Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra

The Ziegler spectrum for derived-discrete algebras

“…The indecomposable objects of D b (mod-Λ) were classified in [8] and the morphisms studied in [10]. This fits inside a more general classification, due to Bekkert and Merklen [3], of indecomposable complexes in the bounded derived category of a gentle algebra and a more general description, in [1], of morphisms between the indecomposable complexes. We build on these results, describing all the indecomposable complexes of K(Proj-Λ).…”

“…We begin with a description of the string complexes in K(Proj-Λ), which is an extension of [1,Lem. 7.1]; see also [15,Lem.…”

“…For the vanishing statements in the X k+1 −∞ components, use (1) with n ∈ Z ≤0 , using the one-dimensionality of the nonzero Hom-spaces to start the induction. The same argument applied to the X l −∞ components for l = k + 1 mod r shows Hom K (Z k a,b , −) = 0 on those components.…”

“…The non-vanishing statements are contained in Lemma 3.13. For the vanishing statements, one needs to argue once from string combinatorics and [1,Thm. 3.15] for each X ±∞ and Y ±∞ component and then use the triangles from Lemma 3.14 for an induction.…”

“…In the case of the derived-discrete algebras Λ(n, n, 0), n ≥ 1 the Ziegler spectrum had already been described by Z. Han [16] using covering theory. Our methods are very different, replying heavily on the description of morphisms in [1] and techniques around the Ziegler spectrum and the functor category. We will require ideas and results coming from various directions; some we summarise and for all we give references to where the details can be found.…”