# $X_i \equiv a_i \pmod{P}$ for some $a_i \in \mathcal{O}$ given a prime ideal $P$ of $\mathcal{O}[X_1, \ldots, X_n]/(f_1, ..., f_n)$

Let $$\mathcal{O}$$ be a complete local ring with maximal ideal $$\mathfrak{m}$$. Let $$R = \mathcal{O}[X_1, \ldots, X_n]/(f_1, ..., f_n)$$ such that $$\det( \partial f_i/ \partial X_j ) \notin P$$, where $$\mathfrak{m} \subset P$$ is a prime ideal of $$R$$ such that $$R_P/P R_P \cong \mathcal{O}/\mathfrak{m}$$. How can I show that there exists $$a_i \in \mathcal{O}$$ such that each $$X_i \equiv a_i \pmod{P}$$?

• With your hypothesis, what is an example of $P$ with $R/P\neq \mathcal{O}/\mathfrak{m}$ without localizing? Commented Jul 24, 2020 at 22:04

This is the multivariate form of Hensel's Lemma. The isomorphism $$R_\mathfrak{P}/\mathfrak{P}R_\mathfrak{P} \to \mathcal{O}/\mathfrak{m}$$ is the same as a choice of $$n$$ elements $$\overline{a_i} \in \mathcal{O}/\mathfrak{m}$$ (they're just the images of $$X_i$$) and what you want is a solution of the polynomial system of equations $$\forall i: f_i(X_1,\ldots,X_n) - f_i(a_1,\ldots,a_n) = 0$$ to which you already know an approximate solution, namely $$(\overline{a_1},\ldots,\overline{a_n})$$. The condition for the determinant is exactly the non-degeneracy condition for the approximate solution that you need to apply Hensel lifting.