LINEARIZATION OF LOCAL COHOMOLOGY MODULES 9
The n-hypercubes corresponding to the local cohomology modules supported
on squarefree monomial ideals can be better described in topological terms by using
the cellular structure of the complexes used in their computation.
As before, let I
=
(x01
, •.• ,
x
0
•)
~
R be a minimal system of generators of a
squarefree monomial ideal I. Consider the Cech complex:
Vertices:
To describe the vertices of the n-hypercubes corresponding to the
local cohomology modules
HJ(R)
we only have to notice the following:
Applying the functor *HomR( -, E
0
)
to the Cech complex Cj in the case
a
=
0
=
(0, ... , 0)
E
{0, 1
}n, we obtain the complex:
* (
v )
do
d1
d.
-1 HomRC1,Eo:
o--c--cs-- ..
·--C--0.
This complex may be identified with the augmented relative simplicial chain com-
plex C.(~; C), where ~ is the full simplicial complex whose vertices
{x1, ... ,
X
8 }
are labelled by the minimal system of generators of I.
In general, for any a
E
{0,
1}n,
the terms of the complex *HomR(Cj, Ea)
are:
*HomR
(R
[2_] ,
Ea)
=
{C
if (3
2:
a,
x/3
0 otherwise.
Namely, from the augmented relative simplicial chain complex
C.(~;
C), we are
taking out the pieces corresponding to the faces
a1-/3
:=
{x1, ...
,xs} \{xi
I
(Ji
=
1}
E ~
such that (3
t_
a.
Let Ta
:= {
a1-/3
E ~
I
(3
t_
a} be a simplicial subcomplex of
~.
Then, the
complex *HomR( Cj, Ea) may be identified with the augmented relative simplicial
chain complex C.(~, T0
;
C) associated to the pair (~, T0
).
By taking homology,
the vertices of the n-hypercubes corresponding to the local cohomology modules
HJ(R)
are:
(1t/(R))a
=
Hr(*HomR(Cj,Ea))
=
Hr-l(~,Ta;C)
=
Hr-2(T0 ;C),
where the last assertion comes from the fact that
~
is contractible.
Linear maps:
By using the description of the vertices we notice the following:
The morphism of complexes *HomR(Cj, Ea)) ---+* HomR(Cj, Ea+c-J, in-
duced by the quotient map Ea ---+ Ea+c-; is nothing but the morphism of com-
plexes C.(~,T
0
;C)---+ C.(~,Ta+c-;;C) induced by the inclusion T0 ~ Ta+c-;· By
taking homology, the linear maps
ui
of the n-hypercubes corresponding to the local
cohomology modules
HJ(R)
are:
lli:
Hr-2(Ta; C) ---+ Hr-2(Ta+c-;; C),
induced by the inclusion T0
~
Ta+c-;.
Collecting the previous results we get:
PROPOSITION
5.2. Then-hypercubes corresponding to local cohomology modules
HJ(R) supported on squarefree monomial ideals
I~
R are:
Vertices:
(1t'j(R))
0
~ Hr-2(Ta; C).
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