Auslander-Reiten duality for subcategories

Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for $\mathcal{P}^{<\infty}(\Lambda)$, the subcategory consisting of finitely generated modules with finite projective dimension over an artin algebra $\Lambda$, and also, the subcategory of Gorenstein projectove modules of \rm{mod}\mbox{-}\Lambda, denoted by \rm{Gprj}\mbox{-}\Lambda. In this paper, we give a method to compute the Auslander-Reiten translation in $\mathcal{P}^{<\infty}(\Lambda)$ whenever $\Lambda$ is a $1$-Gorenstein algebra. In addition, we characterize when the Auslander-Reiten translation in \rm{Gprj}\mbox{-}\Lambda is the first syzygy and provide many algebras having such property.Comment: A major revision, Title, abstract and introduction completely changed, adding some new result

From subcategories to the entire module categories

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$.Comment: Accepted for publication in Forum Mathematicu

Measuring topological invariants in photonic systems

Motivated by the recent theoretical and experimental progress in implementing topological orders with photons, we analyze photonic systems with different topologies and present a scheme to probe their topological features. Specifically, we propose a scheme to modify the boundary phases to manipulate edge state dynamics. Such a scheme allows one to measure the winding number of the edge states. Furthermore, we discuss the effect of loss and disorder on the validity of our approach.Comment: 5 pages, 5 figure

On relative Auslander algebras

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent

On the derived dimension of abelian categories

We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that if \CX is a sufficiently nice subcategory of an abelian category, then derived dimension of \CA is at most \CX-dim\CA, provided \CX-dim\CA is greater than one. We provide some applications

Stability of Fractional Quantum Hall States in Disordered Photonic Systems

The possibility of realizing fractional quantum Hall liquids in photonic systems has attracted a great deal of interest of late. Unlike electronic systems, interactions in photonic systems must be engineered from non-linear elements and are thus subject to positional disorder. The stability of the topological liquid relies on repulsive interactions. In this paper we investigate the stability of fractional quantum Hall liquids to impurities which host attractive interactions. We find that for sufficiently strong attractive interactions these impurities can destroy the topological liquid. However, we find that the liquid is quite robust to these defects, a fact which bodes well for the realization of topological quantum Hall liquids in photonic systems.Comment: 11 pages, 5 figure

Homotopy category of N-complexes of projective modules

In this paper, we show that the homotopy category of N-complexes of projective R-modules is triangle equivalent to the homotopy category of projective T_{N-1}(R)- modules where T_{N-1}(R) is the ring of triangular matrices of order N-1 with entries in R. We also define the notions of N-singularity category and N-totally acyclic complexes. We show that the category of N-totally acyclic complexes of finitely generated projective R-modules embeds in the N-singularity category, which is a result analogous to the case of ordinary chain complexes.Comment: 2

Homotopy category of projective complexes and complexes of Gorenstein projective modules

Let $R$ be a ring with identity and \C(R) denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted \KPC, is always well generated and is compactly generated provided \KPR is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor \iota : \K(\RGPrj)\hookrightarrow \KR has a right adjoint $\iota_{\rho}$, where \K(\RGPrj) is the homotopy category of Gorenstein projective $R$ modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If $R$ has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories \K(\RGPrj) and \K(\RGInj), that restricts to an equivalence between \KPR and \KIR, whenever $R$ is commutative, noetherian and admits a dualising complex

Derived equivalences of functor categories

Let \Mod \CS denote the category of \CS-modules, where \CS is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for \Mod \CS. This will have several interesting applications. In the second part, we apply our techniques to get some interesting recollements of derived categories in different levels. We specialize our results to path rings as well as graded rings

Recollements of Cohen-Macaulay Auslander algebras and Gorenstein derived categories

Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a \mathbb{D}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} ) level recollement, writing $A$ in terms of $B$ and $C$, then we get a \mathbb{D}^-({\rm{Mod\mbox{-}}} ) level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we show that being Gorenstein as well as being of finite Cohen-Macaulay type, are invariants with respect to \mathbb{D}^{{\rm{b}}}_{{{\mathcal{G}p}}}({\rm{{mod\mbox{-}}}}) level recollements of virtually Gorenstein algebras, where $\mathbb{D}^{{\rm{b}}}_{{{\mathcal{G}p}}}$ denotes the Gorenstein derived category.Comment: This paper has been withdrawn by the author due to some mistake