Contemporary Mathematics

Volume 331, 2003

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 C R, the local cohomolgy

modules H}(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

Vx-

modules and the category Perv (en) of perverse sheaves. Let

T

be the union

of the coordinate hyperplanes in

en ,

endowed with the stratification given

by the intersections of its irreducible components and denote by Perv

T

(en)

the subcategory of Perv (en) of complexes of sheaves of finitely dimensional

vector spaces on

en

which are perverse relatively to the given stratification

ofT. This category has been described in terms of linear algebra by Galligo,

Granger and Maisonobe. If M

is

a local cohomology module H}(R) supported

on a monomial ideal, one can see that the equivalent perverse sheaf belongs to

Perv

T

(en). 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 H} ( 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. Mustata.

1.

Introduction

Let

k

be a field of characteristic zero and

R

=

k[x1, ... , Xn] the polynomial

ring in

n

variables. For any ideal

I

C

R,

the local cohomolgy modules

H}(R)

are known since long to have a module structure over the Weyl algebra An(k),

more precisely, they are regular holonomic An(k)-modules. This structure has been

fruitfully used in recent years to prove interesting properties of the local cohomolgy

modules

H}(R),

as it was first realized by G. Lyubeznik

[9],

and later by several

other authors (see e.g.

[1], [2], [11]

and

[12]),

particularly in the case of monomial

ideals.

Assume now that

k

is the field of complex numbers and let

X

=

en'

let

0

X

be the sheaf of holomorphic functions in

en,

and let

1J

X

be the sheaf of differential

2000 Mathematics Subject Classification. Primary 13045; Secondary 32C38.

Key words and phrases. Local cohomology, monomial ideals, V-modules.

The first author was partially supported by the University of Nice.

The second author has been partially supported by DGCYT BFM2001-3584.

@2003 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/331/05899