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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[x 1 , . . . , xn ] the
polynomial ring in n variables. For any ideal I # R, the local cohomolgy
modules H i
I (R) are known to be regular holonomic An (k)­modules. If k is
the field of complex numbers, by the Riemann­Hilbert correspondence there is
an equivalence of categories between the category of regular holonomic DX ­
modules and the category Perv (C n ) of perverse sheaves. Let T be the union
of the coordinate hyperplanes in C n , endowed with the stratification given
by the intersections of its irreducible components and denote by Perv T (C n )
the subcategory of Perv (C n ) of complexes of sheaves of finitely dimensional
vector spaces on C n 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 H i
I (R) supported
on a monomial ideal, one can see that the equivalent perverse sheaf belongs to
Perv T (C n ). Our main purpose in this note is to give an explicit description of


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


Collections: Mathematics