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

TL;DR

This paper shows that for certain singularities in algebraic geometry, the truncated dualizing complex simplifies to a complex of vector spaces, leading to Buchsbaum properties under specific conditions.

Contribution

It establishes that $F$-injective and Du Bois singularities with isolated non-Cohen-Macaulay loci have simplified dualizing complexes, extending understanding of their structure.

Findings

Truncated dualizing complex is quasi-isomorphic to a complex of $k$-vector spaces.

Such singularities are Buchsbaum when the non-Cohen-Macaulay locus is isolated.

Stronger results are obtained when the ring has $F$-rational or rational singularities on the punctured spectrum.

Abstract

Let $(R,m,k)$ be an excellent local ring of equal characteristic. Let $j$ be a 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 Bois in characteristic $0$, then the truncated dualizing complex $\tau_{>-j}\omega_R^\bullet$ is quasi-isomorphic to a complex of $k$-vector spaces. As a consequence, $F$-injective or Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. Moreover, when $R$ has $F$-rational or rational singularities on the punctured spectrum, we obtain stronger results.