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
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