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Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?

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No, this is already false for a regular ring. $H^d_{\mathfrak{m}}(R)$ is a dualizing module for $R$, it is not finitely generated. If for instance $R=k[[T_1,\ldots ,T_n]]$, where $k$ is a field, $H^d_{\mathfrak{m}}(R)$ is the topological dual $\mathrm{Hom.cont}_k(R,k)$. See SGA 2, Exposé 4, no. 5 (here).

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