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Generically finite morphisms and formal neighborhoods of arcs

Authors:	Lawrence Ein and Mircea Musta&#;&#;

Institution:	(1) Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, IL 60607-7045, USA;(2) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Abstract:	Let f : X → Y be a morphism of pure-dimensional schemes of the same dimension, with X smooth. We prove that if ${\gamma\in J_{\infty}(X)}$ is an arc on X having finite order e along the ramification subscheme R _f of X, and if its image δ = f _∞(γ) on Y does not lie in J _∞(Y _sing), then the induced map T _γ J _∞(X) → T _δ J _∞(Y) is injective, with a cokernel of dimension e. In particular, if Y is smooth too, and if we denote by ${\widehat{J_{\infty}(X)_{\gamma}}}$ and ${\widehat{J_{\infty}(Y)_{\delta}}}$ the formal neighborhoods of ${\gamma\in J_{\infty}(X)}$ and ${\delta\in J_{\infty}(Y)}$ , then the induced morphism ${\widehat{J_{\infty}(X)_{\gamma}}\to \widehat{J_{\infty}(Y)_{\delta}}}$ is a closed embedding of codimension e.

Keywords:	Space of arcs Formal neighborhood Ramification subscheme
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