Cohen-Macaulay higher conormal and K\"ahler differential modules of squarefree monomial ideals

Abstract

Let $S=k[x_1,\ldots,x_n]$ and let $I=I_\Delta\subsetneq S$ be a nonzero squarefree monomial ideal. Motivated by the classical higher-order K\"ahler differential modules and by the theory of higher conormal modules, we study not only the higher conormal quotients $I/I^q$, but more generally the shifted quotients $I^r/I^q$, $1\le r<q$, in the same $I$-adic conormal filtration, together with their symbolic analogues $I^{(r)}/I^{(q)}$. We prove that, for every $1\le r<q$ with $q\ge3$, the module $I^r/I^q$ is Cohen--Macaulay if and only if $I$ is a complete intersection. In sharp contrast, $I^{(r)}/I^{(q)}$ is Cohen--Macaulay if and only if $\Delta$ 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 $I/I^2$ forces $I^2=I^{(2)}$, and hence $I/I^{(2)}$ is Cohen--Macaulay.