8
JOSEP ALVAREZ MONTANER AND SANTIAGO
ZARZUELA
Applying [1, 3.8] there is a local cohomology module
different from zero and its
characteristic cycle is
CC(H2(R))
=
T* X+ T* X+ T* X+ 2 T* X.
I x(1,1,o)
x(l,o,l) xo,l,lJ x(l,l,lJ
In particular, we only have to study the vertices of then-hypercube
(1iJ(R))o:
for
a=
(1,1,0),(1,0,1),(0,1,1),(1,1,1).
We have the Cech complex:
R
[x11 xJ
R
[x1:2xJ
EB EB
do [
1 ]
Cj:
o-R-R X1X3
~
R
[x1:2X3]
EB EB
R
[x21 xJ
R
[x1:2xJ
The complex *HomR(Cj,E(l,l,l)) is of the form:
do d1
d2
0----0----0---- C3
----
C ---- 0
whe" d,
~
(-
D .
Then,
we
get the vete"
(HJ(R))(l,l,l)
=
H2(*HomR(Cj, E(l,l,l)))
=
C2
.
The complexes *HomR(Cj, Eo:), for a=
(1, 1, 0), (1, 0, 1), (0,
1, 1),
are of the
form:
do d1 d2
0----0---- C ---- C3
----
C ----0
whe;e d,
~
(I 0 -I) and
d,
~
(-
D .
Then, we get the vert ice"
(1iJ(R)
)o:
=
H 2(*HomR(Cj, Eo:))
=
C, for a=
(1, 1, 0), (1, 0, 1), (0, 1, 1).
Computing the
linear maps among these vertices in an adequate basis we get
then-hypercube:
0
/!~
0 0 0
!XX!
c c c
~lJ,~
c
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