Topological invariants of piecewise hereditary algebras
Patrick Le Meur
备注：
The hypotheses of the main theorem were modified: The next now deals mainly with piecewise hereditary algebras which are derived equivalent to a hereditary algebra (instead of all piecewise hereditary algebras in the previous version)
发表信息：
Transactions of the American Mathematical Society 363, 4 (2011) 2143--2170
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.
We show that the main results of Happel-Rickard-Schofield (1988) and Happel-Reiten-Smalo (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.
Let $KΔ$ be the incidence algebra associated with a finite poset $(Δ,\preceq)$ over the algebraically closed field $K$. We present a study of incidence algebras $KΔ$ that are piecewise hereditary, which we denominate PHI algebras. We investigate the strong global dimension, the simply conectedeness and the one-point extension algebras over a PHI algebras. We also give a positive answer to the so-called Skowroński problem for $KΔ$ a PHI algebra which is not of wild quiver type. That is for this kind of algebra we show that $HH^1(KΔ)$ is trivial if, and only if, $KΔ$ is a simply connected algebr…
We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia 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, inspired by this we made of a computer program which shows exactly the cutting sets on the given trivial extension that result on incidence algebras.
A Jordan Hölder theorem is established for derived module categories of piecewise hereditary algebras. 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.
We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.
Following the work of Beilinson, Bernstein and Deligne, we study restriction and induction of t-structures in 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 a recollement 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.
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.
Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global dimension, in the sense of Ringel, of the endomorphism algebra of T. This invariant is expressed using the infimum of the lengths of the sequences of tilting objects successively related by tilting mutations and where the last term is T and the endomorphism algebra of the first term is quasi-tilted. It is also expressed in terms of the hereditary abelian genera…
Let $Λ$ be a finite dimensional algebra and $G$ be a finite group whose elements act on $Λ$ as algebra automorphisms. Under the assumption that $Λ$ has a complete set $E$ of primitive orthogonal idempotents, closed under the action of a Sylow $p$-subgroup $S \leqslant G$. If the action of $S$ on $E$ is free, we show that the skew group algebra $ΛG$ and $Λ$ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra $Λ^S$ is a direct summand of the $Λ^S$-bimodule $Λ$. Using a homological characterization of piecewise hereditary algebras proved by Happel an…
Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting modules, we introduce (m,n)-almost hereditary algebras. These are the algebras with global dimension m+n and such that any indecomposable module has projective dimension at most m, or else injective dimension at most n. We relate these two classes of algebras, among which (m,1)-almost hereditary ones play a special role. For these, we prove that any indecomposabl…
Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for non-hereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In this paper we study the image if the natural functor from the bounded derived category to the cluster category, that is we investigate how far the orbit category is from being the cluster category. We show that for wide classes of non-piecewise hereditary algebras the orbit category is never equal to the clus…
We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. For a stably hereditary algebra or a gentle one-cycle algebra, these considerations show that the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated if and only if the algebra is piecewise hereditary.
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…
The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras 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 …
This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$ can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to …
