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The geometric models for the module category and derived category of anygentle algebra were introduced to realize the objects in module category andderived category by permissible curves and admissible curves respectively. Thepresent paper firstly unifies these two realizations of objects in modulecategory and derived category via same surface for any gentle algebra, by therotation of permissible curves corresponding to the objects in the modulecategory. Secondly, the geometric characterization of the cohomology ofcomplexes over gentle algebras is established by the truncation of projectivepermissible curves. It is worth mentioning that the rotation of permissiblecurves and the truncation of projective permissible curves are mutually inverseprocesses to some extent. As applications, an alternative proof of ``no gaps"theorem as to cohomological length for the bounded derived categories of gentlealgebras is provided in terms of the geometric characterization of thecohomology of complexes. Moreover, we obtain a geometric proof for the strongNakayama conjecture for gentle algebras. Finally, we contain two examples toillustrate our results.

Constructing projective resolution and taking cohomology for gentle algebras in the geometric model