Generically finite morphisms and formal neighborhoods of arcs

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.