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
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