Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

On deformations of crossed products

Let A∗Γ be a crossed product algebra, where A is semisimple, finitely generated over its center and Γ is a finite group. We give a necessary and sufficient condition in terms of the outer action of Γ on A for the existence of a multi-parametric semisimple deformation of the form A((t₁,…,t_n))∗Γ (with the induced outer action). The main tool in the proof is the solution of the so-called twisting problem. We also give an example which shows that the condition is not sufficient if one drops the condition on the finite generation of A over its center.     

Properties and examples of FCR-algebras

An algebra A over a field k is FCR if every finite dimensional representation of A is completely reducible and the intersection of the kernels of these representations is zero. We give a useful characterization of FCR-algebras and apply this to C ^*-algebras and to localizations. Moreover, we show that “small” products and sums of FCR-algebras are again FCR. Received: 25 October 2000     

On sign-coherence of c-vectors

Given a finite dimensional algebra A over an algebraically closed field, we consider the c-vectors such as defined by Fu in [18] and we give a new proof of its sign-coherence. Moreover, we characterise the modules whose dimension vectors are c-vectors as bricks respecting a functorially finiteness condition.     

Graded polynomial identities and codimensions: Computing the exponential growth

Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of , n=1,2,…, the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G×Z₂-graded algebra related to A.     

Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

Semisimple Hopf actions on commutative domains

Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen.     

Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.     

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.     

Hochschild cohomology of triangular string algebras and its ring structure

We compute the Hochschild cohomology groups HH^?(A)${\mathrm{HH}}^{?}(A)$ in case A is a triangular string algebra, and show that its ring structure is trivial.     

Liftings of graded quasi-Hopf algebras with radical of prime codimension

Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p. The purpose of this paper is to classify finite dimensional quasi-Hopf algebras A whose radical is a quasi-Hopf ideal and has codimension p; that is, A with gr(A) in RG(p), where gr(A) is the associated graded algebra taken with respect to the radical filtration on A. The main result of this paper is the following theorem: Let A be a finite dimensional quasi-Hopf algebra whose radical is a quasi-Hopf ideal of prime codimension p. Then either A is twist equivalent to a Hopf algebra, or it is twist equivalent to H(2), H_±(p), A(q), or H(32), constructed in [5] and [8]. Note that any finite tensor category whose simple objects are invertible and form a group of order p under tensor is the representation category of a quasi-Hopf algebra A as above. Thus this paper provides a classification of such categories.     

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.     

Self-pure-generators over Dedekind domains

We prove that all pure submodules of a finite rank torsion-free module A over a Dedekind domain are A-generated (i.e. A is a self-pure-generator) if and only if A has a rank 1 direct summand B such that $\mathbf{type}(B)$ is the inner type of A. This result is applied to describe the direct products of torsion-free groups of finite rank which are self-pure-generators.     

Algebras having bases that consist solely of strongly regular elements

“Locally invertible” algebras, those algebras which have a basis consisting solely of strongly regular elements, are introduced as a generalization of “invertible algebras,” that is, algebras which have a basis consisting solely of units. While this new family properly contains the family of (necessarily unital) invertible algebras, its definition does not assume the existence of a multiplicative identity. Because of this, we consider both unital and non-unital examples of locally invertible algebras. In particular, we show that under a mild condition on the basis of a not necessarily unital R-algebra A, the R-algebras $M_n(A)$ of finite matrix rings over the R-algebra A. Furthermore, many infinite matrix algebras are also locally invertible, but not all. Also it is shown that all semiperfect D-algebras over a division ring D are locally invertible.     

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

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.     

Bounds on the global dimension of certain piecewise hereditary categories

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.     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

On the Cohopficity of the Direct Product of Cohopfian Groups

Suppose both A and B are cohopfian groups. Then A × B is cohopfian if A is either extremely noncommutative and torsion free, or finite Abelian, or finite simple.