Algebraic K-theory of endomorphism rings

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let Y\ra X be a covariant morphism of objects in ${\mathcal C}$. Then $K_n\big(_{\mathcal C}(X\oplus Y)\big)\simeq K_n\big(_{{\mathcal C},Y}(X)\big)\oplus K_n\big(_{\mathcal C}(Y)\big)$ for all $1\le n\in \mathbb{N}$, where $_{{\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factorize through $Y$. Moreover, let $R$ be a ring with identity, and let $e$ be an idempotent element in $R$. If $J:=ReR$ is homological and $_RJ$ has a finite projective resolution by finitely generated projective $R$-modules, then $K_n(R)\simeq K_n(R/J)\oplus K_n(eRe)$ for all $n\in \mathbb{N}$. This reduces calculations of the higher algebraic $K$-groups of $R$ to those of the quotient ring $R/J$ and the corner ring $eRe$, and can be applied to a large variety of rings: Standardly stratified rings, hereditary orders, affine cellular algebras and extended affine Hecke algebras of type $\tilde{A}$.Comment: 21 pages. Representation-theoretic methods are used to study the algebraic K-theory of ring

Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities

In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W=0.Comment: 26 pp., LaTe

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category $\mathscr{H}^{(\rho)}$ of semi-stable coherent sheaves of a fixed slope $\rho$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(\rho)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.Comment: 29 page

Krull Dimension of Tame Generalized Multicoil Algebras

We determine the Krull dimension of the module category of finite dimensional tame generalized multicoil algebras over an algebraically closed field, which are domestic

Recollements of Module Categories

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.Comment: Comments are welcom

Categorical Tinkertoys for N=2 Gauge Theories

In view of classification of the quiver 4d N=2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to a N=2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite-dimensional) representations of the Jacobian algebra $\mathbb{C} Q/(\partial W)$ should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal `generic' subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. There is a family of 'light' subcategories $\mathscr{L}_\lambda\subset rep(Q,W)$, indexed by points $\lambda\in N$, where $N$ is a projective variety whose irreducible components are copies of $\mathbb{P}^1$ in one--to--one correspondence with the simple factors of G. In particular, for a Gaiotto theory there is one such family of subcategories, $\mathscr{L}_{\lambda\in N}$, for each maximal degeneration of the corresponding surface $\Sigma$, and the index variety $N$ may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed-point subcategories to `fixtures' (spheres with three punctures of various kinds) and higher-order generalizations. The rules for `gluing' categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N=2 theories which are not of the Gaiotto class.Comment: 142 pages, 8 figures, 5 table

Lifting and restricting recollement data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in "Parametrizing recollement data", allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a 'right bounded' derived category of a differential graded(=dg) category to be a recollement of 'right bounded' derived categories of dg categories. In Theorem 2 we consider the particular case in which those dg categories are just ordinary algebras.Comment: 29 page

Sounds of Silence : The Reflexivity, Self-decentralization, and Transformation Dimensions of Silence at Work

This article explores silence as a phenomenon and practice in the workplace through a Buddhist-enacted lens where silence is intentionally encouraged. It brings forward a reconsideration of the roles of silence in organizations by proposing emancipatory dimensions of silence—reflexivity, self-decentralization, and transformation. Based on 54 interviews with employees and managers in a Vietnamese telecommunications organization, we discuss the dynamic nature of silence, and the possible coexistence of the constructive and the oppressive aspects of silence in a workplace spirituality context. Instead of studying silence as one-dimensional, we call for an integrated view and argue that studying silence requires consideration of the multiplicity of its interconnected dimensions. By considering silence as a relational and emerging processes constructed around its vagueness and uncertainties, our study reveals the many possible ways silence is organized and organizes and sheds light on silence as a marker of the complexities and paradoxes of organizational life

Die Krull-Gabriel-Dimension der endlich praesentierten Funktoren ueber Artin-Algebren und ihre Anwendung auf exakte Sequenzen

SIGLECopy held by FIZ Karlsruhe; available from UB/TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman