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arXiv · 2504.03414

Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps

Dmitry Kerner
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Abstract

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 P\l oski 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 P\l oski version. We establish these for directed rooted trees.

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Dmitry Kerner. 2025-04-04. Further results on Artin approximation, for group-actions on mapping-germs Maps(X,Y) and for quivers of maps. https://arxiv.org/abs/2504.03414

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