LINEARIZATION OF LOCAL COHOMOLOGY MODULES

3

vector spaces, endowed with linear maps

u· v·

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.