LINEARIZATION OF LOCAL COHOMOLOGY MODULES 5
describe the n-hypercube corresponding to
M
from the c-straight module structure
ofM.
Note that a Nilsson class function
f
=
l:
cp,a,m(x)(logx)mx.B
,B,m
is a solution of a module with variation zero if and only if m
=
0
=
(0, ... , 0)
E
zn
and a
E
zn, i.e.
f
E
0
X [
1
] .
So, in order to determine the vertices of
X1· ·
·Xn
then-hypercube for any module
M
E
V'{;=
0
,
we only have to consider the following
spaces of solutions:
·- Ox
[x~-~-xJ
ea.- [ 1
J
L:a
-1
Ox -
k-
X1 ·
Xk · ·Xn
In particular, we can give another description of the vertices of then-hypercube
of a module with variation zero. They are the vector spaces
Ma
:=
Homvx,o(Mo,ea,o).
This description makes the vertices of the n-hypercube more treatable due to
the fact that ea are also modules with variation zero. In the sequel we will denote
for simplicity Ea the corresponding c-straight module. It is easy to see that
Ea
=
*ER(R/Pa)(l) ~ [
1
]
L:a
=lR -
k
X1 · ·
·Xk · · · Xn
where if
a=
(ab ... ,
an)
and
Pa
denotes the monomial prime ideal (xf
1
, .•• ,
x~n
),
*ER(R/Pa) is the graded injective envelope of R/Pa·
REMARK
4.1. The above facts mean, roughly speaking, that for a module of
variation zero its solutions are algebraic. This indicates how to define the category
of the modules of variation zero in the algebraic context. Namely, let
k
be any field
of characteristic zero and R
=
k[x1, ... ,
xn]·
Then, the category of algebraic D-
modules with variation zero is the category of straight R-modules as a full abelian
subcategory of the category of An(k)-modules (the algebraic D-modules). One has
to point out that this category is not closed under extensions as a subcategory of
the category of algebraic D-modules. It is also clear that one can define over
k
the
category of n-hypercubes whose variation maps are zero. Then, by flat base change,
one can see that by taking algebraic solutions, that is, the k-vector spaces Ma
:=
HomAn(k) (M, Ea), one obtains an equivalence between the category of algebraic D-
modules with variation zero and the category of n-hypercubes over k with variation
zero. Similarly, by flat base change, one could also obtain a similar equivalence in
the case of the formal power series ring k[[x1, ... ,
xnll·
All the results that come in
the sequel could be then formulated in these more general settings, but for simplicity
we shall only consider the analytic case.
We have to point out that, by using [2,
3.1],
the vertices Ma of then-hypercube
corresponding toM are isomorphic to the graded pieces M_a of the corresponding
€-straight module M for all a
E
{0,
1
}n. But, in order to reflect the €-straight
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