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).