The work in this thesis is concerned with three subclasses of the string algebras: domestic string algebras, gentle algebras and derived-discrete algebras (of non-Dynkin type). The various questions we answer are linked by the theme of the Krull-Gabriel dimension of categories of functors. We calculate the Cantor-Bendixson 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 Krull-Gabriel 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 Hom-hammocks involving (possibly infinite) string objects in the homotopy category of complexes of projective modules over a derived-discrete algebra. Using this description, we prove that the Krull-Gabriel dimension of the category of coherent functors from a derived-discrete algebra (of non-Dynkin type) is equal to 2. Since the Krull-Gabriel dimension is finite, it is equal to the Cantor-Bendixson 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 pure-injective complexes in the homotopy category are exactly the string complexes. Finally, we prove that every indecomposable complex in the homotopy category is pure-injective, and hence is a string complex.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:694290 |
| Date | January 2016 |
| Creators | Laking, Rosanna Davison |
| Contributors | Symonds, Peter ; Prest, Michael |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/string-algebras-in-representation-theory(c350436a-db9a-429d-a8a5-470dffc0974f).html |
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