  We give a combinatorial description of a family of indecomposable objects in
the bounded derived categories of a new class of algebras: string almost gentle
algebras. These indecomposable objects are, up to isomorphism, the string and
band complexes introduced by V. Bekkert and H. Merklen. 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.
