Cohen-Macaulay higher conormal and Kähler differential modules of squarefree monomial ideals

Let $S=k[x_1,\dots ,x_n]$ and let $I=I_{\mathrm{\Delta }}⊊S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order Kähler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I\mathrm{/}{I}^q$, but more generally the shifted quotients $I^r\mathrm{/}{I}^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}\mathrm{/}{I}^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge 3$, the module $I^r\mathrm{/}{I}^q$ is Cohen–Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}\mathrm{/}{I}^{(q)}$ is Cohen–Macaulay if and only if $\mathrm{\Delta }$ is a matroid, where loops are allowed. Thus, the Cohen–Macaulayness of a single nonexceptional window forces the Cohen–Macaulayness of every window in the corresponding filtration. The unique exceptional pair is $(r,q)=(1,2)$: at this conormal level, we show that the Cohen–Macaulayness of $I\mathrm{/}{I}^2$ forces $I^2=I^{(2)}$, and hence $I\mathrm{/}{I}^{(2)}$ is Cohen–Macaulay.