Pure subrings of Du Bois singularities are Du Bois singularities
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917510309216256 |
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| author | Godfrey, Charles Murayama, Takumi |
| author_facet | Godfrey, Charles Murayama, Takumi |
| contents | Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$, $S$ has log canonical type singularities, and $K_R$ is Cartier, then $R$ has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kovács and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underlineΩ^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14429 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pure subrings of Du Bois singularities are Du Bois singularities Godfrey, Charles Murayama, Takumi Algebraic Geometry Commutative Algebra 14B05, 14J17 (Primary) 14L30, 14F20, 14F17 (Secondary) Let $R \to S$ be a cyclically pure map of Noetherian $\mathbb{Q}$-algebras. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. Our proof also yields interesting results in prime characteristic and in mixed characteristic. As a consequence, we show that if $R \to S$ is a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$, $S$ has log canonical type singularities, and $K_R$ is Cartier, then $R$ has log canonical singularities. Along the way, we prove a version of the key injectivity theorem of Kovács and Schwede for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. Throughout the paper, we use the characterization of the complex $\underlineΩ^0_X$ and of Du Bois singularities in terms of sheafification with respect to Grothendieck topologies. |
| title | Pure subrings of Du Bois singularities are Du Bois singularities |
| topic | Algebraic Geometry Commutative Algebra 14B05, 14J17 (Primary) 14L30, 14F20, 14F17 (Secondary) |
| url | https://arxiv.org/abs/2208.14429 |