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Let $(R,m,k)$ be an excellent local ring of equal characteristic. Let $j$ bea positive integer such that $H_m^i(R)$ has finite length for every $0\leq i<j$. We prove that if $R$ is $F$-injective in characteristic $p>0$ or Du Boisin characteristic $0$, then the truncated dualizing complex$\tau_{>-j}\omega_R^\bullet$ is quasi-isomorphic to a complex of $k$-vectorspaces. As a consequence, $F$-injective or Du Bois singularities with isolatednon-Cohen-Macaulay locus are Buchsbaum. Moreover, when $R$ has $F$-rational orrational singularities on the punctured spectrum, we obtain stronger results.

The dualizing complex of $F$-injective and Du Bois singularities