Proposition 4

Let $A$ be a regular ring, $J$ an ideal of $A$, and $A_0$ a subring of $A$ such that the canonical homomorphism $A_0 \to A/J$ is bijective.

a) The ideal $J$ is completely secant, the $A_0$-module $J/J^2$ is finitely generated projective, and the ring $A_0$ is regular.

b) Let $\varphi : J/J^2 \to J$ be an $A_0$-linear section of the canonical surjection $J \to J/J^2$. The homomorphism of $A_0$-algebras $S_{A_0}(J/J^2) \longrightarrow A$ extending $\varphi$ extends to an isomorphism of the completion of the graded ring $S_{A_0}(J/J^2)$ onto the separated completion of $A$ for the $J$-adic topology.

a) Let $\mathfrak{p}$ be a prime ideal of $A$ containing $J$. We have $\mathfrak{p} = (\mathfrak{p} \cap A_0) \oplus J$ and hence $\mathfrak{p}^2 = (\mathfrak{p} \cap A_0)^2 \oplus \mathfrak{p}J$, therefore $\mathfrak{p}^2 \cap J = \mathfrak{p}J$. Let $i$ denote the canonical injection of $J$ into $\mathfrak{p}$. From the preceding, the mapping $i \otimes 1_{A/\mathfrak{p}} : J \otimes_A A/\mathfrak{p} \longrightarrow \mathfrak{p} \otimes_A A/\mathfrak{p}$ is injective, and the same is true of $i_p \otimes 1_{\kappa(\mathfrak{p})} : J_p \otimes_{A_p} \kappa(\mathfrak{p}) \longrightarrow \mathfrak{p}A_p \otimes_{A_p} \kappa(\mathfrak{p})$. Nakayama's lemma implies that the ideal $J_p$ of $A_p$ is generated by a subset of a system of coordinates of the regular local ring $A_p$. By Prop. 2 of VIII, § 5, No. 3, the ring $A_p/J_p$ is regular and the ideal $J_p$ is completely secant. Thus $J$ is completely secant, the ring $A_0$, isomorphic to $A/J$, is regular, and the $A_0$-module $J/J^2$ is finitely generated projective by Theorem 1 (No. 2).

b) It follows from Theorem 1 that the canonical homomorphism $\beta_J : S_{A_0}(J/J^2) \longrightarrow \mathrm{gr}_J(A)$ is bijective. Let $f : S_{A_0}(J/J^2) \longrightarrow A$ be the homomorphism of $A_0$-algebras extending the $A_0$-linear mapping $\varphi : J/J^2 \longrightarrow J$. If $A$ is endowed with the $J$-adic filtration and $S_{A_0}(J/J^2)$ with the filtration associated with its graduation, $\beta_J$ is identified with the homomorphism deduced from $f$ by passing to the associated graded rings. Assertion b) then follows from III, § 2, No. 8, Corollary 3 to Theorem 1.