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We give a combinatorial description of a family of indecomposable objects inthe bounded derived categories of a new class of algebras: string almost gentlealgebras. These indecomposable objects are, up to isomorphism, the string andband complexes introduced by V. Bekkert and H. Merklen. With this description,we give a necessary and sufficient condition for a given string complex to haveinfinite minimal projective resolution and we extend this condition for thecase of string algebras. Using this characterization we establish a sufficientcondition for a string almost gentle algebra (or a string algebra) to haveinfinite global dimension.

String and band complexes over SAG algebras