2

JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA

operators in

en

with holomorphic coefficients. By the Riemann-Hilbert correspon-

dence there is an equivalence of categories between the category of regular halo-

nomic 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. Denote by Perv

T (en)

the sub-

category of Perv

(en)

of complexes of sheaves of finitely dimensional vector spaces

on

en

which are perverse relatively to the given stratification ofT

([6,

1.1], and

by Mod(Vx )Ir the full abelian subcategory of the category of regular holonomic

Vx-modules such that their solution complex JR.Homvx

(M, Ox)

is an object of

Perv

T

(en).

Then, the above equivalence gives by restriction an equivalence of cat-

egories between Mod(Vx )Ir and Perv

T(en).

Moreover, this last category has been

described in terms of linear algebra by Galligo, Granger and Maisonobe in

[6].

Let

M

be a An(C)-module. Then,

Man= Ox

0R

M

has a natural structure

of Vx-module. In this way, regular holonomic An(C)-modules may be considered

as regular holonomic Vx-modules and, for those such that

Man

E

Mod(Vx )Ir

(e.g. local cohomology modules supported on monomial ideals), one can describe

the linear structure of the corresponding perverse sheaf in terms of the module

M itself, see

[7].

Our main purpose in this note is to give an explicit description

of this linear structure when M is a local cohomology module supported on a

monomial ideal. It will be expressed in terms of the natural zn_graded structure

of these local cohomology modules, coming from the polynomial ring C[x

1 , ... ,

xn]

by giving deg

Xi

=

c:i,

where c:1

, ... ,

C:n

denotes the canonical basis of

zn.

More generally, we will consider the category of straight modules introduced by

K.

Yanagawa

[13].

For any field k, it is a full abelian subcategory of the category

of zn_graded R-modules, which includes the local cohomology modules

H}(R)

sup-

ported on a monomial ideal J. For k =

e,

it is proven in

[2]

that a slight variation

of this category (the category of c:-straight modules) is equivalent to the full abelian

subcategory of Mod(Vx )Ir whose equivalent perverse sheaves have variation zero.

The proof of this result is based on the fact that the simple objects in both cate-

gories coincide, and that the objects in each category admit similar finite increasing

filtrations, such that the quotients are a finite direct sum of simple objets.

Here, for such a Mod(Vx )Ir-module we want to make precise the linear de-

scription of its equivalent perverse sheaf in terms of the corresponding c:-straight

module. Moreover, in the case of the local cohomology modules supported on

squarefree monomial ideals, we shall give a topological interpretation of this linear

description, recovering, as a consequence, the results on the structure of local co-

homology modules supported on squarefree monomial ideals given by M. Mustata

[10].

For any unexplained terminology concerning the theory of V-modules we shall

use

[3],

[4] or [5].

2. Preliminaries

As said in the introduction, for any regular holonomic module Vx-module in

Mod(Vx)Ir its solution complex JR.Homvx(M,Ox) is an object of PervT(Cn),

and by the Riemann-Hilbert correspondence the functor of solutions establishes an

equivalence of categories between Mod(Vx)Ir and PervT(cn).

In

[6],

the category Perv

T

(en)

has been linearized as follows: Let

Cn

be the

category whose objects are families {Ma}aE{O,l}n of finitely dimensional complex