A Survey on Stanley Depth

At the MONICA conference “MONomial Ideals, Computations and Applications” at the CIEM, Castro Urdiales (Cantabria, Spain) in July 2011, I gave three lectures covering different topics of Combinatorial Commutative Algebra: (1) A survey on Stanley decompositions. (2) Generalized Hibi rings and Hibi ideals. (3) Ideals generated by two-minors with applications to Algebraic Statistics. In this article I will restrict myself to give an extended presentation of the first lecture. The CoCoA tutorials following this survey will deal also with topics related to the other two lectures. Complementing the tutorials, the reader finds in [165] a CoCoA routine to compute the Stanley depth for modules of the form I=J , where J I are monomial ideals. In his famous article “Linear Diophantine equations and local cohomology”, Richard Stanley [181] made a striking conjecture predicting an upper bound for the depth of a multigraded module. This conjectured upper bound is nowadays called the Stanley depth of a module. The Stanley depth is of a rather combinatorial nature while the depth is a homological invariant. The definition of Stanley depth is given in Sect. 1. The conjecture is on so far striking as it compares two invariants of modules of very different nature. At a first glance it seems to be no relation among these two invariants. The conjecture was made in 1982, and to best of my knowledge it is Apel who first studied this conjecture intensively and proved it in some special cases in his papers [12, 13]. His papers appeared in 2003. Stanley decompositions were then considered again in 2006 in my joint paper [95] with Popescu and since then the topic has become very popular with numerous publications regarding different