Local Cohomology of Multi-Rees Algebras with Applications to joint Reductions and Complete Ideals
Tóm tắt
Let
$(R,\mathfrak {m})$
be a Cohen-Macaulay local ring of dimension d and I=(I
1,…,I
d
) be
$\mathfrak {m}-$
primary ideals in R. We prove that
${\lambda }_{R}([H^{d}_{(x_{ii}t_{i}:1\leq i \leq d)}({\mathcal {R}}^{\prime }({\mathcal {F}})]_{\mathbf {n}})$
$< \infty $
, for all
$\mathbf {n} \in \mathbb {N}^{d}$
, where
${\mathcal {F}}=\{{\mathcal {F}}(\mathbf {n}):\mathbf {n}\in \mathbb {Z}^{d}\}$
is an I−admissible filtration and (x
i
j
) is a strict complete reduction of
${\mathcal {F}}$
and
${\mathcal {R}}^{\prime }({\mathcal {F}})$
is the extended multi-Rees algebra of
${\mathcal {F}}.$
As a consequence, we prove that the normal joint reduction number of I,J,K is zero in an analytically unramified Cohen-Macaulay local ring of dimension 3 if and only if
$\overline {e}_{3}(IJK)-[\overline {e}_{3}(IJ)+\overline {e}_{3}(IK)$
$+\overline {e}_{3}(JK)]+\overline {e}_{3}(I)+ \overline {e}_{3}(J)+\overline {e}_{3}(K)=0$
. This generalizes a theorem of Rees on joint reduction number zero in dimension 2. We apply this theorem to generalize a theorem of M. A. Vitulli in dimension 3.
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