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