4

JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA

Modules with variation zero form a full abelian subcategory of Mod(Vx

)rr

that

will be denoted

V'{;=O·

In the sequel, we will denote by

Cn,v=O

the corresponding

full abelian subcategory of

Cn

of n-hypercubes having variation zero.

In [2] it is proven that a slight variation of the category of straight modules

introduced by K. Yanagawa [13] is equivalent to the category

v;=O

of modules with

variation zero. More precisely, let

R

= C[x1

, ...

,xn]

be the polynomial ring with

coefficients in IC and

An(IC)

the corresponding Weyl algebra.

Let M be a graded R-module and

a

E

zn.

As usual, we denote by M(a)

the graded R-module whose underlying R-module structure is the same as that

of

M

and where the grading is given by (M(a))13 =

Ma.+/3·

If a

E

zn,

we set

supp+ (a)=

{i

I

ai

0 }. We recall now the following definition of K. Yanagawa:

DEFINITION

3.2. ([13, 2.7]) A zn-graded module M is said to be straight if

the following two conditions are satisfied:

i) dimk

Ma.

oo for all

a

E

zn.

ii) The multiplication map

Ma.

3 y

f-+

xi3y E

Ma.+/3

is bijective for all

a,

(3

E

zn

with supp+

(a+ (3)

= supp+ (a).

The full subcategory of the category *Mod(R) of zn_graded R-modules which

has as objects the straight modules will be denoted Str. Let 1 = (1, ... , 1)

E

zn.

The shifted local cohomology modules HJ(R)( -1) supported on monomial ideals

I

s;;

R

are straight modules. In order to avoid shiftings, we will consider instead

the following (equivalent) category:

DEFINITION

3.3. We will say that a graded module

M

is c-straight if

M(

-1)

is straight in the above sense. We denote c-Str the full subcategory of *Mod(R)

which has as objects the c-straight modules.

If M is a An(IC)-module, then

Man

:= Ox

0R

M has a natural Vx-module

structure. This allows to define a functor

(-)an:

Mod(An(IC)) ---+ Mod(Vx ).

M ---+

Man

f

---+ id 0

f

On the other hand, any c-straight module M can be endowed with a functorial

An(IC)-module structure extending its R-module structure, see [13]. Then, we have:

THEOREM

3.4. ([2, 4.3])

The functor

(-)an:

c- Str---+

v;=O

is an equivalence of categories.

The following lemma (which in particular gives the fully faithfulness of (-)an)

will be useful in the sequel.

LEMMA

3.5. ([2, 4.4])

Let M, N be .s-straight modules. For all i

2': 0,

we have

functorial isomorphisms

*Extk(M, N)

~

ExtbT

(Man, Nan).

v=O

4. The graded structure of Vx-modules with variation zero

Let

M

E

V'{;=

0

be a regular holonomic Vx-module with variation zero and let

A1

E

c - Str be the corresponding c--straight module. Our aim in this section is to