration of magnetic impurities, it is possible to engineer triplet-pairing odd-frequency superconductors with a total diamagnetic Meissner response which stabilizes the superconducting state. This technique opens up an avenue for designing stable odd-frequency superconductors. © The Author(s) 2023.
Michigan Mathematical Journal (00262285) 73(4)pp. 853-873
An Artin algebra Λis said to be of finite Cohen-Macaulay type if, up to isomorphism, there are only finitely many indecomposable modules in g Λ, the full subcategory of modΛconsisting of all gorenstein projective (right) Λ-modules. In this paper, we study 1-gorenstein algebras of finite Cohen-Macaulay type through mod(g Λ), the category of finitely presented g Λ-modules. Some applications will be provided. In particular, a necessary and sufficient condition is given for T3Λ, the 3 by 3 lower triangular matrices over Λ, to be of finite Cohen-Macaulay type. Finally, the structure of almost split sequences will be described explicitly in a special subcategory of mod(g Λ), denoted by g -1(g Λ). IfΛis self-injective, g -1(g Λ) = mod(g Λ). © 2023 University of Michigan. All rights reserved.
Journal of Pure and Applied Algebra (00224049) 226(2)
In this paper, the notion of cotorsion classes is introduced into the higher homological algebra. Our results motivate the definition, showing that this notion of n-cotorsion classes satisfies usual properties one could expect. In particular, a higher version of Wakamatsu's Lemma is proved. At the last section, connections with wide subcategories are studied. © 2021 Elsevier B.V.
Transactions of the American Mathematical Society (00029947) 375(3)pp. 2113-2145
Our aim in this paper is to introduce the so-called ideal approximation theory into higher homological algebra. To this end, we introduce some important notions from approximation theory into the theory of n-exact categories and prove some results. In particular, the higher version of notions such as ideal cotorsion pairs, phantom ideals, Salce’s Lemma and Wakamatsu’s Lemma for ideals are introduced and studied. Our results motivate the definitions and show that n-exact categories are the appropriate context for the study of higher ideal approximation theory. © 2022 American Mathematical Society.
Nagoya Mathematical Journal (00277630) 248pp. 823-848
Building on the embedding of an n-abelian category M into an abelian category A as an n-cluster-tilting subcategory of A, in this paper, we relate the n-torsion classes of M with the torsion classes of A. Indeed, we show that every n-torsion class in M is given by the intersection of a torsion class in A with M. Moreover, we show that every chain of n-torsion classes in the n-abelian category M induces a Harder–Narasimhan filtration for every object of M. We use the relation between M and A to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in M can be induced by a chain of torsion classes in A. Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes. © (2022) The Authors.
Science China Mathematics (16747283) 65(7)pp. 1343-1362
Let ℳ be an n-cluster tilting subcategory of mod-Λ, where Λ is an Artin algebra. Let S(ℳ) denote the full subcategory of S(Λ) , the submodule category of Λ, consisting of all the monomorphisms in ℳ. We construct two functors from S(ℳ) to mod−ℳ¯, the category of finitely presented additive contravariant functors on the stable category of ℳ. We show that these functors are full, dense and objective and hence provide equivalences between the quotient categories of S(ℳ) and mod−ℳ¯. We also compare these two functors and show that they differ by the n-th syzygy functor, provided ℳ is an nℤ-cluster tilting subcategory. These functors can be considered as higher versions of the two functors studied by Ringel and Zhang (2014) in the case Λ= k[x] / 〈 xn〉 and generalized later by Eiríksson (2017) to self-injective Artin algebras. Several applications are provided. © 2021, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.
Journal of the Mathematical Society of Japan (00255645) 73(2)pp. 329-349
Let A be a right coherent ring and X be a contravariantly finite subcategory of mod-A containing projectives. In this paper, we construct a recollement of abelian categories (mod0-X, mod-X, mod-A), where mod0-X is a full subcategory of mod-X consisting of all functors vanishing on projective modules. As a result, a relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Some examples and applications will be provided. © 2021 The Mathematical Society of Japan
Communications in Algebra (00927872) 49(9)pp. 4028-4037
Let Λ be an artin algebra and (Formula presented.) be a quasi-resolving subcategory of (Formula presented.) which is of finite type. Let (Formula presented.) be the full subcategory of the morphism category (Formula presented.) consisting of all monomorphisms (Formula presented.) in (Formula presented.) such that (Formula presented.) also lies in (Formula presented.). In this paper, we state and prove Brauer-Thrall type theorems for (Formula presented.). As applications, we provide necessary and sufficient conditions for the submodule category (Formula presented.) to be of finite type, whenever Λ is of finite representation type, as well as, for the lower 2 × 2 triangular matrix algebra (Formula presented.) to be of finite CM-type, whenever Λ is of finite CM-type. © 2021 Taylor & Francis Group, LLC.
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.
Bulletin Of The Iranian Mathematical Society (10186301) 47(SUPPL 1)pp. 1-4
Journal of Algebra (00218693) 580pp. 127-157
Let Λ be an Artin algebra. In this paper, the notion of nZ-Gorenstein cluster tilting subcategories will be introduced. It is shown that every nZ-cluster tilting subcategory of mod-Λ is nZ-Gorenstein if and only if Λ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an nZ-Gorenstein cluster tilting subcategory of mod-Λ is an nZ-cluster tilting subcategory of the exact category Gprj-Λ, the subcategory of all Gorenstein projective objects of mod-Λ. Some basic properties of nZ-Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are n-resolving, a higher version of resolving subcategories. © 2021 Elsevier Inc.
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.
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.
Proceedings of the American Mathematical Society (00029939) 148(6)pp. 2379-2396
In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study relative Auslander algebras. In particular, we study the existence of tilting-cotilting modules over such algebras. Some applications will be provided. In particular, it will be shown that two Gorenstein algebras of G-dimension one that are of finite Cohen-Macaulay-type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent. © 2020 American Mathematical Society
Journal of Commutative Algebra (19390807) 12(1)pp. 1-26
The main theme of this paper is to study different "Gorenstein defect categories" and their connections. This is done by studying rings for which Kac. Prj-R/ D Ktac. Prj-R/, that is, rings enjoying the property that every acyclic complex of projectives is totally acyclic. Such studies have been started by Iyengar and Krause over commutative Noetherian rings with a dualizing complex. We show that a virtually Gorenstein Artin algebra is Gorenstein if and only if it satisfies the above mentioned property. Then, we introduce recollements connecting several categories which help in providing categorical characterizations of Gorenstein rings. Finally, we study relative singularity categories that lead us to some more "Gorenstein defect categories". © 2020 Rocky Mountain Mathematics Consortium.
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.
Hasoomi N. ,
Naderi M. ,
Sarabi Asl A. ,
Asadollahi, J. ,
Hemat, S. ,
Hemat, S. ,
Vahad, R. Journal of Radiotherapy in Practice (14603969) (3)pp. 347-353
Background In radiotherapy, electron beam irradiation is an effective modality for superficial tumours. Electron beams have good coverage of tumours which involve the skin, however there is an issue about electron scattering and tissue heterogeneity. This subsequently demands dosimetric analysis of electron beam behaviour, particularly in the treatment of lesions on the scalp requiring the application of treatment to scalp curvatures. There are various methods which are used to treat scalp malignancies including photons and electrons, but, the later needs precise dosimetry before each session of treatment. The purpose of the study was to undertake a detailed analysis of the dosimetry of electron beams when applied to the curved surface of the scalp using Gafchromic® EBT2 films.Methods and materials A rando phantom and Gafchromic® EBT2 films were used for dosimetric analysis. A gafchromic calibration curve was plotted and an in-treatment beam dosimetric analysis was carried out using dosimetry films placed on the scalp. Electron behaviour was assessed by introducing five electron fields in particular curvature regions of scalp.Result There was an acceptable dose range through all five fields and hotspots occurred in the curved borders. In our study, skin doses and doses at the field junctions, with no gaps, were between 78-97% and 80-97%, respectively.Conclusions Electron beams are a good modality for treating one flat field, but in the special topography of the scalp, whole scalp treatment requires precise field matching and dosimetry. In undertaking this detailed dosimetric analysis using a rando phantom and Gafchromic® EBT2 films, it is concluded that this method requires further detailed analysis before using in clinics. © Cambridge University Press 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.
Journal of Algebra and its Applications (17936829) 17(1)
Let R be a ring and be a finite quiver. We give an explicit formula for the injective envelopes and projective precovers in the category Rep(I,R) of bound representations of by left R-modules, where I is a combination of monomial and commutativity relations. Some applications will be provided. In particular, it is shown that if is acyclic and R is an Iwanaga-Gorenstein ring, then so are these bound quiver algebras. © 2018 World Scientific Publishing Company.
Communications in Algebra (00927872) 45(6)pp. 2557-2568
Let R be a ring and Q be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(Q,R) of representations of Q by left R-modules. We also extend our formula to all terms of the minimal injective resolution of RQ. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra RQ is k-Gorenstein if and only if (Formula presented.) and R is a k-Gorenstein ring, where n is the number of vertices of Q. © 2017, Copyright © Taylor & Francis.
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.
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.
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (00418994) 136pp. 257-264
We provide a simple proof for a recent result of Bravo, Gillespie and Hovey, showing that over a left coherent ring for which the projective dimension of flat right modules is finite, the class of Gorenstein projective right modules is precovering. As an immediate application, we provide a description for the Gorenstein defect category over such rings. © 2016, Universita di Padova. All rights reserved.
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.
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.
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.
Journal of Algebra (00218693) 399pp. 423-444
Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(PrjC(R)), is compactly generated provided K(PrjR) is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever R is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor ι:K(GPrjR){right arrow, hooked}K(R) has a right adjoint ιρ, where K(GPrjR) is the homotopy category of Gorenstein projective R modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If R has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories K(GPrjR) and K(GInjR), that restricts to an equivalence between K(PrjR) and K(InjR), whenever R is commutative, noetherian and admits a dualising complex. © 2013 Elsevier Inc..
Communications in Algebra (15324125) 42(5)pp. 1953-1964
In this article we provide arguments for constructing Kaplansky classes in the category of complexes out of a Kaplansky class of modules. This leads to several complete cotorsion theories in such categories. Our method gives a unified proof for most of the known cotorsion theories in the category of complexes and can be applied to the category of quasi-coherent sheaves over a scheme as well as the category of the representations of a quiver. © 2014 Copyright Taylor and Francis Group, LLC.
Kyoto Journal of Mathematics (21562261) 54(3)pp. 693-702
We give an upper bound on the dimension of the bounded derived category of an abelian category.We show that if x is a sufficiently nice subcategory of an abelian category, then the derived dimension of A is at most x-dimA, provided that x-dimA is greater than one.We provide some applications. © 2014 by Kyoto University.
Mathematical Research Letters (10732780) 21(1)pp. 19-31
In this paper, we study the representation dimension as well as the derived dimension of the path algebra of an Artin algebra over a finite and acyclic quiver.
Journal of Algebra (00218693) 353(1)pp. 93-120
Let A be an associative ring with identity, K(FlatA) the homotopy category of flat modules and K p(FlatA) the full subcategory of pure complexes. The quotient category K(FlatA)/K p(FlatA), called here the pure derived category of flats, was introduced by Neeman. In this category flat resolutions are unique up to homotopy and so can be used to compute cohomology. We develop theories of Tate and complete cohomology in the pure derived category of flats. These theories extend naturally to sheaves over semi-separated noetherian schemes, where there are not always enough projectives, but we do have enough flats. As applications we characterize rings with finite sfli and schemes which are locally Gorenstein. © 2011 Elsevier Inc.
Forum Mathematicum (09337741) 24(2)pp. 273-287
Using the notion of flat covers and proper flat resolutions, we study modules with periodic flat resolutions. It follows that, equivalently, we may study modules with periodic homology. We specialize our results to the category of modules over integral group ring ℤF, where F is an arbitrary group. Among other results, we show that if a group F is in a certain class of groups, then F has periodic homology of period q after some steps with the periodicity isomorphisms of homology groups induced by the cap product with an element in H q(F;C), where C is the cotorsion envelope of the trivial F-module ℤ, if and only if it has periodic cohomology of period q after some steps with the periodicity isomorphisms of cohomology groups induced by the cup product with an element in H q(F; ℤ). © de Gruyter 2012.
Communications in Algebra (00927872) 39(3)pp. 888-904
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{script}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 {script}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. © Taylor & Francis Group, LLC.
Journal of Algebra (00218693) 335(1)pp. 18-35
Recently the notions of sfliΓ, the supremum of the flat lengths of injective Γ-modules, and silfΓ, the supremum of the injective lengths of flat Γ-modules have been studied by some authors. These homological invariants are based on spli and silp invariants of Gedrich and Gruenberg and it is shown that they have enough potential to play an important role in studying homological conjectures in cohomology of groups. In this paper we will study these invariants. It turns out that, for any group Γ, the finiteness of silfΓ implies the finiteness of sfliΓ, but the converse is not known. We investigate the situation in which sfliΓ<∞ implies silfΓ<∞. The statement holds for example, for groups Γ with the property that flat Γ-modules have finite projective dimension. Moreover, we show that the Gorenstein flat dimension of the trivial ZΓ-module Z, that will be called Gorenstein homological dimension of Γ, denoted GhdΓ, is completely related to these invariants. © 2011 Elsevier Inc.
Journal of Algebra (00218693) 346(1)pp. 101-115
Let R be a ring and Q be a quiver. We study the homotopy categories K(PrjQ) and K(InjQ) consisting, respectively, of projective and injective representations of Q by R-modules. We show that, for certain quivers, these triangulated categories are compactly generated and provide explicit descriptions of compact generating sets. Moreover, in case R is commutative and noetherian with a dualizing complex D, the dualizing functor D⊗R-:K(PrjR) → K(InjR) is extended to a triangulated functor K(PrjQ)→K(InjopQ) which is an equivalence of triangulated categories. This functor, establishes an equivalence on K(PrjQ) and K(InjQ), whenever Q is finite. © 2011 Elsevier Inc.
Kyoto Journal of Mathematics (0023608X) 51(4)pp. 811-829
Recently a notion of support and a construction of local cohomology functors for [TR5] compactly generated triangulated categories were introduced and studied by Benson, Iyengar, and Krause. Following their idea, we assign to any object of the category a new subset of Spec(R), again called the (big) support. We study this support and show that it satisfies axioms such as exactness, orthogonality, and separation. Using this support, we study the behavior of the local cohomology functors and show that these triangulated functors respect boundedness. Then we restrict our study to the categories generated by only one compact object. This condition enables us to get some nice results. Our results show that one can get a satisfactory version of the local cohomology theory in the setting of triangulated categories, compatible with the known results for the local cohomology for complexes of modules. © 2011 by Kyoto University.
Taiwanese Journal of Mathematics (10275487) 14(4)pp. 1677-1687
We extend some major theorems in commutative algebra to the class of modules that are not necessarily finitely generated. The novelty of our extension is that the hypothesis of finite generation over R is replaced by one over S, where R and S are commutative Noetherian local rings and there is a local homomorphism φ{symbol}: R -→ S. Among the results that we extend are: Intersection Theorem and Intersection dimension formula.
Journal of Pure and Applied Algebra (00224049) 213(9)pp. 1795-1803
Using Auslander's G-dimension, we assign a numerical invariant to any group Γ. It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ. We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups. © 2009 Elsevier B.V. All rights reserved.
Journal of Algebra (00218693) 319(6)pp. 2626-2651
We develop and study Tate and complete cohomology theory in the category of sheaves of OX-modules. Different approaches are included. We study the properties of these theories and show their power in reflecting the Gorensteinness of the underlying scheme. The connection of these two theories will be discussed. © 2007 Elsevier Inc. All rights reserved.
Journal of Pure and Applied Algebra (00224049) 210(3)pp. 771-787
We introduce and study a complete cohomology theory for complexes, which provides an extended version of Tate-Vogel cohomology in the setting of (arbitrary) complexes over associative rings. Moreover, for complexes of finite Gorenstein projective dimension a notion of relative Ext is introduced. On the basis of these cohomology groups, some homological invariants of modules over commutative noetherian local rings, such as Martsinkovsky's ξ-invariants and relative and Tate versions of Betti numbers, are extended to the framework of complexes with finite homology. The relation of these invariants with their prototypes is explored. © 2006 Elsevier Ltd. All rights reserved.
Algebra Colloquium (02191733) 14(1)pp. 155-166
There is a complete cohomology theory developed over a commutative noetherian ring in which injectives take the role of projectives in Vogel's construction of complete cohomology theory. We study the interaction between this complete cohomology, that is referred to as I-complete cohomology, and Vogel's one and give some sufficient conditions for their equivalence. Using I-complete functors, we assign a new homological invariant to any finitely generated module over an arbitrary commutative noetherian local ring, that would generalize Auslander's delta invariant. We generalize the results about the δ-invariant to arbitrary rings and give a sufficient condition for the vanishing of this new invariant. We also introduce an analogue of the notion of the index of a Gorenstein local ring, introduced by Auslander, for arbitrary local rings and study its behavior under flat extensions of local rings. Finally, we study the connection between the index and Loewy length of a local ring and generalize the main result of [11] to arbitrary rings. © 2007 AMSS CAS & SUZHOU UNIV.
Communications in Algebra (00927872) 35(12)pp. 4328-4329
Transactions of the American Mathematical Society (00029947) 358(5)pp. 2183-2203
In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing. ©2005 American Mathematical Society.
Communications in Algebra (00927872) 34(8)pp. 3009-3022
The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga-Gorenstein rings. In particular, we show that an associative ring R is Iwanaga-Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the 'Ext' functor. Copyright © Taylor & Francis Group, LLC.
Kyoto Journal of Mathematics (0023608X) 46(2)pp. 357-365
Let R be a (not necessary finite dimensional) commutative noetherian ring and let C be a semi-dualizing module over R. There is a generalized Gorenstein dimension with respect to C, namely GC-dimension, sharing the nice properties of Auslander's Gorenstein dimension. In this paper, we establish the Faltings' Annihilator Theorem and it's uniform version (in the sense of Raghavan) for local cohomology modules over the class of finitely generated R-modules of finite GC-dimension, provided R is Cohen-Macaulay. Our version contains variations of results already known on the Annihilator Theorem.
Forum Mathematicum (09337741) 18(6)pp. 951-966
Let A and B be two rings, M be a left B, right A bimodule and T = ( MA B0) be the formal triangular matrix ring. It is known that the category of right T-modules is equivalent to the category Ω of triples (X, Y)f, where X is a right A-module, Y is a right B-module and f : Y ⊗B M → X is a right A-homomorphism. Using this alternative description of right T-modules, in the first part of this paper, we study the vanishing of the extension functor 'Ext' over T. To this end, we first describe explicitly the structure of (right) T-modules of finite projective (respectively, injective) dimension. Using these results, we shall characterize respectively modules in Auslander's G-class, Gorenstein injective modules, cotorsion modules and tilting and cotilting modules over T. As another application we investigate the structure of the flat covers of right T-modules. © de Gruyter 2006.
Journal of Algebra (00218693) 299(2)pp. 480-502
Motivated by the classical structure of Tate cohomology, we develop and study a Tate cohomology theory in a triangulated category C. Let E be a proper class of triangles. By using E-projective, as well as E-injective objects, we give two alternative approaches to this theory that, in general, are not equivalent. So, in the second part of the paper, we study triangulated categories in which these two theories are equivalent. This leads us to study the categories in which all objects have finite E- Gprojective as well as finite E- Ginjective dimension. These categories will be called E-Gorenstein triangulated categories. We give a characterization of these categories in terms of the finiteness of two invariants: E- silp C, the supremum of the E-injective dimension of E-projective objects of C and E- spli C, the supremum of the E-projective dimension of E-injective objects of C, where finiteness of each of these invariants for a category implies the finiteness of the other. Finally, we show that over E-Gorenstein triangulated categories, the class of objects of finite E-projective dimension and the class of E- Ginjective objects form an E-complete cotorsion theory. © 2005 Elsevier Inc. All rights reserved.
Rocky Mountain Journal of Mathematics (00357596) 35(4)pp. 1069-1076
We extend a criterion of Gerko for a ring to be Cohen-Macaulay to arbitrary, not necessarily local, Noetherian rings. Our version reads as follows: The Noetherian ring R is Cohen-Macaulay if and only if, for all finitely generated R-modules M, CM-dimRM is finite. © 2005 Rocky Mountain Mathematics Consortium.
Communications in Algebra (00927872) 33(10)pp. 3439-3446
Let (R,ℳ) be a commutative Noetherian local ring. We exhibit certain modules T over R which test G-dimension of a finitely generated R-module M with finite G-dimension in the following sense: if Extjg(M,T) = 0 for all j ≥ i, where i is a positive integer, then G-dimR M < i. Modules with the property like T will be called Gorenstein test modules (G-test modules for short). It is known that R itself is a G-test module. We show that k, the residue field of R, also tests G-dimension. Some more examples of G-test modules are introduced.
Bulletin of the Australian Mathematical Society (00049727) 71(2)pp. 337-346
Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen-Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula. Copyright Clearance Centre, Inc.
Communications in Algebra (00927872) 32(10)pp. 3969-3979
We define two new homological invariants for a finitely generated module M over a commutative Noetherian local ring R, its Buchsbaum dimension B-dim R M, and its Monomial conjecture dimension MC-dimR M. It will be shown that these new invariants have certain nice properties we have come to expect from homological dimensions. Over a Buchsbaum ring R, every finite module M has B-dimR M < ∞; conversely, if the residue field has finite B-dimension, then the ring R is Buchsbaum. Similarly R satisfies the Hochster Monomial Conjecture if only if MC-dimR k is finite, where k is the residue field of R. MC-dimension fits between the B-dimension and restricted flat dimension Rfd of Christensen et al. [Christensen, L. W., Foxby, H.-B., Frankild, A. (2002). Restricted homological dimensions and Cohen-Macaulayness. J. Algebra 251(1):479-502]. B-dimension itself is finer than CM-dimension of Gerko [Gerko, A. A. (2001). On homological dimensions. Sb. Math. 192(7-8): 1165-1179] and we have equality if CM-dimension is finite. It also satisfies an analog of the Auslander-Buchsbaum formula. Copyright © 2004 by Marcel Dekker, Inc.
Journal of Algebra (00218693) 273(1)pp. 384-394
A new homological dimension, called GCM-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. GCM-dimension (short for Generalized Cohen-Macaulay dimension) characterizes Generalized Cohen-Macaulay rings in the sense that: a ring R is Generalized Cohen-Macaulay if and only if every finitely generated R-module has finite GCM-dimension. This dimension is finer than CM-dimension and we have equality if CM-dimension is finite. Our results will show that this dimension has expected basic properties parallel to those of the homological dimensions. In particular, it satisfies an analog of the Auslander-Buchsbaum formula. Similar methods will be used for introducing quasi-Buchsbaum and Almost Cohen-Macaulay dimensions, which reflect corresponding properties of their underlying rings. © 2004 Elsevier Inc. All rights reserved.
Communications in Algebra (00927872) 32(11)pp. 4415-4432
Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions. Copyright © 2004 by Marcel Dekker, Inc.
Journal of Algebra (00218693) 281(1)pp. 264-286
Indian Journal of Pure and Applied Mathematics (00195588) 35(5)pp. 621-627
Using the theory of flat covers, we introduce a set of primes dual to the set of associated primes. It turns out that this set has properties similar (or rather dual) to those of the set of associated primes. With the aid of this set of primes, we obtain a necessary and sufficient condition for a cotorsion module M, over a ring whose injective hull is flat, to have a nonzero injective cover.
Journal of the Australian Mathematical Society (14467887) 75(3)pp. 313-324
Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module H Ii(M) is I-cofinite. Furthermore, for any system of ideals φ of R, we study the cofiniteness problem in the context of general local cohomology modules.
Japanese Journal of Mathematics (02892316) 29(2)pp. 285-296
The paper contributes to the question whether the set of associated prime ideals of the local cohomology module (M) is finite for all ideals I of a local ring (R,m) and a finitely generated generalized Cohen-Macaulay R-module M. We prove that it will be enough to solve the problem for i = 2 resp. 3 for ideals with two resp. certain ideals with three generators. This extends Hellus' result, see [5], of a Cohen-Macaulay ring. Moreover, in the case of M a Cohen-Macaulay module there is another sufficient criterion for the finiteness of associated prime ideals of (M) related to certain cofiniteness conditions. Finally, we discuss several examples of the literature related to the problem.© 2003, The Mathematical Society of Japan. All rights reserved.
Communications in Algebra (00927872) 30(2)pp. 859-867
Communications in Algebra (00927872) 30(8)pp. 3813-3823
Colloquium Mathematicum (00101354) 87(1)pp. 129-136
Let A be a Noetherian ring, let M be a finitely generated A-module and let φ be a system of ideals of A. We prove that, for any ideal a in φ, if, for every prime ideal p of A, there exists an integer k(p), depending on p, such that ak(p) kills the general local cohomology module Hjφ (Mp) for every integer j less than a fixed integer n, where φp := {ap : a ε φ}, then there exists an integer k such that akHjφ(M) = 0 for every j < n. © 2001, Instytut Matematyczny. All rights reserved.
