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The dualizing complex of F-injective and Du Bois singularities

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Abstract

Let \((R,\mathfrak {m}, k)\) be an excellent local ring of equal characteristic. Let j be a positive integer such that \(H_\mathfrak {m}^i(R)\) has finite length for every \(0\le i <j\). We prove that if R is F-injective in characteristic \(p>0\) or Du Bois in characteristic 0, then the truncated dualizing complex 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 generalizing Ishida (The dualizing complexes of normal isolated Du Bois singularities. Algebraic and topological theories, 387–390, 1984) and Ma (Math Ann 362:25–42, 2015).

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Notes

  1. Or dually, that has finite length for \(0 \le i < j\) (provided a dualizing complex exists). Under mild conditions this is equivalent to saying that the non-Cohen–Macaulay locus on \(\mathrm {Spec}(R)\) has codimension j.

  2. Equivalently, \(\tau ^{< d,*} \mathbf{R}\Gamma _\mathfrak {m}(R)[1] \in D(R)\) is the cone of the canonical composite map \(\mathbf{R}\Gamma _\mathfrak {m}(R) \rightarrow H^d_\mathfrak {m}(R)[-d] \rightarrow (H^d_\mathfrak {m}(R)/0^*_{H^d_\mathfrak {m}(R)})[-d]\).

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Acknowledgements

We would like to thank Shunsuke Takagi for several useful discussions and for bringing these questions to our attention. We would also like to thank Sándor Kovács, Zsolt Patakfalvi and Sean Sather-Wagstaff for useful discussions. Finally we thank Lance Miller, Kazuma Shimomoto and the referees for comments on previous drafts of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109, USA

    Bhargav Bhatt

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Linquan Ma & Karl Schwede

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  1. Bhargav Bhatt
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  2. Linquan Ma
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  3. Karl Schwede
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Correspondence to Karl Schwede.

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Bhargav Bhatt, was supported by NSF Grants DMS #1501461 and DMS #1522828 and by a Packard Fellowship. Linquan Ma, was supported by NSF Grant DMS #1600198, NSF CAREER Grant DMS #1252860/1501102 and a Simons Travel Grant. Karl Schwede was supported by the NSF FRG Grant DMS #1265261/1501115, NSF CAREER Grant DMS #1252860/1501102 and a Sloan Fellowship.

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Bhatt, B., Ma, L. & Schwede, K. The dualizing complex of F-injective and Du Bois singularities. Math. Z. 288, 1143–1155 (2018). https://doi.org/10.1007/s00209-017-1929-5

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