Abstract
In the present paper, we introduce the concepts of Prufer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prufer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prufer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prufer sheaves.
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