10 JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA
Linear maps: We have the following commutative diagram:
where
Vi
is induced by the inclusion
Ta
:;;;
Ta+E,.
REMARK
5.3. By using the previous result and Theorem 4.2 we recover the
result on the module structure of the local cohomology modules
HJ(R)
given by
M. Mustata
[10].
Namely, if (- )* denotes the dual of a C-vector space, the graded
pieces of
HJ(R)
are:
[H[(R)]-cx
~
(Hr-2(Ta;
C))*~
jjr-
2(Ta;
C), a
E
{0,
l}n,
and the multiplication map
Xi:
[H[(R)]-a-E,
-----
[H[(R)]-cx
is determined by the
following commutative diagram:
where
Vi
is induced by the inclusion
Ta
:;;;
Ta+E;.
References
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J.
Pure Appl. Algebra, 150 (2000), 1-25.
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subspaces and monomial ideals, Adv. Math. 174 (2003), 35-56.
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[4]
J.-
E. Bjork, Analytic V-modules and applications, Mathematics and its Applications, Vol.
247, Kluwer Academic Publishers, Dordrecht, 1993.
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New York, 1987.
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1983), Asterisque, 130 (1985), 240-259.
[8] S. Goto and K. Watanabe, On Graded Rings, II (lln- graded rings), Tokyo
J.
Math., 1-2
(1978), 237-261.
[9] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of V-
modules to commutative algebra), Invent. Math., 113 (1993), 41-55.
[10] M. Mustata, Local Cohomology at Monomial Ideals,
J.
Symbolic Comput., 29 (2000), 709-
720.
[11] U. Walther, Algorithmic computation of local cohomology modules and the cohomological
dimension of algebraic varieties,
J.
Pure Appl. Algebra, 139 (1999), 303-321.
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(2001), 1631-1634.
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