In the present paper, we introduce the concepts of Pr\"{ u}fer sheaves and adic 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 of coherent sheaves. Moreover, we describe the relationship between Pr\"{ u}fer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Pr\"{ u}fer sheaves.
Categories: math.RT