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
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