6

JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA

module structure of M, we will consider the following description, which is the

main result of this section.

THEOREM 4.2. Let M

E

v;;=O

be a regular holonomic Vx-module with vari-

ation zero and M

E

r:: -

Str

be the corresponding r::-straight module.

Denote by

(M-a)* the dual of the C.-vector space defined by the piece of M of

multidegree -a,

for all

a

E

{0,

1

}n.

Then, there are isomorphisms

such that the following diagram commutes:

where

(Xi)*

is the dual of the multiplication by

xi.

PROOF. By using the isomorphism given in Lemma 3.5, we only

have to de-

scribe the C.-vector space *HomR(M, Ea) in order to prove the

existence of the

isomorphisms.

Any map f E *HomR(M, Ea) is determined by the pieces f-!3 , {3 E {0, 1}n

due to the fact that M and Ea are r::-straight modules. Notice that

f

-/3

=

0, for all

{3

a,

due to the fact that !Ja is the unique associated prime of Ea. On the other

side, for all i such that

ai

=

0 we have

1

i.e. f

-a-E:

:= -

o f-a o

Xi.

By using this construction in an analogous way we

t

Xi

can

describe f-!3 for all {3

2:

a. In particular, the map f E *HomR(M, Ea) is

determined by the piece f-a· Namely, we have the isomorphism:

Homc(M_a, [Ea]-a)

f-a

~

*HomR(M, Ea)

)

f

Finally, since [Ea]-a is the

C.-vector space spanned by

_!:_,

the multiplication by

xn

x

0

gives an isomorphism:

Homc(M-a, [Ea]-a)

f-a

~

Homc(M_a,

C)= (M-a)*

·· ·

Xa f-a

where we consider C as the C.-vector space spanned by 1.

Once the vertices of the n-hypercube are determined, we only have to

describe

the linear map

Ui:

*HomR(M, Ea)

-----**

HomR(M, Ea+EJ

induced by

the natural quotient maps Ea

-----*

Ea+c,.

Let f E

*HomR(M, Ea) be a morphism described by the linear map f-a E

Homc(M_a, [Ea]-a)· Then, the corresponding morphism

f

E

*HomR(M, Ea+EJ