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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)