Download or read book Minimal Free Resolutions, Hilbert Functions and the Graded Betti Numbers written by Rana Rizkallah Sabbagh and published by . This book was released on 2009 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis discusses the connection between the Hilbert function, graded Betti numbers and minimal free resolutions. In our first chapter we let R be a Noether ian local ring and M its maximal ideal. We define in general, the minimal free r esolution of R/I where I is an ideal in R and give properties and examples. In t he next chapter, we let R to be the polynomial ring in n variables of a field K and I a homogeneous ideal, we define and explain the relationship in details of the Hilbert function of K[x1, ..., xn]/I and its minimal graded free resolution. In particular, explain how Hilbert was able to compute the Hilbert function fro m the graded free resolution, show that the Hilbert polynomial exists, and show that the Hilbert series of such a module has a very nice form using the resoluti on. We finally explain in our last chapter the graded Betti number and their rel ation with minimal free resolutions and Hilbert functions.