Equivalence of germs (of mappings and sets) over k vs that over K
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| Format: | Preprint |
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2025
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| _version_ | 1866911627319705600 |
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| author | Kerner, Dmitry |
| author_facet | Kerner, Dmitry |
| contents | Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence.
On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C.
These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form:
* for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k;
* for the classical groups of (right/left-right/contact) equivalence of Singularity Theory;
* for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic.
The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k.
Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff their zero sets are ambient isomorphic".
This study involves the contact equivalence of maps with singular targets, which seems to be not well-established. We write down the relevant part of this theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivalence of germs (of mappings and sets) over k vs that over K Kerner, Dmitry Algebraic Geometry Commutative Algebra Complex Variables Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence. On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C. These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: * for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k; * for the classical groups of (right/left-right/contact) equivalence of Singularity Theory; * for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic. The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k. Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff their zero sets are ambient isomorphic". This study involves the contact equivalence of maps with singular targets, which seems to be not well-established. We write down the relevant part of this theory. |
| title | Equivalence of germs (of mappings and sets) over k vs that over K |
| topic | Algebraic Geometry Commutative Algebra Complex Variables |
| url | https://arxiv.org/abs/2504.10922 |