Kademeli Halkalar, Modüller ve Çokkatlılık

Abstract

This thesis is based on a research over graded rings and graded modules as well as some related ideas which are of particular importance especially to the areas Algebra and Geometry. An important concept related to graded modules, so-called Hilbert functions, occurs as an invariant which measures size of homogeneous components. In this thesis we study the concept of multiplicity, which can be read from Hilbert functions, in many aspects. This thesis consists of five chapters. In the introductory chapter we give some important notions and results on commutative rings and their modules which will be used in the sequel. In the second chapter, we introduce the notions of graded rings and graded modules. We also give some particular graded rings which will be used later in other chapters. Finally we deal with the question as to whether a homogeneous decomposable submodule of a graded module has a primary decomposition in which every primary term is homogeneous. In the third chapter, by considering multiplicity systems, we give an account of a theory for “general multiplicity symbols” initiated by D. J. Wright. Also, we obtain a limit formula given for multiplicty symbol by C. Lech. In the fourth chapter, we study Hilbert and Hilbert-Samuel functions over a graded ring R0[x1, …, xs], where x1, …, xs are homogeneous elements of degree 1. Then, as an application of Hilbert functions theory, we give another limit formula for multiplicity symbol which is proved by P. Samuel. Moreover, we demonstrate how Hilbert-Samuel functions are used in Algebraic Geometry to determine the dimension of an affine variety. In the last chapter, we give some properties of Koszul complexes which provide us with homological methods for studying multiplicities. As one of the main results of this chapter, we establish a connection between two multiplicity symbols using Euler-Poincaré characteristics which are defined with the help of Koszul complexes.