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Let $(R,\m)$ be a commutative Noetherian local ring and $I$ be an ideal of $R$. In this paper it is shown that if ${\rm cd}(I,R)=t>0$ and the $R$-module $\Hom_R(R/I, H^t_I(R))$ is finitely generated, then \begin{align*} t={\rm sup}\,\{\dim \widehat{\widehat{R}_{\mathfrak P}}/\mathfrak Q:~&\mathfrak P\in V(I\widehat{R}),\,\,\mathfrak Q\in \mAss_{\widehat{\widehat{R}_{\mathfrak P}}}\widehat{\widehat{R}_{\mathfrak P}}\,\,{\rm and}\\ &\mathfrak P\widehat{\widehat{R}_{\mathfrak P}}=\Rad(I\widehat{\widehat{R}_{\mathfrak P}}+\mathfrak Q)\}. \end{align*} Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R\subset R[X]$ will be included.
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참고문헌 (18)
참고문헌 신청
K. Bahmanpour / 2017 / A note on Lynch's conjecture / Comm. Algebra 45(6):2738~2745
K. Bahmanpour / 2008 / On the cofiniteness of local cohomology modules / Proc. Amer. Math. Soc. 136(7):2359~2363
K. Bahmanpour / 2009 / Cofiniteness of local cohomology modules for ideals of small dimension / J. Algebra 321(7):1997~2011
K. Bahmanpour / 2013 / Minimaxness and cofiniteness properties of local cohomology modules / Comm. Algebra 41(8):2799~2814
M. P. Brodmann / 1998 / Local Cohomology: an algebraic introduction with geometric applications / Cambridge University Press
K. Bahmanpour / 2008 / On the cofiniteness of local cohomology modules / Proc. Amer. Math. Soc. 136(7):2359~2363
K. Bahmanpour / 2009 / Cofiniteness of local cohomology modules for ideals of small dimension / J. Algebra 321(7):1997~2011
K. Bahmanpour / 2013 / Minimaxness and cofiniteness properties of local cohomology modules / Comm. Algebra 41(8):2799~2814
M. P. Brodmann / 1998 / Local Cohomology: an algebraic introduction with geometric applications / Cambridge University Press
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