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Abstract The extensions of rings provide a useful tool for obtain- ing algebraic theorems and results and additional struc- tures for both rings, prime ideals of both rings, modules over both rings with some functors and algebras over both rings. The second level of prime ideals; in general of ideal theory, occupies an important place in the algebraic geom- etry. Ring extensions, like field extensions, can be considered from two points of view. One can look upward from a ring to its extensions or downward to its subrings. The work in this thesis provides an example of the upward point of view. There are many different ways of constructing the ring extensions and their applications; several of these were thought to be different in some papers, see [5], [9], [11], [14], [17], [18], [25] and [26]. Graded ring extensions permit us to construct struc- ture sheaves extensions, on the micro-structure sheaves, on the graded prime spectrum Spec g ( G ( R )) of the associ- ated graded ring G ( R ), when this space is endowed with the Zariski graded topology. This thesis is devoted to the study of filtered and graded Procesi extensions of filtered and graded rings. After a ii brief study of general features concerning filtered and graded Procesi extensions at the level of graded Rees rings and their associated graded rings, we turn to the behavior of filtered and graded Procesi extensions towards the micro- affine schemes. We show that these extensions behave well from the geometric point of view. The thesis consists of three chapters : Chapter 1 : Preliminaries This chapter provides the preliminaries and the back ground material to be used in subsequent chapters. We provide a brief survey of the basic definitions and elemen- tary results concerning extensions of a ring, prime spec- trum of a ring, graded rings and filtered rings as well as sheaves and schemes. Chapter 2 : Filtered and Graded Procesi Exten- sions of Rings In this chapter we continue the study of filtered and graded Procesi extensions of filtered and graded rings in- troduced in [17]. In the first section of this chapter, we define a new filtra- tion F ′′ S of S : F ′′ S = { F ′′ n S } n ∈ Z ; with F ′′ n S = φ ( F n R ) .S R . As in [17], we study, over the filtered level, the passage of iii various ring theoretic properties between R,S and concern with the relationship between the filtration F ′′ S on S and those studied in [17]. In the second section, with respect to F ′′ S , we prove that G ( S ) is graded Procesi extension of G ( R ) and, for any n ∈ Z , ̃ S/X n ̃ S = ̄ ̃ S ( n ) is a graded Procesi extension of ̄ ̃ R as in Proposition 2.2.5. Chapter 3 : Procesi Extensions of Filtered and Graded Rings Applied to the Micro-Affine Schemes In this chapter we turn to the behavior and graded Pro- cesi extensions towards to the micro-affine schemes . A number of results concerning these concepts are given. In the first two sections we introduce a survey, some- times with proofs, on the graded spectrum of the asso- ciated graded ring G ( R ) of R . Some results about the micro-affine schemes are concerned, see [6], [10], [15], [21] and [26]. This survey represents a solid foundation for our results in this chapter. In the third section we prove that the filtered Procesi extension φ : R → S , for every n ∈ Z , induces a graded Procesi extension ( Y = Spec g ( G ( S )) , ̃ O ( n ) Y ) −→ ( X = Spec g ( G ( R )) , ̃ O ( n ) X ) of affine schemes. iv By considering the inverse limit in the graded sense and the idea of micro-affine structure sheaves, we pay our at- tention to deduce that ( Y,O μ Y ) → ( X,O μ X ) , of filtered micro-affine schemes, is a filtered Procesi ex- tension. The main results of Chapters 2 and 3 seem to be original and have been published in the International Mathematical Forum (Bulgaria), Vol.7, 2012, no. 26, 1279 -1288. |