Category Of Complexes Research Articles

Energetic features of antiparallel stacking and hydrogen bonding interactions in two coordination complexes bearing 1,10-phenanthroline-2,9-dicarboxylic acid

Two new metal–organic complexes, [Co(phen)(PDA)H2O]·CH3CN·H2O (1), (phen = 1,10-phenanthroline, PDA = 1,10-phenanthroline-2,9-dicarboxylate) and [Ce2(PDA)3(H2O)2]·3H2O (2) have been synthesized and characterized by elemental analysis, infrared spectroscopy, and single-crystal X-ray crystallography. Structural characterization by single-crystal X-ray diffraction shows that in 1 the cobalt ion forms a discrete seven-coordinated complex with a distorted pentagonal bipyramidal geometry around the metal centre. These discrete units are further packed into 3-D supramolecular assemblies by different kinds of the noncovalent interactions. Complex 2 consists of a 1D coordination polymeric structure with the cerium ion coordinated by PDA and water in the form of left-handed and right-handed helices. The extended aromatic systems of PDA and phen have a strong tendency to establish antiparallel π···π stacking interactions generating interesting assemblies in the solid state of 1. They have been studied theoretically using DFT calculations and characterized by means of the quantum theory of “atoms-in-molecules” (QTAIM) and the noncovalent interaction (NCI) plot methods. This study is useful to understand in more detail the energetic and structural stability of these categories of metal-organic complexes. The results reported herein suggest that the antiparallel π-stacking interactions involving the neutral and, in principle innocent, phen auxiliary ligand are energetically very relevant in the solid state of 1.

New model structures and projective (injective) cotorsion pairs

Let [Formula: see text] be either the category of R-modules or the category of chain complexes of R-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer n, and hence, one can get new algebraic triangulated categories. Second, it is shown that any n-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let M be an R-module with finite flat dimension and n a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then M is injective.

Singular Equivalences Induced by Bimodules and Quadratic Monomial Algebras

We investigate the problem when the tensor functor by a bimodule yields a singular equivalence. It turns out that this problem is equivalent to the one when the Hom functor given by the same bimodule induces a triangle equivalence between the homotopy categories of acyclic complexes of injective modules. We give conditions on when a bimodule appears in a pair of bimodules, that defines a singular equivalence with level. We construct an explicit bimodule in a combinatorial manner, which yields a singular equivalence between a quadratic monomial algebra and its associated algebra with radical square zero. Under certain conditions which include the Gorenstein cases, the bimodule does appear in a pair of bimodules defining a singular equivalence with level.

The homotopy type of elliptic arrangements

We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.

Morphisms in the infinite radical of the bounded derived category

We study sufficient conditions for a morphism in the category of complexes of fixed size Cn(projA), with n≥2 and A an artin algebra, to belong to the infinite radical of K−,b(projA). As an application we consider A a Nakayama gentle algebra, whose ordinary quiver is an oriented cycle and we prove that the irreducible morphisms in Cn(projA) are irreducible in K−,b(projA) or belong to the infinite radical of K−,b(projA). We characterize the irreducible morphisms between indecomposable complexes in Cn(projA) that belong to the infinite radical of K−,b(projA) for A a Nakayama gentle algebra not selfinjective. For the case of a Nakayama gentle selfinjective algebra, we characterize the irreducible morphisms between indecomposable complexes in Cn(projA) that belong to the infinite radical of the bounded homotopic category in terms of their left and right degrees.

Reducers and K 0 with support

Let R be a commutative noetherian ring, be a Serre subcategory of the category of finitely generated R-modules and be the category of finitely generated projective R-modules. We define invariants based on the categories of chain complexes from homologically supported in with support between 0 and n () and the K-group with support similar to the stable range in classical K-theory. We introduce a notion called a reducer which we use to express the class of a complex in terms of classes of complexes of smaller amplitude and use these to study and bound the above defined invariants by standard invariants like arithmetic rank, grade and projective dimension. We also give conditions for to be isomorphic to

Lie algebras arising from 1-cyclic perfect complexes

Let A be the path algebra of a Dynkin quiver Q over a finite field, and P be the category of projective A-modules. Denote by C1(P) the category of 1-cyclic complexes over P, and n˜+ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in C1(P). In this paper, we prove the existence of Hall polynomials in C1(P), and then establish a relationship between the Hall numbers for indecomposable objects therein and those for A-modules. Using Hall polynomials evaluated at 1, we define a Lie bracket in n˜+ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if Q is bipartite, n˜+ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type An, n˜+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different n˜+. We also characterize the Lie algebras n˜+ by generators and relations. When Q is of type A, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable A-module.

CONNECTIVITY AND PURITY FOR LOGARITHMIC MOTIVES

AbstractThe goal of this article is to extend the work of Voevodsky and Morel on the homotopyt-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel’s connectivity theorem and show a purity statement for$({\mathbf {P}}^1, \infty )$-local complexes of sheaves with log transfers. The homotopyt-structure on${\operatorname {\mathbf {logDM}^{eff}}}(k)$is proved to be compatible with Voevodsky’st-structure; that is, we show that the comparison functor$R^{{\overline {\square }}}\omega ^*\colon {\operatorname {\mathbf {DM}^{eff}}}(k)\to {\operatorname {\mathbf {logDM}^{eff}}}(k)$ist-exact. The heart of the homotopyt-structure on${\operatorname {\mathbf {logDM}^{eff}}}(k)$is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn-Saito-Yamazaki and Rülling.

Model Structures on the Category of Complexes of Quiver Representations

In this paper, we study the category \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) of complexes of representations of quiver \(\mathcal {Q}\) with values in an abelian category \(\mathfrak {A}\). We develop a method for constructing some model structures on \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of \(\mathcal {Q}\) in \(\mathrm{Mod\hbox {-}}R\).

Windows for cdgas

We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the category of perfect complexes on the two GIT quotients. More generally, we obtain semi-orthogonal decompositions which show that these categories differ by a certain number of copies of the derived category of the derived fixed locus. The derived equivalence for the Mukai flop is recovered as a very special case.

Revisiting the de Rham-Witt complex

The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p$. We introduce a category of cochain complexes equipped with an endomorphism $F$ (of underlying graded abelian groups) satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L \eta_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison in $A \Omega$-cohomology theory.

The Recreational Capacity of the Zones of the Ukrainian National Natural Parks

Purpose. To determine the recreational capacity of the territories of several National Natural Parks (NNPs) of Ukraine, as well as the recreational capacity of the tourist trails of these parks. Methods. The methodological basis of the study is based on a critical analysis of the current approaches to the assessment of a recreational facilities of the Ukrainian NNPs. Various published data sources as well as the data, regarding to the recreational renewal capacity of the territories of the NNPs acquired during own research was used. Results. The area of the reserved core that is larger than a one third of the entire area of the park may be seen in only few parks. Most of the NPPs have an insufficient size of the protected areas. This indicate an unstable and severely disturbed ecological conditions, which threatens the conservation of the biological diversity in these parks. That’s why the increase of the protected areas of these NNPs is highly advisable. In addition to all above, the area of the NNP special use zone should be smaller than the area of the regulated recreational area, which plays a certain environmental role; it can be decreased in a favor of the stationary recreational zone if needed. Unfortunately, these conditions are not met for the most of the studied NPPs. That’s why it is advisable to optimize the functional zoning of the NNPs, which will effectively protect the unique natural complexes and will let use them for the recreational purposes. The capacity, the degree of the resistance to the recreational loads, the level of improvement and the capacity of the tourist routes in each of the spoken NNPs were analyzed. The main criteria for choosing the studied NNPs were their representativeness or representation in terms of a variety of the accommodating regions, the availability of the data based on the physical and geographical characteristics, the length of the tourist routes and the statistcs of the visitors per year. The recreational capacity for the summer season of some NPPs of Ukraine was calculated, while taking into account the standards for each type of the natural complex for the zones of stationary and regulated recreation. Potentially, these zones may be expanded replacing the parts of the reserved special use zone. An assessment of the capacity of some tourist trails of the barely improved NPPs for an unstable and stable category of natural landscape complexes was given. Conclusions. The recreational potential of the researched Ukrainian NPPs is currently underutilized, but with the increase in the flow of the recreants, the recreational areas that are poorly equipped as well as the trails may lead to a deterioration in the state of the natural ecosystems. The functional zones of the stationary and of the regulated parts of the recreational areas of the NNPs may be expanded by migrating some types of the recreational activities to the special use zone of the parks. It is vital to expand the list of the recreational services for recreants, to strive the level of improvement of the tourist trails of the NNPs. If the natural healing resources are present on the territories of the NNPs, it is advisable to organize some centers for the sanatorium treatment, medical and socio-psychological rehabilitation, which will contribute to the creation and an improvement of the medical and the health infrastructure, the activation of recreational activities and the increase in the efficiency of the functioning of the NNP.

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a family of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.

The resolution of the universal Abel map via tropical geometry and applications

Let g and n be nonnegative integers and A=(a0,…,an) a sequence of n+1 integers summing up to d. Let M‾g,n+1 be the moduli space of (n+1)-pointed stable curves of genus g and J‾μ,g→M‾g,1 be the Esteves' universal Jacobian, where μ is a universal genus-g polarization of degree d. We give an explicit resolution of the universal Abel map αA,μ:M‾g,n+1⇢J‾μ,g, taking a pointed curve (X,p0,…,pn) to OX(∑0≤i≤naipi). The blowup of M‾g,n+1 giving rise to the resolution is inspired by the resolution of the tropical analogue of the map αA,μ (in the category of generalized cone complexes). As an application, we describe the double ramification cycle in terms of the universal sheaf inducing the resolution of the map αA,μ.

Algebraic K-theory of quasi-smooth blow-ups and cdh descent

We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason's blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodsky's cdh topology, which we use to give a direct proof of Cisinski's theorem that Weibel's homotopy invariant K-theory satisfies cdh descent.

The acyclic closure of an exact category and its triangulation

For any exact category A with splitting idempotents, a maximal exact category T(A) containing A as a biresolving subcategory, is constructed. Important types of exact categories, including n-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form T(A). The quotient category T(A)/A in the sense of Grothendieck always exists and carries a triangulated structure. More generally, it is proved that any biresolving subcategory A of an exact category M gives a triangulated localization M/A. For example, the unbounded derived category D(A) of an exact category A is obtained directly, with no passage through a homotopy category of complexes. As applications, some recent developments related to Gorenstein projectivity, non-commutative crepant resolutions, singularity categories, and Cohen-Macaulay representations are extended and improved in the new framework. For example, the concept of non-commutative resolution of a noetherian Frobenius category is extended to arbitrary exact categories, which leads to an overarching connection with representation dimension of exact categories and n-tilting.

Free [formula omitted]-complexes and p-DG modules

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring Fp[G] of an elementary abelian p-group G in terms of commutative algebra. This extends results of Carlsson for p=2 to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over Fp[G] into the derived category of p-DG modules over a polynomial ring.

On the Digital Cohomology Modules

In this paper, we consider the digital cohomology modules of a digital image consisting of a bounded and finite subset of Zn and an adjacency relation. We construct a contravariant functor from the category of digital images and digital continuous functions to the category of unitary R-modules and R-module homomorphisms via the category of cochain complexes of R-modules and cochain maps, where R is a commutative ring with identity 1R. We also examine the digital primitive cohomology classes based on digital images and find the relationship between R-module homomorphisms of digital cohomology modules induced by the digital convolutions and digital continuous functions.

A Model Structure on Categories Related to the Categories of Complexes

We prove a Hinich-type theorem on the existence of a model structure on a category related by adjunction to the category of differential graded modules over a graded commutative ring.

Relative projectivity and the Green correspondence for complexes

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group G and the same for a subgroup H, often the normalizer of a p-subgroup of G. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.