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In the present paper, we introduce the concepts of Pr\"{u}fer sheaves andadic sheaves over a weighted projective line of genus one or an elliptic curve,show that Pr\"{u}fer sheaves and adic sheaves can classify the category ofcoherent sheaves. Moreover, we describe the relationship between Pr\"{u}fersheaves and generic sheaves, and provide two methods to construct genericsheaves by using coherent sheaves and Pr\"{u}fer sheaves.

Pr\"ufer sheaves and generic sheaves over the weighted projective lines and elliptic curves