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Bulletin Of The Iranian Mathematical Society (10186301) 51(2)
Let (S,n) be a commutative noetherian local ring and let ω∈n be non-zero divisor. This paper is concerned with the category of monomorphisms between finitely generated Gorenstein projective S-modules, such that their cokernels are annihilated by ω. We will observe that this category, which will be denoted by Mon(ω,G), is an exact category in the sense of Quillen. More generally, it is proved that Mon(ω,G) is a Frobenius category. Surprisingly, it is shown that not only the category of matrix factorizations embeds into Mon(ω,G), but also its stable category as well as the singularity category of the factor ring R=S/(ω), can be realized as triangulated subcategories of the stable category Mon̲(ω,G). © The Author(s) under exclusive licence to Iranian Mathematical Society 2025.
Bahlekeh, A. ,
Fotouhi, F.S. ,
Hamlehdari M.A. ,
Salarian S. ,
Bahlekeh, A. ,
Fotouhi, F.S. ,
Hamlehdari M.A. ,
Salarian, S. Forum Mathematicum (9337741) 37pp. 1185-1200
Let (S, n) be a commutative noetherian local ring and let ω ∈ n be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by Mon(ω, P) and Mon(ω, G), are both Frobenius categories with the same projective objects. It is also proved that the stable category Mon(ω, P) is triangle equivalent to the category of D-branes of type B, DB(ω), which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories Mon(ω, P) and Mon(ω, G) are closely related to the singularity category of the factor ring R = S/(ω). Precisely, there is a fully faithful triangle functor from the stable category Mon(ω, G) to Dsg(R), which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to Mon(ω, P), guarantees the regularity of the ring S. © 2024 Walter de Gruyter GmbH. All rights reserved.
Kyoto Journal of Mathematics (21543321) 63(1)pp. 1-22
Let (R, m) be a d-dimensional commutative complete Noetherian local ring and A be a Noetherian R-algebra. Motivated by the notion of Cohen-Macaulay Artin algebras of Auslander and Reiten, we say that A is Cohen-Macaulay if there is a finitely generated A-bimodule w that is maximal Cohen-Macaulay over R such that the adjoint pair of functors (w circle times?' -, Hom?' (w, -)) induces quasi-inverse equivalences between the full subcategories of finitely generated A'-modules consisting of modules of finite projective dimension, P degrees degrees(A'), and the modules of finite injective dimension, I degrees degrees(A'), whenever A' = A, Aop. It is proved that such a module w is unique, up to isomorphism, as a A'-module. It is also shown that A is a Cohen-Macaulay algebra if and only if there is a semidualizing A-bimodule w of finite injective dimension and P degrees degrees(A') and I degrees degrees(A') are contained in the Auslander and Bass classes, respectively. We prove that Cohen-Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if A is a Cohen-Macaulay algebra, then for any system of parameters x = x1, ... , xd of R, the Artin algebra A/xA is Cohen-Macaulay as well. Assume that w is a semidualizing A-bimodule of finite injective dimension that is maximal Cohen-Macaulay as an R-module. It will turn out that A being a Cohen-Macaulay algebra is equivalent to saying that the pair (CM(A'),I degrees degrees(A')) forms a hereditary complete cotorsion theory and the pair (CM(A'op), P degrees degrees(A')) forms a Tor-torsion theory, where CM(A') is the class of all finitely generated A'-modules admitting a right resolution by modules in addw. Finally, it is shown that Cohen-Macaulayness ascends from R to RF and RQ, where F is a finite group and Q is a finite acyclic quiver.
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.
Bahlekeh, Abdolnaser ,
Bahlekeh, A. ,
Fotouhi, F.S. ,
Nateghi, A. ,
Salarian, S. Bulletin Of The Malaysian Mathematical Sciences Society (01266705) 46(3)
Let (S, n) be a commutative noetherian local ring and omega is an element of n be non-zerodivisor. This paper deals with the behavior of the category Mon(omega, P) consisting of all monomorphisms between finitely generated projective S-modules with cokernels annihilated by omega. We introduce a homotopy category HMon(omega, P), which is shown to be triangulated. It is proved that this homotopy category embeds into the singularity category of the factor ring R = S/(omega). As an application, not only the existence of almost split sequences ending at indecomposable non-projective objects of Mon(omega, P) is proved, but also the Auslander-Reiten translation, tau Mon(-), is completely recognized. Particularly, it will be observed that any non-projective object of Mon(omega, P) with local endomorphism ring is invariant under the square of the Auslander-Reiten translation.
Algebras and Representation Theory (15729079) 25(3)pp. 595-613
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 SW(X) of the stable category of X with the singularity category Dsg(A) of A. We construct a fully faithful triangle functor from SW(X) to Dsg(A) , and we prove that it is dense if and only if the bounded derived category Db(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 SW(X) in terms of generation in X. © 2021, The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature.
Algebras and Representation Theory (15729079) 23(5)pp. 1983-2011
Brauer and Thrall conjectured that a finite-dimensional algebra over a field of bounded representation type is actually of finite representation type and a finite-dimensional algebra (over an infinite field) of infinite representation type has strongly unbounded representation type. The first conjecture was proved in full generality, and the second conjecture was proved under the additional assumption that the field be algebraically closed. These results are our motivation for studying (generalized) orders of bounded and strongly unbounded lattice type. To each lattice over an order we assign a numerical invariant, h̲-length, measuring Hom modulo projectives. We show that an order of bounded lattice type is actually of finite lattice type, and if there are infinitely many non-isomorphic indecomposable lattices of the same h̲-length, then the order has strongly unbounded lattice type. For a hypersurface R = k[[x0,..,xd]]/(f), we show that R is of bounded (respectively, strongly unbounded) lattice type if and only if the double branched cover R♯ of R is of bounded (respectively, strongly unbounded) lattice type. This is an analog of a result of Knörrer and Buchweitz-Greuel-Schreyer for rings of finite mCM type. Consequently, it is proved that R has strongly unbounded lattice type whenever k is infinite. © 2019, Springer Nature B.V.
Journal of Algebra (00218693) 523pp. 15-33
Let (R,m) be a commutative complete Gorenstein local ring and let Λ be a Gorenstein order, that is to say, Λ is a maximal Cohen–Macaulay R-module and HomR(Λ,R) is a projective Λ-module. The main theme of this paper is to study the representation-theoretic properties of generalized lattices, i.e. those Λ-modules which are Gorenstein projective over R. It is proved that Λ has only finitely many isomorphism classes of indecomposable lattices if and only if every generalized lattice is the direct sum of finitely generated ones. It is also turn out that, if R is one-dimensional, then a generalized lattice M which is not the direct sum of copies of a finite number of lattices, contains indecomposable sublattices of arbitrarily large finite h_-length, an invariant assigned to each generalized lattice which measures Hom modulo projectives. © 2018
Mathematische Zeitschrift (14328232) 293(3-4)pp. 1673-1709
Let (R, m, k) be a complete Cohen–Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen–Macaulay module, called h̲-length. Among other results, it is proved that, for a given balanced big Cohen–Macaulay R-module M with an m-primary cohomological annihilator, if there is a bound on the h̲-length of all modules appearing in CM-support of M, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer–Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen–Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen–Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen–Macaulay modules is settled. Namely, it is shown that R is of finite CM-type if and only if R is an isolated singularity and the category of all fully decomposable balanced big Cohen–Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen–Macaulay artin algebras admitting a dualizing bimodule ω, as defined by Auslander and Reiten. It will turn out that, ω-Gorenstein projective modules with bounded CM-support are fully decomposable. In particular, a Cohen–Macaulay algebra Λ is of finite CM-type if and only if every ω-Gorenstein projective module is of finite CM-type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel–Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay modules. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
Kyoto Journal of Mathematics (21543321) 59(1)pp. 237-266
Let (R, m) be a d-dimensional commutative Noetherian local ring. Let M denote the morphism category of finitely generated R-modules, and let S be the full subcategory of M consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category S can be described explicitly within mod R, the category of finitely generated R-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in S. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories H and G, consisting of all morphisms which are maximal Cohen–Macaulay R-modules and Gorenstein projective morphisms, respectively, may be computed within mod R via G-covers. The corresponding result for the subcategory of epimorphisms in H is also obtained. © 2019 by Kyoto University
Communications in Algebra (00927872) 45(1)pp. 121-129
In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and Q a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra RQ, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented. © 2017, Copyright © Taylor & Francis Group, LLC.
Quarterly Journal of Mathematics (00335606) 67(3)pp. 387-404
Let (R, m, k) be a commutative noetherian local ring of Krull dimension d. We prove that the cohomology annihilator ca(R) of R contains a power of m if and only if, for some n≥ 0, the nth syzygies in mod R are constructed from syzygies of k by taking direct sums/summands and a fixed number of extensions. These conditions yield that R is an isolated singularity such that the bounded derived category Db(R) and the singularity category Dsg(R) have finite dimension, and the converse holds when R is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and d extensions. This result is exploited to investigate several ascent and descent problems between R and its completion R. © 2016 Published by Oxford University Press. All rights reserved.
Algebra Colloquium (02191733) 23(1)pp. 97-104
Let R and S be Artin algebras and Γ be their triangular matrix extension via a bimodule SMR. We study totally acyclic complexes of projective Γ-modules and obtain a complete description of Gorenstein projective Γ-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras. © 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
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.
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.
Journal of Algebra (00218693) 427pp. 252-263
A recent result of Araya asserts that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring R, then it holds for R. This note extends this result to left Gorenstein R-algebras Λ, whenever R is a commutative Gorenstein ring. This, in particular, implies that any finitely generated self-orthogonal Gorenstein projective Λ-module is projective, provided Λ is an isolated singularity and dim. R≥. 2. Also, some examples of bound quiver algebras satisfying the Auslander-Reiten conjecture are presented. © 2015.
Kyoto Journal of Mathematics (21543321) 55(1)pp. 129-141
Let γ be a finite group, and let A be any Artin algebra. It is shown that the group algebra Λγ is virtually Gorenstein if and only if ΛΓ' is virtually Gorenstein, for all elementary abelian subgroups Γ of Γ We also extend this result to cover the more general context. Precisely, assume that Γ is a group in Kropholler's hierarchy HS, Γ is a subgroup of Γ of finite index, and Ris any ring with identity. It is proved that, in certain circumstances, that RΓ is virtually Gorenstein if and only if RΓ' is so. © 2015 by Kyoto University.
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..
Proceedings of the American Mathematical Society (10886826) 141(3)pp. 753-762
Let X be a Noetherian scheme, K(FlatX) be the homotopy category of flat quasi-coherent OX-modules and Kp(FlatX) be the homotopy category of all flat complexes. It is shown that the pair (Kp(FlatX), K (dg- CofX)) is a complete cotorsion theory in K(FlatX), where K (dg-CofX) is the essential image of the homotopy category of dg-cotorsion complexes of flat modules. Then we study the homotopy category K(dg-Cof X). We show that in the affine case, this homotopy category is equal with the essential image of the embedding functor j*: K(ProjR) → K(FlatR) which has been studied by Neeman in his recent papers. Moreover, we present a condition for the inclusion K(dg-Cof X) ⊆ K(Cof X) to be an equality, where K(Cof X) is the essential image of the homotopy category of complexes of cotorsion flat sheaves. © 2012, American Mathematical Society.
Journal of Algebra and its Applications (17936829) 12(6)
This note is devoted to the study of cotorsion theory in the category of quiver representations. Let Q be a quiver, x be a class of representations, and let Vx denote the class of all modules each of which appears in a vertex of an element of x. If the pair (⊥Vx, V x) (respectively, (Vx, Vx⊥)) is a cotorsion theory, we investigate conditions under which the pair ( ⊥x, x) (respectively, (x, x⊥)) is a cotorsion theory and vice versa. We show that in certain cases, completeness can be transferred between these cotorsion theories. © 2013 World Scientific Publishing Company.
Archiv der Mathematik (0003889X) 100(3)pp. 231-239
Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore's conjecture: Assume that, for all x ∈ (Γ-Γ′), either there is an integer n such that 1 ≠ xn ∈ Γ′ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler's hierarchy LHF. In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index. © 2013 Springer Basel.
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.
Advances in Mathematics (10902082) 226(2)pp. 1096-1133
We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves on a semi-separated noetherian scheme, and study these complexes using the pure derived category of flat quasi-coherent sheaves. We prove that a scheme is Gorenstein if and only if every acyclic complex of flat quasi-coherent sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jørgensen's proof of the existence of Gorenstein projective precovers. © 2010 Elsevier Inc.
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
Journal of Pure and Applied Algebra (00224049) 207(1)pp. 99-108
Let (R, m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If the GC-dimension of M / a M is finite for all ideals a generated by an R-regular sequence of length at most d - t then either the GC-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given. © 2005 Elsevier Ltd. All rights reserved.
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.
Communications in Algebra (00927872) 34(5)pp. 1625-1630
In this article, by using the theory of Gorenstien dimension, we study the uniform annihilation theorem for local cohomology modules over a (not necessary finite dimensional) Noetherian Gorenstein ring.
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.
Proceedings of the American Mathematical Society (10886826) 132(8)pp. 2215-2220
In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.
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.
Proceedings of the American Mathematical Society (10886826) 131(8)pp. 2329-2335
Let R be a commutative Noetherian ring with nonzero identity and let the injective envelope of R be flat. We characterize these kinds of rings and obtain some results about modules with nonzero injective cover over these rings.
Communications in Algebra (00927872) 30(10)pp. 4821-4826
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.
Kyoto Journal of Mathematics (0023608X) 39(4)pp. 607-618
Communications in Algebra (00927872) 27(12)pp. 6191-6198
Nagoya Mathematical Journal (21526842) 151pp. 37-50
The first part of the paper is concerned, among other things, with a characterization of filter regular sequences in terms of modules of generalized fractions. This characterization leads to a description, in terms of generalized fractions, of the structure of an arbitrary local cohomology module of a finitely generated module over a notherian ring. In the second part of the paper, we establish homomorphisms between the homology modules of a Koszul complex and the homology modules of a certain complex of modules of generalized fractions. Using these homomorphisms, we obtain a characterization of unconditioned strong d-sequences.
Acta Mathematica Hungarica (15882632) 81(1-2)pp. 109-119
Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0:A I = 0:A(x1,..., xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].
Journal of the Korean Mathematical Society (03049914) 34(4)pp. 949-957
The purpose of this paper is to establish connection between certain complex of modules of generalized fractions and the concept of cosequence in commutative algebra. The main theorem of the paper leads to characterization, in terms of modules of generalized fractions, of regular (co) sequences.
