Classification of abelian hereditary directed categories satisfying Serre duality

In an ongoing project to classify all hereditary abelian categories, we provide a classification of -finite directed hereditary abelian categories satisfying Serre duality up to derived equivalence.

In order to prove the classification, we will study the shapes of Auslander-Reiten components extensively and use appropriate generalizations of tilting objects and coordinates, namely partial tilting sets and probing of objects by quasi-simples.

     

Serre duality for noncommutative projective schemes

We prove the Serre duality theorem for the noncommutative projective scheme when is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

     

Torsion pairs in recollements of abelian categories

For a recollement (A,B, C) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in B; and the converse holds true under certain conditions.     

Gorenstein global dimension of recollements of abelian categories

Abstract

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras.     

The quiver of projectives in hereditary categories with Serre duality

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to for a so-called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form ZQ.     

The Serre duality theorem for a non-compact weighted CR manifold

It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented.

     

From triangulated categories to abelian categories: cluster tilting in a general framework

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.      

Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

From recollement of triangulated categories to recollement of abelian categories

In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T)  T,j j (T) T.     

半群环的右Noether性

研究了一般半群的右Noetber性，得到了几个刻划定理．     

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors ${\gamma }_W$ with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then ${\gamma }_W$ coincides with the right derived functor ${\mathrm{R}\mathrm{\Gamma }}_W$ of the section functor ${\mathrm{\Gamma }}_W$ with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for ${\gamma }_W$ with W being an arbitrary subset.     

Hereditary triangular matrix comonads

We recognize Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less than or equal to 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Construction and enumeration of left dihedral codes satisfying certain duality properties

Let ${\mathbb{F}}_q$ be the finite field of q elements and let $D_{2n}=〈x,y|x^n=1,y^2=1,yxy=x^{n?1}〉$ be the dihedral group of 2n elements. Left ideals of the group algebra ${\mathbb{F}}_q[D_{2n}]$ are known as left dihedral codes over ${\mathbb{F}}_q$ of length 2n, and abbreviated as left $D_{2n}$-codes. Let $\gcd (n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over ${\mathbb{F}}_q$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over ${\mathbb{F}}_q$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over ${\mathbb{F}}_q$, and present several numerical examples to illustrative our applications.     

Quotients of partial abelian monoids and the Riesz decomposition property

Partial abelian monoids (PAMs) are structures (), where is a partially defined binary operation with domain , which is commutative and associative in a restricted sense, and 0 is the neutral element. PAMs with the Riesz decomposition properties and binary relations with special properties on PAMs are studied. Relations with abelian groups, dimension equivalence and K ₀ for AF C^*-algebras are discussed. Received September 17, 2000; accepted in final form March 13, 2002.     

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor

This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.