Extreme behavior of lex ideals on Betti numbers

This paper mainly deals with the finitely generated graded modules of the polynomial ring k[x1, x2, ..., xn]. Free resolutions is an important tool to understand the structure of these modules. Betti numbers are an useful invariant that encodes the free resolutions. Our concentration accumulates on proving that the lex ideals provides an upper bound for Betti numbers of the graded ideals with the same Hilbert function in the polynomial ring k[x1, x2, ..., xn]. The material of this thesis is contemporary classical and includes the detailed study of the material that is scattered throughout the sources cited in the bibliography list.