Finite separating sets and quasi-affine quotients
Nagata’s famous counterexample to Hilbert’s fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-aff...
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| Format: | Journal article |
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Elsevier
2012
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| _version_ | 1826285576005877760 |
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| author | Dufresne, E |
| author_facet | Dufresne, E |
| author_sort | Dufresne, E |
| collection | OXFORD |
| description | Nagata’s famous counterexample to Hilbert’s fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some known examples and in a new construction. |
| first_indexed | 2024-03-07T01:30:54Z |
| format | Journal article |
| id | oxford-uuid:938f8fe9-30e0-433f-81b8-a37f024c659a |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:30:54Z |
| publishDate | 2012 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:938f8fe9-30e0-433f-81b8-a37f024c659a2022-03-26T23:33:09ZFinite separating sets and quasi-affine quotientsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:938f8fe9-30e0-433f-81b8-a37f024c659aSymplectic Elements at OxfordElsevier2012Dufresne, ENagata’s famous counterexample to Hilbert’s fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from affine. Indeed, invariant rings are always quasi-affine, and there always exist finite separating sets. In this paper, we give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and we give a criterion to recognize separating algebras. The method and criterion are used on some known examples and in a new construction. |
| spellingShingle | Dufresne, E Finite separating sets and quasi-affine quotients |
| title | Finite separating sets and quasi-affine quotients |
| title_full | Finite separating sets and quasi-affine quotients |
| title_fullStr | Finite separating sets and quasi-affine quotients |
| title_full_unstemmed | Finite separating sets and quasi-affine quotients |
| title_short | Finite separating sets and quasi-affine quotients |
| title_sort | finite separating sets and quasi affine quotients |
| work_keys_str_mv | AT dufresnee finiteseparatingsetsandquasiaffinequotients |