Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.
Let R=k(Q,I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q,I). We show that if R is a simply connected right multipeak algebra which is chord-free and -free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic...
A class of stratified posets is investigated and their incidence algebras are studied in connection with a class of non-shurian vector space categories. Under some assumptions on we associate with a bound quiver (Q, Ω) in such a way that . We show that the fundamental group of (Q, Ω) is the free group with two free generators if is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition is three-partite a fundamental domain of this covering is...
A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if is a three-partite subamalgam of a tiled order then it is of tame lattice type if and only if the reduced Tits quadratic form associated with in [26] is weakly non-negative. The result generalizes a criterion for tameness of such orders given by Simson [28] and gives an...
We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.
Associative algebras of fixed dimension over algebraically closed fields of fixed characteristic are considered. It is proved that the class of algebras of tame representation type is axiomatizable. Moreover, finite axiomatizability of this class is equivalent to the conjecture that the algebras of tame representation type form a Zariski-open subset in the variety of algebras.
Let k be a field of characteristic different from 2. We consider two important tame non-polynomial growth algebras: the incidence k-algebra of the garland 𝒢₃ of length 3 and the incidence k-algebra of the enlargement of the Nazarova-Zavadskij poset 𝒩 𝓩 by a greatest element. We show that if Λ is one of these algebras, then there exists a special family of pointed Λ-modules, called an independent pair of dense chains of pointed modules. Hence, by a result of Ziegler, Λ admits a super-decomposable...
Given a quiver Q, a field K and two (not necessarily admissible) ideals I, I' in the path algebra KQ, we study the problem when the factor algebras KQ/I and KQ/I' of KQ are isomorphic. Sufficient conditions are given in case Q is a tree extension of a cycle.
Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do...
Assume that k is a field of characteristic different from 2. We show that if Γ is a strongly simply connected k-algebra of non-polynomial growth, then there exists a special family of pointed Γ-modules, called an independent pair of dense chains of pointed modules. Then it follows by a result of Ziegler that Γ admits a super-decomposable pure-injective module if k is a countable field.
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