JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA
Modules with variation zero form a full abelian subcategory of Mod(Vx
will be denoted
In the sequel, we will denote by
full abelian subcategory of
of n-hypercubes having variation zero.
In  it is proven that a slight variation of the category of straight modules
introduced by K. Yanagawa  is equivalent to the category
of modules with
variation zero. More precisely, let
be the polynomial ring with
coefficients in IC and
the corresponding Weyl algebra.
Let M be a graded R-module and
As usual, we denote by M(a)
the graded R-module whose underlying R-module structure is the same as that
and where the grading is given by (M(a))13 =
0 }. We recall now the following definition of K. Yanagawa:
3.2. ([13, 2.7]) A zn-graded module M is said to be straight if
the following two conditions are satisfied:
oo for all
ii) The multiplication map
is bijective for all
= 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)
The shifted local cohomology modules HJ(R)( -1) supported on monomial ideals
are straight modules. In order to avoid shiftings, we will consider instead
the following (equivalent) category:
3.3. We will say that a graded module
is c-straight if
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
M has a natural Vx-module
structure. This allows to define a functor
Mod(An(IC)) ---+ Mod(Vx ).
---+ id 0
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 . Then, we have:
3.4. ([2, 4.3])
is an equivalence of categories.
The following lemma (which in particular gives the fully faithfulness of (-)an)
will be useful in the sequel.
3.5. ([2, 4.4])
Let M, N be .s-straight modules. For all i
4. The graded structure of Vx-modules with variation zero
be a regular holonomic Vx-module with variation zero and let
c - Str be the corresponding c--straight module. Our aim in this section is to