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

Question

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 $X_i \equiv a_i \pmod{P}$?

Answer 1

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.