Local cohomology modules with support in 2-regular monomial ideals (Q1002434)

Let \(K\) be a field, \(V\) a finite set of variables and \({\mathcal Q}\subseteq S:=K[V]\) a 2-regular (i.~e., its Castelnuovo-Mumford regularity is 2) monomial ideal. The paper is on the local cohomology modules of \(S\) supported in \({\mathcal Q}\): Write \[ {\mathcal Q}={\mathcal Q}_1\cap \dots \cap {\mathcal Q}_l \] where each \({\mathcal Q}\) is generated by a set of variables \(Q_i\subseteq V\), write \[ {\mathcal Q}^{(i)}:={\mathcal Q}_1\cap \dots \cap {\mathcal Q}_i. \] For any \(i=2,\dots ,l\) and any \(j\) one has a short exact sequence \[ 0\to H^j_{{\mathcal Q}^{(i-1)}}(S)\oplus H^j_{{\mathcal Q}_i}(S)\to H^j_{{\mathcal Q}^{(i)}}(S)\to H^{j+1}_{{\mathcal Q}^{(i-1)}+{\mathcal Q}_i}(S)\to 0. \] \(H^j_{\mathcal Q}(S)\) is non-zero if and only if either {\parindent=6mm \begin{itemize}\item[(i)]\(j=\#Q_k\) for some \(k\in \{ 1,\dots ,l\} \): In this case \(\dim_K(H^j_{\mathcal Q}(S))_{-\alpha (Q_k)}=1\) or \item[(ii)] \(j+1=\# (D_{k-1}\cup P_k)\) for some \(k\in \{ 2,\dots ,l\} \): In this case \(\dim_K(H^j_{\mathcal Q}(S))_{-\alpha(D_{k-1}\cup P_k)}\) is the number of occurences of \(D_{k-1}\cup P_k\) in the list \(D_1\cup P_2,\dots ,D_{l-1}\cup P_l\). \end{itemize}} (The decompositions \(Q_i=D_i\cup P_i\) are from a previous paper from the second author; \(\alpha \) refers to the \(\mathbb Z^n\)-grading in an obvious way). The sets of associated primes of such local cohomology are computed: In the above situation set \[ {\mathcal A}_{i,{\mathcal Q}}=\{ (Q_j)\mid \#Q_j=i\} \] \[ {\mathcal B}_{i,{\mathcal Q}}=\{ (D_{k-1},P_k),\# D_{k-1}+\# P_k=i+1\} . \] By definition, an element \((D_{k-1},P_k)\in {\mathcal B}_{i,{\mathcal Q}}\) belongs to \(\widetilde{{\mathcal B}_{i,{\mathcal Q}}}\) if \((D_{k-1},P_k)=(Q_j)+(Q_k)\) for some \(j\) and \(k\) such that \(\# Q_j,\# Q_k<i\). Using this notation, \[ \text{Ass}(H^i_{\mathcal Q}(S))={\mathcal A}_{i,{\mathcal Q}}\cup \widetilde{{\mathcal B}_{i,{\mathcal Q}}}. \]