Stanley Depth and Sequentially Cohen- Macaulay Lexsegment Ideals

Abstract

In the first chapter we give some basic definitions from commutative algebra. We give some results obtained in recent years for the Stanley depth of multigraded S-modules, where S = K[x1 , . . . , xn ] is a polynomial ring in n indeteminantes with coefficients in a field K. We also give some results regarding the progress towards the Stanley’s conjecture. In the second chapter, we show that if I ⊂ J be monomial ideals of a polynomial algebra S over a field. Then the Stanley depth of J/I is smaller or equal to the √ √ Stanley depth of J/ I. We give also an upper bound for the Stanley depth of the intersection of two primary monomial ideals Q, Q , which is reached if Q, Q √ √ are irreducible, ht(Q + Q ) is odd and Q, Q have no common variables. These results are proved in my paper [23]. In the third chapter, we give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley’s conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables. These results are proved in my paper [24]. In the forth chapter, we give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of an s-uniform complete bipartite hypergraph. In this chapter we also give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph. These results are proved in our papers [26] and [27]. In the fifth chapter, the associated primes of an arbitrary lexsegment ideal I ⊂ S are determined. As application it is shown that S/I is a pretty clean module, therefore, S/I is sequentially Cohen-Macaulay and satisfies the Stanley’s conjecture. These results are proved in my paper [25].