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).
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