The dualizing complex of \(F\)-injective and Du Bois singularities
Let \((R, \mathfrak{m}, k)\) be an excellent ring of equal characteristic. The paper under review studies several complexes associated to the normalized dualized complex of \(R\), when \(R\) is \(F\)-injective in prime characteristic or when \(R\) defines a Du Bois singularity in characteristic zero. Specifically the authors assume that either (1) \(R\) is F-injective of prime characteristic or (2) \(R\) is essentially of finite type over a field of characteristic zero and is Du Bois. Now fix an integer \(j>0\) and assume that the local cohomology modules \(H^i_{\mathfrak{m}}(R)\) have finite length for all \(0\leq i<j\). The main result of the paper shows that, under these conditions, the trucation of the normalized dualizing complex \(\tau_{>-j}\omega_R^\cdot\) is quasi-isomorphic to a complex of \(k\)-vector spaces. This theorem recovers a result of Ma that asserts that \(F\)-injective singularities with isolated non-Cohen-Macaulay locus are [\textit{L. Ma}, Math. Ann. 362, No. 1--2, 25--42 (2015; Zbl 1398.13003)]. Additionally, it answers a question of Schwede and Takagi, by showing that Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum, generalizing work of \textit{M. N. Ishida} [in: Algebraic and topological theories. Papers from the symposium dedicated to the memory of Dr. Takehiko Miyata held in Kinosaki, October 30- November 9, 1984. Tokyo: Kinokuniya Company Ltd.. 387--390 (1986; Zbl 0800.32007)]. Moreover, the paper contains a more elementary approach of the main result in the case where the characteristic is prime and the field \(k\) is perfect. It also contains generalizations of the main result by involving some additional complexes related to normalized dualizing complex \(\omega_R^{\cdot}\).
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