Category Of Complexes Research Articles

Rouquier dimension versus global dimension

We give an example of a commutative coherent ring of infinite global dimension such that the category of perfect complexes has finite Rouquier dimension.

TCR3d 2.0: expanding the T cell receptor structure database with new structures, toolsand interactions.

Recognition of antigens by T cell receptors (TCRs) is a key component of adaptive immunity. Understanding the structures of these TCR interactions provides major insights into immune protection and diseases, and enables design of therapeutics, vaccinesand predictive modeling algorithms. Previously, we released TCR3d, a database and resource for structures of TCRs and their recognition. Due to the growth of available structures and categories of complexes, the content of TCR3d has expanded substantially in the past 5 years. This expansion includes new tables dedicated to TCR mimic antibody complex structures, TCR-CD3 complexesand annotated Class I and II peptide-MHC complexes. Additionally, tools are available for users to calculate docking geometries for input TCR and TCR mimic complex structures. The core tables of TCR-peptide-MHC complexes have grown by 50%, and include binding affinity data for experimentally determined structures. These major content and feature updates enhance TCR3d as a resource for immunology, therapeuticsand structural biology research, and enable advanced approaches for predictive TCR modeling and design. TCR3d is available at: https://tcr3d.ibbr.umd.edu.

Unearthing the Real-Time Excited State Dynamics from Antenna to Rare Earth Ions Using Ultrafast Transient Absorption.

The conventional energy transfer pathway in organic lanthanide complexes is purported to be from the excited singlet state of the chromophore to the triplet state and subsequently directly to the emitting state of the trivalent lanthanide ion. In this work, we found that the energy transfer occurs from the triplet state to the nearest energy level, instead of directly to the emitting state of the lanthanide ion. The triplet decay rate for different lanthanide ions follows an energy gap law from the triplet level to the receiving level of the lanthanide ion. Three different categories of complexes were synthesized and inspected using different techniques, demonstrating the universality of our findings. This work renews the insights to conventional findings, highlighting the importance of the energy gap between the triplet state and the nearest lanthanide energy level in optimization of light harvesting. The rationale of ligand design of chromophores should be reconsidered, leading to various applications of lanthanide complexes with enhanced quantum yield and brightness.

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

Derived category of weak chain [formula omitted]-complexes

In this paper, we define the derived category of weak chain U-complexes, and we give a characterization of any weak chain U-complex as an object in the right bounded homotopy category of weak chain U-complexes of projective modules. Then, we show that the homology functor between the category of chain complexes and weak chain U-complexes is a natural isomorphism by constructing an adjoint pair between these two categories.

The N-stable category

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N-complexes, one must find an appropriate candidate for the N-analogue of the stable category. We identify this “N-stable category” via the monomorphism category and prove Buchweitz’s theorem for N-complexes over a Grothendieck abelian category. We also compute the Serre functor on the N-stable category over a self-injective algebra and study the resultant fractional Calabi–Yau properties.

Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories, and use these to study model structures on categories of chain complexes C h ∗ ( E ) Ch_{*}(\mathcal {E}) which are induced by cotorsion pairs on E \mathcal {E} . As a special case we show that under very general conditions the categories C h + ( E ) Ch_{+}(\mathcal {E}) , C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) , and C h ( E ) Ch(\mathcal {E}) are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When E \mathcal {E} is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which C h ( E ) Ch(\mathcal {E}) and C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) are homotopical algebra contexts, thus making them suitable settings for derived geometry.

K-flatness in Grothendieck categories: application to quasi-coherent sheaves

AbstractLet $$(\mathcal {G},\otimes )$$ ( G , ⊗ ) be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of $$K(\mathcal {G})$$ K ( G ) by the K-flat complexes is always a well generated triangulated category. Under the further assumption that $$\mathcal {G}$$ G has a set of $$\otimes$$ ⊗ -flat generators we can show more: (i) The category is in recollement with the $$\otimes$$ ⊗ -pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of $$\otimes$$ ⊗ -pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.

The Auslander-Reiten quiver of perfect complexes for a self-injective algebra

We consider the homotopy category of perfect complexes for a finite dimensional self-injective algebra over a field, identifying many aspects of perfect complexes according to their position in the Auslander-Reiten quiver. Short complexes lie close to the rim. We characterize the position in the quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and truncated projective resolutions. We describe completely the quiver components that contain projective modules (complexes of length 1). We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We show that homology stabilizes, in a certain sense, away from the rim of the quiver.

Bounded $t$-structures on the category of perfect complexes

Bounded $t$-structures on the category of perfect complexes

Structure, stability, reactivity and bonding in noble gas compounds.

Noble gases (Ngs) are recognized as the least reactive elements due to their fully filled valence electronic configuration. Their reluctance to engage in chemical bond formation necessitates extreme conditions such as low temperatures, high pressures, and reagents with high reactivity. In this Perspective, we discuss our endeavours in the theoretical prediction of viable Ng complexes, emphasizing the pursuit of synthesizing them under nearly ambient conditions. Our research encompasses various bonding categories of Ng complexes and our primary aim is to comprehend the bonding mechanisms within these complexes, utilizing state-of-the-art theoretical tools such as natural bond orbital, energy decomposition, and electron density analyses. These complex types manifest distinct bonding scenarios. In the non-insertion type, the donor-acceptor interaction strength hinges on the polarizing ability of the binding atom, drawing the electron density of the Ng towards itself. In certain instances, especially with heavier Ng elements, this interaction reaches a magnitude where it can be considered a covalent bond. Conversely, in most insertion cases, the Ng prefers to share electrons to form a covalent bond on one side while interacting electrostatically on the other side. In rare cases, both bonds may be portrayed as electron-shared covalent bonds. Furthermore, a host cage serves as an excellent platform to explore the limits of achieving Ng-Ng bonds (even for helium), under high pressure.

The Auslander-Reiten quiver of the category of m−periodic complexes

Let A be an additive k-category and C≡m(A) be the category of m-periodic complexes. For any integer m>1, we study conditions under which the compression functor Fm:Cb(A)→C≡m(A) preserves or reflects irreducible morphisms. Moreover, we find sufficient conditions for the functor Fm to be a Galois G-covering in the sense of [3]. If in addition A is a dualizing category and modA has finite global dimension then C≡m(A) has almost split sequences. In particular, for a finite dimensional algebra A with finite strong global dimension we determine how to build the Auslander-Reiten quiver of the category C≡m(projA). Furthermore, we study the behavior of sectional paths in C≡m(projA), when A is a finite dimensional algebra over a field k.

On K-absolutely pure complexes

In this paper, we examine the class of K-absolutely pure complexes. These are the complexes which are right orthogonal in the homotopy category K(R) to the acyclic complexes of pure-projective modules. The class K-abspure of these complexes is preenveloping in K(R); in fact, a Bousfield localization exists for the embedding K-abspure⊆K(R) and the quotient K(R)/K-abspure is equivalent to the homotopy category of acyclic complexes of pure-projective modules. We examine the role of K-absolutely pure complexes in the pure derived category Dpure(R) and show that K-abspure is the isomorphic closure of the class of K-injective complexes therein. We explore the relevance of strongly fp-injective modules in the study of K-absolutely pure complexes and characterize the K-absolutely pure complexes of strongly fp-injective modules. Finally, we show that a K-absolutely pure complex of strongly fp-injective modules admits a K-injective complex of injective modules as a K(PInj)-preenvelope, in the case where the ring is left coherent. The notion of K-absolute purity is dual to the notion of K-flatness in the homotopy category, in a way analogous to the duality between (strongly) fp-injectivity and flatness in the module category.

Chain duality for categories over complexes

We show that the additive category of chain complexes parametrized by a finite simplicial complex K forms a category with chain duality. This fact, never fully proven in the original reference (Ranicki, 1992), is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincaré duality to global Poincaré duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on K -based chain complexes.

Generalized tilting theory in functor categories

AbstractThis paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories $\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$ with $\mathcal{C}$ an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory $\mathcal{T}$ of $\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$ is constructed. Some applications of these two results include the equivalence of Grothendieck groups $K_0(\mathcal{C})$ and $K_0(\mathcal{T})$ , the existences of a new abelian model structure on the category of complexes $\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$ , and a t-structure on the derived category $\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$ .

Equivariant [formula omitted]-modules for the cyclic group C2

For the cyclic group C2 we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring Z/ℓ_, for ℓ a prime. This is fairly simple for ℓ odd, but for ℓ=2 depends on a new splitting theorem. As corollaries of the splitting theorem we compute the associated Picard group and the Balmer spectrum for compact objects in the derived category, and we obtain a complete classification of finite modules over the C2-equivariant Eilenberg–MacLane spectrum HZ/2_. We also use the splitting theorem to give new and illuminating proofs of some facts about RO(C2)-graded Bredon cohomology, namely Kronholm's freeness theorem [12,11] and the structure theorem of C. May [13].

Irreducible morphisms in the bounded derived category via the category of complexes of fixed size

We study which irreducible morphisms in the category of complexes of fixed size Cn(projA) with n≥2 and A an artin algebra, are irreducible in C[0,n](projA) or either in Cn+1(projA). Moreover, we determine which of them are also irreducible in the bounded derived category.

The Balmer spectrum of certain Deligne–Mumford stacks

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by the action of a finite group $G$ and show that the Balmer spectrum of the tensor triangulated category of perfect complexes on $X$ is homeomorphic to the space of homogeneous prime ideals in the group cohomology ring $H^*(G,A)$.

English

We investigate the triangulated hull of the orbit categories of the perfect derived category and the bounded derived category of a ring concerning the power of the suspension functor. It turns out that the triangulated hull will correspond to the full subcategory of compact objects of certain triangulated categories of periodic complexes. This specializes to Stai and Zhao's result when the ring is a finite dimensional algebra with finite global dimension over a field. As the first application, if $A,B$ are flat algebras over a commutative ring and they are derived equivalent, then the corresponding derived categories of $n$-periodic complexes are triangle equivalent. As the second application, we get the periodic version of the Koszul duality.

Reconstruction of Projective Curves from the Derived Category

We prove that integral projective curves with Cohen–Macaulay singularities over an algebraically closed field of characteristic zero are determined by their triangulated category of perfect complexes. This partially extends a theorem of Bondal and Orlov to the case of singular projective varieties and in particular shows that all integral complex projective curves are determined by their category of perfect complexes.