How to compute the Stanley depth of a module

In this paper we introduce an algorithm for computing the Stanleydepth of a finitely generated multigraded module M over the polynomialring K[X1,...,Xn]. As an application, we give an example of a module whoseStanley depth is strictly greater than the depth of its syzygy module. In particular,we obtain complete answers for two open questions raised by Herzog.Moreover, we show that the question whether M has Stanley depth at leastr can be reduced to the question whether a certain combinatorially definedpolytope P contains a Zn-lattice point

An algorithm to compute the Hilbert depth

We present an algorithm which computes the Hilbert depth of a graded module based on a theorem of Uliczka. Connected to a Herzog's question we see that the Hilbert depth of a direct sum of modules can be strictly bigger than the Hilbert depth of all the summands.Comment: to appear in: Journal of Symbolic Computatio

Exterior depth and exterior generic annihilator numbers

We study the exterior depth of an $E$-module and its exterior generic annihilator numbers. For the exterior depth of a squarefree $E$-module we show how it relates to the symmetric depth of the corresponding $S$-module and classify those simplicial complexes having a particular exterior depth in terms of their exterior shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung and Conca, Herzog and Hibi. In addition to a combinatorial interpretation of the annihilator numbers we show how they are related to the symmetric Betti numbers and the Cartan-Betti numbers, respectively. We finally conclude with an example which shows that neither the symmetric nor the exterior generic annihilator numbers are minimal among the annihilator numbers with respect to a sequence.Comment: 22 pages; added proof of Thm. 4.6, extended Ex. 2.11

Some remarks on the Stanley's depth for multigraded modules

We show that the Stanley's conjecture holds for any multigraded $S$-module $M$ with \sdepth(M)=0, where $S=K[x_1,...,x_n]$. Also, we give some bounds for the Stanley depth of the powers of the maximal irrelevant ideal in $S$.Comment: 6 page

On the Stanley depth of edge ideals of line and cyclic graphs

We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for $n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient of two monomial ideals, in combinatorial terms.Comment: 8 pages. Will appear in Romanian Journal of Mathematics and Computer Scienc