LINEARIZATION OF LOCAL COHOMOLOGY MODULES
3
vector spaces, endowed with linear maps

Ma
~
Ma+e; , Ma
~
Ma+e;
for each
a
E {0, 1 }n such that
ai
=
0. These maps are called canonical (resp.,
variation) maps, and they are required to satisfy the conditions:
UiUj
=
UjUi, ViVj
=
VjVi, UiVj
=
VjUi
and
ViUi
+
id
is invertible.
Such an object will be called ann-hypercube. A morphism from ann-hypercube
{Ma}a to an n-hypercube {Na}a is a set of linear maps Ua:
Ma
---+
Na}a,
commuting with the canonical and variation maps (see
[7]).
It is proved in [loc.cit.]
that there is an equivalence of categories between Perv
T
(en)
and
Cn.
We have to point out that the functor Mod(Vx
)fr
----t
Cn
is a contravariant
exact functor and, given an object
M
of Mod(Vx
)fr,
its corresponding n-hypercube
is constructed as follows:
Consider
en=
IJ~= 1
ei,
with
ei
=
e
for 1
:S
i
:S
n,
let
Ki
= ~+
c ei
and set
Vi= ei \ Ki.
For any
a=
(a1, ... ,
an)
E
{0, l}n denote
S
._
rn7=1
v;Ox
C .-"
r o ·
L..ak=l !Ck xfli#k V; X
Denoting with a subscript 0 the stalk at the origin, one has:
i) The vertices of the n-hypercube associated to
M
are the vector spaces
Ma
:=
Homv0 (Mo,Sa,o).
ii) The linear maps
ui
are those induced by the natural quotient maps
Sa
---+
Sa+e; ·
iii) The linear maps
Vi
are the partial variation maps around the coordinate
hyperplanes,
i.e.
for any
cp
E
Homv0 (Mo,Sa,o)
one has
(vi
o
ui)(cp)
=
cli(cp) - cp,
where
cli
is the partial monodromy around the hyperplane
Xi=
0.
The following is proved as well in
[7]:
iv) If
CC(M)
=
L
me
T.L
en
is the characteristic cycle of
M,
then for all
a
E
{0, l}n one has the equality dimcMa
=
ma.
REMARK 2.1. Since
M
0
is a regular holonomic Dx,o-module in order to de-
termine the solutions Homvx,o
(Mo, Sa,
o) we only have to consider those Nilsson
class functions in
Sa,
0
,
i.e.
those finite sums
f =
L
'P!3,m(x)(logx)mxi3,
/3,m
where 'Pt3,m(x) E
e{x},
f3
E
e
and mE
(Z+)n
contained in
Sa,o
(see
[3]
for details).
So, if
N
denotes the set of Nilsson class functions then:
Ma
:=
Homvx,o(Mo,Na),
where
Na
=
N
n
Sa,o·
3. Vx-modules with variation zero
Among the objects of the category Mod(Vx
)fr
we will be interested in those
having the following property:
DEFINITION 3.1. We say that an object
M
of Mod(Vx
)fr
has variation zero
if the morphisms
Vi:
Homvx,o
(Mo, Sa+e;,
o)
----t
Homvx,o
(Mo, Sa,
o) are zero for
alll
:S
i
:S
nand all
a
E {0, l}n with
ai
=
0.
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