Piecewise Hereditary Algebras Research Articles

Algebra extensions and derived-discrete algebras

Let be an algebra homomorphism between finite dimensional algebras. We prove that if is split, the derived-discreteness of A implies the derived-discreteness of B; if is separable and the right A-module BA is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.

Sections in the bounded derived category of piecewise hereditary algebras

Let [Formula: see text] be a finite-dimensional algebra over an algebraically closed field. We study sections in the Auslander–Reiten quiver of the categories of complexes of fixed size and in the Auslander–Reiten quiver of the bounded derived category. One of the aims of this work is to show where the indecomposable [Formula: see text]-modules can be found in the Auslander–Reiten quiver of the derived category, whenever [Formula: see text] is a piecewise hereditary algebra. We also prove that we can obtain a transjective component of the bounded derived category from a section. As an application of the results of sections, we find a bound on the strong global dimension of some piecewise hereditary algebras of tree type. Moreover, we also determine the strong global dimension of some algebras taking into account their ordinary quivers with relations.

Piecewise hereditary incidence algebras of Dynkin and extended Dynkin type

Let be the incidence algebra of a poset . We study some aspects of incidence algebras that are piecewise hereditary, which we denominate PHI algebras. We describe the quiver with relations of the PHI algebras of Dynkin type. Also we introduce a new family of PHI algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowroński. In this description, the important method was the one of cutting sets on trivial extensions. We implemented this method in the computer algebra system GAP/QPA to shows exactly the quivers with relations resulted.

$(m,n)$-Quasitilted and $(m,n)$-almost hereditary algebras

Motivated by the study of $(m,n)$-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension 2 by a sequence of (co)tiltings involving $n-1$ tilting modules and $m-1$ cotilting modules, we intr

The strong global dimension of piecewise hereditary algebras

Let A be a finite-dimensional piecewise hereditary algebra over an algebraically closed field. This text investigates the strong global dimension of A. This invariant is characterised in terms of the lengths of sequences of tilting mutations relating A to a hereditary abelian category, in terms of the generating hereditary abelian subcategories of the derived category of A, and in terms of the Auslander–Reiten structure of that derived category.

Brauer-Thrall Type Theorems for Derived Module Categories

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of a complex (an algebra) are introduced. Cohomological range leads to the concepts of derived bounded algebra and strongly derived unbounded algebra naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and the algebras of finite global cohomological length respectively.

Derived Brauer–Thrall Type Theorem I for Artin Algebras

We define the global cohomological range for artin algebras, and define the derived bounded algebras to be the algebras with finite global cohomological range, then we prove the first Brauer–Thrall type theorem for bounded derived categories of artin algebras, i.e., derived bounded algebras are precisely the derived finite algebras. Moreover, the main theorem establishes that the derived bounded artin algebras are just piecewise hereditary algebras of Dynkin type, and can be also characterized as those artin algebras with derived dimension zero, which can be regarded as a generalization of the results of Han–Zhang [11, Theorem 1] and Chen–Ye–Zhang [4, Theorem] in the context of finite-dimensional algebras over algebraically closed fields, respectively.

FINITISTIC DIMENSIONS AND PIECEWISE HEREDITARY PROPERTY OF SKEW GROUP ALGEBRAS

AbstractLet Λ be a finite-dimensional algebra andGbe a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete setEof primitive orthogonal idempotents, closed under the action of a Sylowp-subgroupS≤G. If the action ofSonEis free, we show that the skew group algebra ΛGand Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛSis a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for ΛGto be piecewise hereditary.

Jordan–Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.

Formula omitted]-structures via recollements for piecewise hereditary algebras

Following the work of Beilinson, Bernstein and Deligne, we study restriction and induction of t -structures on triangulated categories with respect to recollements. For derived categories of piecewise hereditary algebras we give a necessary and sufficient condition for a bounded t -structure to be induced from recollements by derived categories of algebras. As a corollary we prove that for hereditary algebras of finite representation type all bounded t -structures can be obtained in this way.

Combinatorial classification of piecewise hereditary algebras

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension.

Topological invariants of piecewise hereditary algebras

We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering whose group of automorphisms is free and depends only on the derived category of the algebra. As a corollary, we prove that the algebra is simply connected if and only if its first Hochschild cohomology vanishes.

Homological properties of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over an algebraically closed field k. We will investigate homological properties of piecewise hereditary algebras Λ. In particular we give lower and upper bounds of the strong global dimension, show the behavior of the strong global dimension under one point extensions and tilting. Moreover we show that the “pieces” of mod Λ have Auslander–Reiten sequences.

Skew group algebras of piecewise hereditary algebras are piecewise hereditary

We show that the main results of Happel–Rickard–Schofield (1988) and Happel–Reiten–Smalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if G is a finite group acting on a piecewise hereditary algebra A over an algebraically closed field whose characteristic does not divide the order of G , then the resulting skew group algebra A [ G ] is also piecewise hereditary.

A homological characterization of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite.

On the periodicity of Coxeter transformations and the non-negativity of their Euler forms

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as - A + - 1 A - t , where A + , A - are closely associated with the upper and lower triangular parts of A.

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

Derived and Stable Equivalence Classification of Twisted Multifold Extensions of Piecewise Hereditary Algebras of Tree Type

DERIVED PICARD GROUPS OF FINITE DIMENSIONAL HEREDITARY ALGEBRAS

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

Hochschild cohomology of piecewise hereditary algebras

Hochschild cohomology of piecewise hereditary algebras

Piecewise hereditary algebras

Piecewise hereditary algebras