String and Band Complexes Over String Almost Gentle Algebras

We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of two new classes of algebras: string almost gentle (SAG) algebras and SUMP algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by Bekkert and Merklen (Algebras Rep Theory 6:285–302, 2003). With this description, we give a necessary and sufficient condition for a given string complex to have infinite minimal projective resolution and we extend this condition for the case of string algebras. Using this characterization we establish a sufficient condition for a string almost gentle algebra (or a string algebra) to have infinite global dimension.