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Contemporary Mathematics Linearization of Local Cohomology Modules

Summary: Contemporary Mathematics
Linearization of Local Cohomology Modules
Josep `Alvarez Montaner and Santiago Zarzuela
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


Source: Alvarez Montaner, Josep - Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya


Collections: Mathematics