Representation dimension and tilting

Let Λ be a finite dimensional k-algebra over an algebraically closed field k and let _ΛT be a splitting tilting module of projective dimension at most 1. Let Γ=End_ΛT. If the representation dimension of Λ is at most 3 then the main result asserts that the representation dimension of Γ does not exceed that of Λ.     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Wild algebras have one-point extensions of representation dimension at least four

We show that any wild algebra has a one-point extension of representation dimension at least four, and more generally that it has an n-point extension of representation dimension at least n+3. We give two explicit constructions, and obtain new examples of small algebras of representation dimension four.     

Applications of stratifying systems to the finitistic dimension

Given an Ext-injective stratifying system of Λ-modules satisfying that the projective dimension of Y is finite, we prove that the finitistic dimension of the algebra Λ is equal to the finitistic dimension of the category . Moreover, using the theory of stratifying systems we obtain bounds for the finitistic dimension of Λ. In particular, we get the optimal bound 2n-2 for the finitistic dimension of a standardly stratified algebra with n simples.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

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.     

The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

Submodule categories of wild representation type

Let Λ be a commutative local uniserial ring with radical factor field k. We consider the category S(Λ) of embeddings of all possible submodules of finitely generated Λ-modules. In case Λ=Z/〈pⁿ〉, where p is a prime, the problem of classifying the objects in S(Λ), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that Λ has Loewy length at least seven. We show that S(Λ) is controlled k-wild with a single control object I∈S(Λ). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X)_I of the endomorphism ring of some object X∈S(Λ) modulo the ideal End(X)_I of all maps which factor through a finite direct sum of copies of I.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

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.     

Ringel duality and Auslander–Dlab–Ringel algebras

We introduce a new class of quasi-hereditary algebras, containing in particular the Auslander–Dlab–Ringel (ADR) algebras. We show that this new class of algebras is preserved under Ringel duality, which determines in particular explicitly the Ringel dual of any ADR algebra. As a special case of our theory, it follows that, under very restrictive conditions, an ADR algebra is Ringel dual to another one. The latter provides an alternative proof for a recent result of Conde and Erdmann, and places it in a more general setting.     

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.     

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.     

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 modΛ have Auslander–Reiten sequences.     

The number of Gabriel-Roiter submodules

Let k be an algebraically closed field and Λ a tilted algebra of a path algebra of a Dynkin quiver (of type A_n, D_n, E₆, E₇ or E₈). We show that every sincere indecomposable Λ-module has at most three Gabriel-Roiter submodules.