The Stacks project

Lemma 52.4.1. Let $I = (f_1, \ldots , f_ r)$ be an ideal of a Noetherian ring $A$. If $\text{cd}(A, I) = 1$, then there exist $c \geq 1$ and maps $\varphi _ j : I^ c \to A$ such that $\sum f_ j \varphi _ j : I^ c \to I$ is the inclusion map.

Proof. Since $\text{cd}(A, I) = 1$ the complement $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$ is affine (Local Cohomology, Lemma 51.4.8). Say $U = \mathop{\mathrm{Spec}}(B)$. Then $IB = B$ and we can write $1 = \sum _{j = 1, \ldots , r} f_ j b_ j$ for some $b_ j \in B$. By Cohomology of Schemes, Lemma 30.10.5 we can represent $b_ j$ by maps $\varphi _ j : I^ c \to A$ for some $c \geq 0$. Then $\sum f_ j \varphi _ j : I^ c \to I \subset A$ is the canonical embedding, after possibly replacing $c$ by a larger integer, by the same lemma. $\square$

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