Faltings' Finiteness Dimension of Local Cohomology Modules Over Local Cohen–Macaulay Rings

Bahmanpour, Kamal ; Naghipour, Reza

Canadian mathematical bulletin, Tome 60 (2017) no. 2, pp. 225-234

Résumé

Let $(R,\mathfrak{m})$ denote a local Cohen–Macaulay ring and $I$ a non-nilpotent ideal of $R$ . The purpose of this article is to investigate Faltings’ finiteness dimension ${{f}_{I}}(R)$ and the equidimensionalness of certain homomorphic images of $R$ . As a consequence we deduce that ${{f}_{I}}(R)=\max \{1,\text{ht}I\}$ , and if $\text{mAs}{{\text{s}}_{R}}(R/I)$ is contained in $\text{As}{{\text{s}}_{R}}(R)$ , then the ring $R/I+{{\bigcup }_{n\ge 1}}(0{{:}_{R}}{{I}^{n}})$ is equidimensional of dimension dim $R-1$ . Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H_{I}^{\text{ht}I}(R)$ , in the case where $(R,\mathfrak{m})$ is a complete equidimensional local ring.

Détail

DOI : 10.4153/CMB-2016-092-9

Mots-clés : 13D45, 14B15, Cohen Macaulay ring, equidimensional ring, finiteness dimension, local cohomology