Krull–Gabriel Dimension of 1-domestic String Algebras

Prest, Mike; Puninski, Gena

doi:10.1007/s10468-006-9028-z

Cited by 8 publications

(7 citation statements)

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“…In particular, for n = 2, we have the following special case of Schröer's conjecture: for an algebra A of infinite representation, KG(A) = 2 if and only if m≥1 (rad ∞ A ) m = 0. It has been confirmed for the following classes of algebras: the tilted algebras of Euclidean type [22,23], the algebras stably equivalent to tame hereditary algebras [23], the algebras with directing indecomposable projective modules [65], the enveloping algebras of restricted Lie algebras [19] (more generally, the infinitesimal group schemes [20]) in odd characteristic, the strongly simply connected algebras [61,69], the 1-domestic string algebras [44,45,51], and recently the tame generalized multicoil algebras [31].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

Skowroński

2015

Algebr Represent Theor

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

Skowroński

2015

Algebr Represent Theor

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“…Recently Kasjan and Pastuszak [10] proved the same for strongly simply connected algebras of nonpolynomial growth. In the light of all this and of results such as [24], [22], [27], [28] for domestic string algebras, a more reasonable conjecture now is that this dimension, like, conjecturally, Krull-Gabriel dimension, detects the difference between domestic and nondomestic representation type.…”

Section: Pure-injective Modulesmentioning

confidence: 99%

Modules with irrational slope over tubular algebras

Harland¹,

Prest²

2015

Proceedings of the London Mathematical Society

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“…By Proposition 4.1, the interval (ξ/x = 0) in the lattice of all pp-formulae over A is freely generated by two chains L 1 = L 1 ∪ {x = 0} and L 2 = L 2 ∪ {x = 0, ξ }, where L 1 = {ϕ u | u ∈ S(π ) and πβα ≤ u} and [15,Prop. 4.10], there exists a natural surjection from the set ( p 1 , p 2 ) of cuts on L 1 and L 2 to the set of isomorphism types of indecomposable pure-injective A-modules in (ξ/x = 0).…”

Section: The Existence Of a Superdecomposable Modulementioning

confidence: 99%

“…It seems to be even more complicated to prove that there is no superdecomposable pure-injective module over a domestic string algebra. Its nonexistence would follow if the conjecture about the finiteness of the Krull-Gabriel dimension (see [15,Conj. 1.2]) of any domestic string algebra were proven.…”

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How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

Puninski¹

2008

Journal of Pure and Applied Algebra

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We construct an element in a direct product of finite dimensional modules over a string algebra such that the pure-injective envelope of this element is a superdecomposable module.A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For example, let R be the endomorphism ring of a countable dimensional vector space V , and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ∼ = e R ⊕ (1 − e)R as a right module over itself and R ∼ = e R ∼ = (1 − e )R . Furthermore, every nontrivial decomposition of R as a right module is of this form; therefore R is superdecomposable.For more examples, let R = k X, Y be a free algebra over a field k, and let E = E(R R ) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that R R has no nonzero uniform submodules. Therefore the same is true for E, and hence E is a superdecomposable injective module.More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pure-injective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pure-injective if M is a direct summand of a direct product of finite dimensional A-modules.If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pure-injective A-module has an indecomposable direct summand. But recently, Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pure-injective module (note that every string algebra is tame). The main drawback of his proof is that it depends on the cardinality of the ground field and is highly nonconstructive. Specifically, it is based on a quite ingenious construction of Ziegler [26] which seems to work only for countable fields.

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Krull–Gabriel Dimension of 1-domestic String Algebras

Abstract: We calculate, confirming a conjecture of Schröer, the Krull-Gabriel dimension of the category of modules over any domestic string algebra, as well as the Cantor-Bendixson rank of each point of its Ziegler spectrum. We also determine the topology on this space. 1 2

Cited by 8 publications

References 23 publications

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

The Krull-Gabriel Dimension of Cycle-Finite Artin Algebras

Modules with irrational slope over tubular algebras

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

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