The work in this thesis is concerned with three subclasses of the string algebras: domestic string algebras, gentle algebras and deriveddiscrete algebras (of nonDynkin type). The various questions we answer are linked by the theme of the KrullGabriel dimension of categories of functors.We calculate the CantorBendixson rank of the Ziegler spectrum of the category of modules over a domestic string algebra. Since there is no superdecomposable module over a domestic string algebra, this is also the value of the KrullGabriel dimension of the category of finitely presented functors from the category of finitely presented modules to the category of abelian groups. We also give a description of a basis for the spaces of homomorphisms between pairs of indecomposable complexes in the bounded derived category of a gentle algebra. We then use this basis to describe the Homhammocks involving (possibly infinite) string objects in the homotopy category of complexes of projective modules over a deriveddiscrete algebra. Using this description, we prove that the KrullGabriel dimension of the category of coherent functors from a deriveddiscrete algebra (of nonDynkin type) is equal to 2. Since the KrullGabriel dimension is finite, it is equal to the CantorBendixson rank of the Ziegler spectrum of the homotopy category and we use this to identify the points of the Ziegler spectrum. In particular, we prove that the indecomposable pureinjective complexes in the homotopy category are exactly the string complexes.Finally, we prove that every indecomposable complex in the homotopy category is pureinjective, and hence is a string complex.
Date of Award  31 Dec 2016 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Michael Prest (Supervisor) & Peter Symonds (Supervisor) 

String Algebras in Representation Theory
Laking, R. (Author). 31 Dec 2016
Student thesis: Phd