On the finitistic dimension conjecture II: Related to finite global dimension

In this paper, we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras, on the one hand, and the finitistic dimensions of their subalgebras, on the other hand. This is a continuation of the project in [J. Pure Appl. Algebra 193 (2004) 287-305]. For an artin algebra A we denote by gl.dim(A), fin.dim(A) and rep.dim(A) the global dimension, finitistic dimension and representation dimension of A, respectively. The Jacobson radical of A is denoted by rad(A). The main results in the paper are as follows: Let B be a subalgebra of an artin algebra A such that rad(B) is a left ideal in A. Then (1) if gl.dim(A)?4 and rad(A)=rad(B)A, then fin.dim(B)<∞. (2) If rep.dim(A)?3, then fin.dim(B)<∞. The results are applied to pullbacks of algebras over semi-simple algebras. Moreover, we have also the following dual statement: (3) Let ?:B?A be a surjective homomorphism between two algebras B and A. Suppose that the kernel of ? is contained in the socle of the right B-module B_B. If gl.dim(A)?4, or rep.dim(A)?3, then fin.dim(B)<∞. Finally, we provide a class of algebras with representation dimension at most three: (4) If A is stably hereditary and rad(B) is an ideal in A, then rep.dim(B)?3.     

On the composite of irreducible morphisms in almost sectional paths

We study here when the composite of n irreducible morphisms in almost sectional paths is non-zero and lies in ℜⁿ⁺¹.     

Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6.     

Support variety theory is invariant under singular equivalences of Morita type

Using a new equivalent definition of support varieties in the sense of Snashall and Solberg [23], we show that both the (Fg) condition and support varieties are preserved under singular equivalences of Morita type. In particular, support variety theory is invariant under stable equivalences of Morita type.     

On the asymptotics of numbers of observations in random regions determined by order statistics

In this paper, we consider random variables counting numbers of observations that fall into regions determined by extreme order statistics and Borel sets. We study multivariate asymptotic behavior of these random variables and express their joint limiting law in terms of independent multinomial and negative multinomial laws. First, we give our results for samples with deterministic size; next we explain how to generalize them to the case of randomly indexed samples.     

The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra S_n in the title which is an algebra obtained from a polynomial algebra P_n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P_n. The algebra S_n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S_n is 2n; but the weak and the global dimensions of S_n are n. The prime and maximal spectra of S_n are found, and the simple S_n-modules are classified. It is proved that the algebra S_n is central, prime, and catenary. The set I_n of idempotent ideals of S_n is found explicitly. The set I_n is a finite distributive lattice and the number of elements in the set I_n is equal to the Dedekind number d_n.     

On tame weakly symmetric algebras having only periodic modules

We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.Received: 25 February 2002     

Friezes and a construction of the Euclidean cluster variables

Let Q be a Euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated with any object in the cluster category of Q. In particular, this algorithm allows us to compute all the cluster variables in the cluster algebra associated with Q. It also allows us to compute the sum of the Euler characteristics of the quiver Grassmannians of any module M over the path algebra of Q.     

Generic variables in acyclic cluster algebras

Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q). We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig’s dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set G(Q) is a Z-basis of A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.     

Polynomially bounded cohomology and discrete groups

We establish the homological foundations for studying polynomially bounded group cohomology, and show that the natural map from PH^*(G;Q) to H^*(G;Q) is an isomorphism for a certain class of groups.     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism with X and Y_i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1≤i≤r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.     

Some characterisations of supported algebras

We give several equivalent characterisations of left (and hence, by duality, also of right) supported algebras. These characterisations are in terms of properties of the left and the right parts of the module category, or in terms of the classes L₀ and R₀ which consist respectively of the predecessors of the projective modules, and of the successors of the injective modules.     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

On a separation of orbits in the module variety for string special biserial algebras

Given a pair M,M^′ of finite-dimensional modules over a string special biserial algebra Λ, a fully verifiable criterion, expressed in terms of a finite set of simple linear algebra invariants, deciding if M and M^′ lie in the same orbit in module variety, equivalently, if M and M^′ are isomorphic, is formulated and proved.     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

On the critical ideals of graphs

We introduce some determinantal ideals of the generalized Laplacian matrix associated to a digraph G, that we call critical ideals of G. Critical ideals generalize the critical group and the characteristic polynomials of the adjacency and Laplacian matrices of a digraph. The main results of this article are the determination of some minimal generator sets and the reduced Gröbner basis for the critical ideals of the complete graphs, the cycles and the paths. Also, we establish a bound between the number of trivial critical ideals and the stability and clique numbers of a graph.     

n-representation infinite algebras

From the viewpoint of higher dimensional Auslander–Reiten theory, we introduce a new class of finite dimensional algebras of global dimension n, which we call n-representation infinite. They are a certain analog of representation infinite hereditary algebras, and we study three important classes of modules: n-preprojective, n-preinjective and n  -regular modules. We observe that their homological behaviour is quite interesting. For instance they provide first examples of algebras having infinite Ext¹${\mathrm{Ext}}^1$-orthogonal families of modules. Moreover we give general constructions of n-representation infinite algebras.     

On the maximal cardinality of half-factorial sets in cyclic groups

We consider the function μ(G), introduced by W. Narkiewicz, which associates to an abelian group G the maximal cardinality of a half-factorial subset of it. In this article, we start a systematic study of this function in the case where G is a finite cyclic group and prove several results on its behaviour. In particular, we show that the order of magnitude of this function on cyclic groups is the same as the one of the number of divisors of its cardinality. This work was supported by the Austrian Science Fund FWF (Project P16770-N12) and by the Austrian-French Program ``Amadeus 2003–2004'.