Characteristic cycles of local cohomology modules of monomial ideals
This paper studies the local cohomology modules \(\text{H}_I^i(R)\), where \(I\) is a square free monomial ideal in \(R=k[x_1,\dots,x_n]\) (or \(R=kx_1,\dots,x_n\)), \(k\) a field of characteristic 0. Such an ideal \(I\) has a minimal primary decomposition of the form \(I=I_1\cap\cdots\cap I_m\), where the \(I_i\) are ``face ideals'', i.e. ideals of the form \((x_{i_1},\dots,x_{i_h})\). The \(\text{H}^i_I(R)\) have a natural structure of finitely generated \({\mathcal D}=D(R,k)\)-module. Furthermore these local cohomology modules are holonomic \(\mathcal D\)-modules (i.e. of dimension \(n\)), hence have a characteristic cycle \(CC(\text{H}^i_I(R))\). The characteristic cycle \(CC(\text{H}^i_I(R))\) is first described if \(I\) is a face ideal. Then a closed form for \(CC(\text{H}^i_I(R))\) is given for \(I\) a square free monomial ideal in terms of local cohomology modules \(CC(\text{H}^j_{I_d}(R))\) where \(I_d\) ranges over the face ideals occurring in the primary decomposition of \(I\), and sums of such face ideals (which will again be face ideals). This is used to determine when \(\text{H}^i_I(R)=0\) and to find the cohomological dimension \(\text{cd}(R,I)\) in terms of the primary decomposition of \(I\). The invariants \(\lambda_{p,i}(R/I)\) introduced by \textit{G. Lyubeznik} [in Invent. Math. 113, 41-55 (1993; Zbl 0795.13004)] are found and interpreted as the multiplicities of the characteristic cycle of \(\text{H}^p_m(\text{H}^{n-i}_I(R))\) (where \(m\) is the graded maximal ideal of \(R\)). Finally this material is related to the \(f\)-vectors of the associated simplicial complex of a Stanley-Reisner ring.
scientific article; zbMATH DE number 1488530