Abstract
In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.
Highlights
Let K be a field, and set S = K[ x1, . . . , xn ]
Depth(mk ) = 1 and by ([6] Corollary 24), we know that the Stanley depth of any monomial ideal is at least one
Do we believe that Conjecture 1 is true for every polymatroidal ideal I, but we have a prediction for the limit value of the Stanley depth of powers of I
Summary
The minimum dimension of a Stanley space in a Stanley decomposition D is called the Stanley depth of D , and is denoted by sdepth(D). We say that a Zn -graded S-module M satisfies Stanley’s inequality if depth( M) ≤ sdepth( M ). Stanley [2] conjectured that the above inequality holds for every finitely generated, S-module. Stanley’s conjecture has been disproved by Duval, Goeckner, Klivans, and Martin [4] They constructed a non-partitionable Cohen-Macaulay simplicial complex, and using a result of Herzog, Soleyman Jahan, and Yassemi ([5] Corollary 4.5), deduced that the Stanley-Reisner ring of this simplicial complex did not satisfy Stanley’s inequality. Of particular interest is the validity of Stanley’s inequality for high powers of monomial ideals In this survey article, we review the recent developments in this regard. We complement his survey by collecting the results obtained since with a focus on powers of monomial ideals
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