On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

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.     

Totally acyclic complexes over noetherian schemes

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.     

Characterizing local rings via homological dimensions and regular sequences

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 G_C-dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d−t then either the G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.     

Synthesis and characterization of a new nanomagnetic coordination composite from Fe<Subscript>3</Subscript>O<Subscript>4</Subscript> and Cu(II) complex: as an efficient catalyst in oxidation of benzyl alcohol

New nanomagnetic coordination compound, Cu(salal)@DA@ Fe₃O₄, was synthesized by bonding between the Cu(II) complex and Fe₃O₄ nanoparticles. The Cu complex has two aldehyde groups. The surface of Fe₃O₄ nanoparticles was modified by the dopamine molecules and the amine group of dopamine is free. Therefore, the Cu complex is covalently anchored to Fe₃O₄ nanoparticle by the formation of imine bonds between the aldehyde and amine groups. On the other hand, a Cu-Schiff base complex is immobilized on nano-Fe₃O₄ by dopamine as a bridge. Consequently, the homogeneous Cu complex is easily converted to the heterogeneous-magnetic compound in this project. The Cu complex, Fe₃O₄ and new nanocomposite were characterized by general techniques such as FTIR, TGA, XRD, FESEM, MAP and EDS. The average crystallite size of Fe₃O₄ and Cu(salal)@DA@ Fe₃O₄ were calculated by Scherrer’s formula and they are 18.52 and 24.69 nm, respectively. These results indicated the average crystallite size of Fe₃O₄ nanomaterials is slightly increased by surface modification by Cu complex. The FESEM images show a tiny spherical mushroom morphology for both nanocompounds, and the EDS analysis confirms the presence of Fe, Cu, C and O in the nanomagnetic coordination composite. The catalytic properties of these compounds were studied and compared to oxidation of benzyl alcohol by 30% H₂O₂ at room temperature. The results show that the catalytic properties Cu complex and Fe₃O₄ were enhanced by cooperation of both compounds in a nanocoordination composite.     

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Algebras and Representation Theory - Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this...     

Cohomology theories based on Gorenstein injective modules

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.

     

Approximation of fuzzy integrals using fuzzy bernstein polynomials

In this paper, we approximate the integration of continuous fuzzy real number valued function of one and two variables. To do this, we use Bernstein-type fuzzy polynomials. Moreover, we obtain the error estimates for these approximations in terms of the modulus of continuity.     

Cohomology theories for complexes

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.