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
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