LINEARIZATION OF LOCAL COHOMOLOGY

MODULES

7

induced by the quotient map

Ea. __, Ea.+ei

is described by the linear map

f -a.-r;;i

E

Homc(M_a.-r;;., [Ea.+eil-a.-r;;J defined as:

...!..

M

Xi

M

f-a [ ]

"i [ ]

-a.-r;;i

~

-a

~

Ea. -a.

--~

Ea. -a.-r;;i

lid

[Ea.+eJ

-a.-r;;i

In

particular we have the commutative diagram:

*HomR(M, Ea.+r;;.)

)

7

f-a. .........................

f -a.-r;;i

Homc(M_a., [Ea.]-a.) Homc(M_a.-r;;., [Ea.+r;;J-a.-r;;.)

Then we are done because the following diagram is

commutative:

(M-a.)*

(xi}*

xa.

f-a.

············)

"'

f-a.

················

f -a.-r;;i

Homk(M-a., [Ea.]-a.) Homk(M-a.-r;;.,

[Ea.+eil-a.-r;;.)

Namely, we have:

(

)*(Xa.J-a) a.J a.+ei J a.+ei J

Xi

=X

-a.Xi

=X

-1

-a.Xi

=X -a.-e·.

Xi •

0

5. Local cohomology modules: Topological

interpretation

Let

I

= ( xa.

1

, ... ,

xa.•)

~

R,

a:i

E {

0,

1}

n,

be a minimal system of

generators of

a

squarefree monomial ideal

I.

Consider the Cech complex

Cj:

o~R~

ffi

R[-i-]

~---~R[

1

]~o.

"J7 X

It

Xa.1 ... xa.s

l:"'i1:"0s

By using the equivalence of categories given in Section 3 and the contravariance

of the functor

Mod(Dx

)Ir

~

Cn,

then-hypercube corresponding to a local coho-

mology module

Hl(R)

supported on

I

may be described as follows:

• Vertices:

(1-l!(R))a.

=

Hr(*HomR(Cj, Ea.)).

• Linear

maps:

Ui:

Hr(*HomR(Cj, Ea.))

~

Hr(*HomR(Cj, Ea.+r;;.)),

in-

duced by the

quotient map

Ea.

~

Ea.+ei.

To illustrate this computations we present the following:

EXAMPLE

5.1. Let R =

C[xb x2, x3].

Consider the ideal:

• I=

(x1x2,x1x3,x2x3).