On classes defining a homological dimension

A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{homological dimension} if for any object in $\mathcal A$ the length of any $\mathcal F$-resolution is uniquely determined. In the present paper we investigate classes satisfying this property.Comment: to appear in Contribution to Module Theory, de Gruyter 200

Derived dualities induced by a 1-cotilting bimodule

In this paper we characterize the modules and the complexes involved in the dualities induced by a 1-cotilting bimodule in terms of a linear compactness condition. Our result generalizes the classical characterization of reflexive modules with respect to Morita dualities. The linear compactness notion considered, permits us to obtain finiteness properties of the rings and modules involved

Reflexivity in Derived Categories

An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works

Pr\"ufer modules over Leavitt path algebras

Let $L_K(E)$ denote the Leavitt path algebra associated to the finite graph $E$ and field $K$. For any closed path $c$ in $E$, we define and investigate the uniserial, artinian, non-noetherian left $L_K(E)$-module $U_{E,c-1}$. The unique simple factor of each proper submodule of $U_{E,c-1}$ is isomorphic to the Chen simple module $V_{[c^\infty]}$. In our main result, we classify those closed paths $c$ for which $U_{E,c-1}$ is injective. In this situation, $U_{E,c-1}$ is the injective hull of $V_{[c^\infty]}$.Comment: 24 pages. Submitted for publication July 2017. Comments are welcome

Extensions of simple modules over Leavitt path algebras

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\infty$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[c^\infty]}$. Further, when $E$ is row-finite, for each irrational infinite path $p$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[p]}$. For Chen simple modules $S,T$ we describe ${\rm Ext}_{L_K(E)}^1(S,T)$ by presenting an explicit $K$-basis. For any graph $E$ containing at least one cycle, this description guarantees the existence of indecomposable left $L_K(E)$-modules of any prescribed finite length.Comment: updated: dedication to Alberto Facchini on the occasion of his 60th Birthday added in front matte

Pr\ufcfer modules over Leavitt path algebras

Let LK(E) denote the Leavitt path algebra associated to the finite graph E and field K. For any closed path c in E, we define and investigate the uniserial, artinian, non-Noetherian left LK(E)-module U_{E,c 121}. The unique simple factor of each proper submodule of U_{E,c 121}is isomorphic to the Chen simple module V_[c 1e]. In our main result, we classify those closed paths c for which U_{E,c 121} is injective. In this situation, U_{E,c 121} is the injective hull of V_[c 1e]

A classification theorem for t-structures

We give a classification theorem for a relevant class of t-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of t-structures whose hearts have at most nfixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the t-tree, a new technique which generalizesthe filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by Happel, Reiten and Smal\uf8. The last section provides applications to classical n-tilting objects, examples of t-trees for modules over a path algebra, and new developments on compatible t-structures

Injective modules over the Jacobson algebra K<X,Y∣XY=1>K< X,Y | XY = 1 >

For a field $K$, let $\mathcal{R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle$. We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal{R}$-modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal{R}$. Our approach involves realizing $\mathcal{R}$ up to isomorphism as the Leavitt path $K$-algebra of an appropriate graph $\mathcal{T}$, which thereby allows us to utilize important machinery developed for that class of algebras.Comment: 16 page