Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$. This allows us to compute the depth of $S\mathrm{/}I$ in terms of its skeleton ideals. We apply these techniques to show that Stanley’s conjecture on Stanley decompositions of $S\mathrm{/}I$ holds provided it holds whenever $S\mathrm{/}I$ is Cohen–Macaulay. We also discuss a conjecture of Soleyman-Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution.