Purity of branch and critical locus
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arXiv
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| Format: | Preprint |
| Publié: |
2010
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| _version_ | 1866917206669918208 |
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| author | Källström, Rolf |
| author_facet | Källström, Rolf |
| contents | To a dominant morphism $X/S \to Y/S$ of Nœtherian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1003_5872 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Purity of branch and critical locus Källström, Rolf Algebraic Geometry Commutative Algebra 14A10 (Primary), 32C38, 17B99 Secondary) To a dominant morphism $X/S \to Y/S$ of Nœtherian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $Ω_{X/Y}$, $Ω_{X/S}$, and $Ω_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms. |
| title | Purity of branch and critical locus |
| topic | Algebraic Geometry Commutative Algebra 14A10 (Primary), 32C38, 17B99 Secondary) |
| url | https://arxiv.org/abs/1003.5872 |