Generic Sheaves On Elliptic Curves And Interrelated Researches

The theory of schemes was initiated by Grothendieck in early sixties of last century,which marked a new stage of algebraic geometry.Since quasi-coherent sheaves and coherent sheaves on a scheme behave as modules and finite-generated modules on a ring,respectively.Quasi-coherent sheaves and coherent sheaves on a scheme become popular objects to be studied in algebraic geometry and related subjects.In this thesis for Ph.D degree,we study quasi-coherent sheaves on an elliptic curve and the simple extension category of coherent sheaves category on a smooth projective curve.It mainly consists of the following five parts.In the first chapter,we give a detailed introduction of recent developments related to this thesis,and make a systemic exposition of our main results.In the second chapter,we list some concepts and properties which are closely related to this thesis,which give a necessary preparation for the following chapters.In the third chapter,we discuss generic sheaves on an elliptic curve.At first, we define generic sheaves on an elliptic curve and extend the notions of rank,Euler characteristic and slope of coherent sheaves to those of generic sheaves.Then we prove that the slope of a generic sheaf is in Q∪{∞}.Base on this,we determine all generic sheaves on an elliptic curve:for each q∈Q∪{∞},there exists an unique generic sheaf G_q of slope q up to isomorphism.In particular,we point out that the rational function sheafκis the generic sheaf of slope∞,and for each q∈Q,there exists an exact auto-equivalenceΦ_q∞ of the bounded derived category of coherent sheaves category of the elliptic curve,such that the generic sheaf G_q=Φ_q∞(κ). Finally,by studying relationship between generic sheaves and coherent sheaves on an elliptic curve,we show that coherent sheaves can be classified by generic sheaves.In the fourth chapter,we introduce a method to construct generic sheaves on an elliptic curve.According to the relationship between the rational function sheaf and generic sheaves which has been showed in the third chapter,we need only discuss the construction of the rational function sheaf.We firstly introduce an effective method to construct torsionfree divisible sheaves on an elliptic curve. Then we deduce a construction of the rational function sheafκby pointing outκis the unique indecomposable torsionfree divisible sheaf up to isomorphism. Finally,we show that the full subcategory of semi-stable sheaves of slope∞can be characterized by the rational function sheaf by using the construction.In the fifth chapter,we discuss the simple extension category of coherent sheaves category on a smooth projective curve.We define the notion of the simple extension of an abelian category,and then give a necessary condition for an object to be indecomposable in the simple extension of a Hom-finite abelian category. Moreover,we show that there are only two types of simple objects in the simple extension of a Hom-finite abelian category.Furthermore,we study the simple extension category of coherent sheaves category on a smooth projective curve over an algebraically closed field.We classify all the indecomposable objects,and then take the advantage of the relationship among these indecomposable objects to describe the whole structure of this simple extension category.