On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth

Abstract

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 pureinjective module if k is a countable field.

Strongly simply connected algebras with super-decomposable pure-injective modules

“…In case the lattice P Θ R contains a wide sublattice L, we say that the width of P Θ R is undefined. The above definition is a special case of a general definition of a wide lattice, see [24,25] or Section 3 of [17]. Assume that C is a set.…”

“…We give a concrete example of a string algebra satisfying the thesis of the theorem. This algebra plays an important role in [16] (see also [17] and [23]). Assume that Q = and Λ = kQ/I, where I = δα, γβ .…”

“…of dense chains of Θ-pointed Λ-modules where Θ = P (x 3 ) (see Lemma 3.10 of [17] for the details). Observe that Θ is a simple module, so homomorphisms χ M (X) , χ M (Y ) are monomorphisms, for any X ∈ C 1 , Y ∈ C 2 .…”

“…Recall that an independent pair of dense chains of pointed modules is some special family of pointed modules that allows to formulate a handy sufficient condition for the existence of a super-decomposable pure-injective module (see Section 2 for the details). This notion is introduced in [29] and generalized in [17]. It is successfully used in the series of papers [16,17,23,18] which is devoted to the problem of the existence of super-decomposable pure-injective modules for strongly simply connected algebras, see [38,21].…”

On Wild Algebras and Super-Decomposable Pure-Injective Modules

“…In this paper M. Ziegler proves in Lemma 7.8 (2) a fundamental criterion for such modules to exist, asserting that if the ring R is countable, then R possesses a super-decomposable pureinjective module if and only if the width of the lattice of all pp-formulas is undefined, see [16,Chapter 10] or [17, 7.3] for all the definitions. The later statement can be formulated in terms of the lattice of all pointed finitely-presented R-modules, see [22] and [9] for more details. We call the renowned result of Ziegler given in [31,Lemma 7.8 (2)] the Ziegler criterion.…”