Algebraic and homological properties of polymatroidal ideals

Official URL: http://risc01.sabanciuniv.edu/record=b1817044 (Table of Contents)

Monomial ideals are widely studied in commutative algebra. In this thesis, we study a special class of monomial ideals called polymatroidal ideals which admit many nice algebraic and homological properties. They are distinguished by the fact that they satisfy "exchange property" and their powers have linear resolutions. Another important property of polymatroidal ideals is that their monomial localization at any monomial prime ideal is again a polymatroidal ideal. In [1], Bandari and Herzog gave a conjecture that if all monomial localizations of a monomial ideal I have linear resolution then I is polymatroidal. In chapter 4, we discuss persistence and stability properties of polymatroidal ideals and we see that their index of depth stability and the index of stability for the associated prime ideals are bounded by their analytic spread. Finally, we examine the strong persistence property of polymatroidal ideals.