Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps
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| Format: | Preprint |
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2025
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| _version_ | 1866913776510435328 |
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| author | Kerner, Dmitry |
| author_facet | Kerner, Dmitry |
| contents | Consider (analytic, resp. algebraic) map-germs, Maps((k^n,o),(k^m,o)). These germs are traditionally studied up to the right, let-right and contact equivalences. Below G is one of these groups. An important tool in this study is the Artin approximation: any formal G-equivalence of maps is approximated by ordinary (i.e. analytic, resp. algebraic) G-equivalence. We consider maps of (analytic, resp. algebraic) scheme-germs, with arbitrary singularities, Maps(X,Y), and establish stronger versions of this property (for G): the Strong Artin approximation and the Płoski approximation. As a preliminary step we study the contact equivalence for maps with singular targets.
In many cases one works with multi-germs of spaces, and with their ``muti-maps". More generally, ``quivers of map-germs" occur in various applications. The needed tools are the Strong Artin approximation for quivers and the Płoski version. We establish these for directed rooted trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03414 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps Kerner, Dmitry Commutative Algebra Algebraic Geometry Consider (analytic, resp. algebraic) map-germs, Maps((k^n,o),(k^m,o)). These germs are traditionally studied up to the right, let-right and contact equivalences. Below G is one of these groups. An important tool in this study is the Artin approximation: any formal G-equivalence of maps is approximated by ordinary (i.e. analytic, resp. algebraic) G-equivalence. We consider maps of (analytic, resp. algebraic) scheme-germs, with arbitrary singularities, Maps(X,Y), and establish stronger versions of this property (for G): the Strong Artin approximation and the Płoski approximation. As a preliminary step we study the contact equivalence for maps with singular targets. In many cases one works with multi-germs of spaces, and with their ``muti-maps". More generally, ``quivers of map-germs" occur in various applications. The needed tools are the Strong Artin approximation for quivers and the Płoski version. We establish these for directed rooted trees. |
| title | Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2504.03414 |