Linearization of local cohomology modules

Abstract

Let k be a field of characteristic zero and R = k[x1, . . . , xn] the polynomial ring in n variables. For any ideal I ⊂ R, the local cohomolgy modules Hi I(R) are known to be regular holonomic An(k)-modules. If k is the field of complex numbers, by the Riemann-Hilbert correspondence there is an equivalence of categories between the category of regular holonomic DX modules and the category Perv (Cn) of perverse sheaves. Let T be the union of the coordinate hyperplanes in Cn, endowed with the stratification given by the intersections of its irreducible components and denote by Perv T (Cn) the subcategory of Perv (Cn) of complexes of sheaves of finitely dimensional vector spaces on Cn which are perverse relatively to the given stratification of T . This category has been described in terms of linear algebra by Galligo, Granger and Maisonobe. If M is a local cohomology module Hi I(R) supported on a monomial ideal, one can see that the equivalent perverse sheaf belongs to Perv T (Cn). Our main purpose in this note is to give an explicit description of the corresponding linear structure, in terms of the natural Zn-graded structure of Hi I(R). One can also give a topological interpretation of this linear structure, recovering as a consequence the results on the structure of local cohomology modules supported on squarefree monomial ideals given by M. Mustaţa.