We discuss the widely open conjecture of Stanley concerning the socalled Stanley decompositions of Zn-graded modules. This conjecture predicts a combinatorial description of the depth of such modules. We also discuss its relationship to prime filtrations and show that Stanley’sc onjecture implies a another conjecture of Stanley on partitionable simplicial complexes. Finally I report on recent work of my student Soleyman Jahan who showed, using Alexander duality, that for squarefree monomial ideals Stanley’s conjecture can be reinterpreted to give a conjecture relating the regularity of such ideals to data of its Stanley decomposition. This leads to a new conjecture predicting a combinatorial description of regularity of general Zn-graded modules.

Lecture #13400

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