Lemma 30.23.11. Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. If $V(\mathcal{I}) = V(\mathcal{J})$ is the same closed subset of $X$, then $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(X, \mathcal{J})$ are equivalent.

**Proof.**
First, assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. Let $I, J \subset A$ be the ideals corresponding to $\mathcal{I}, \mathcal{J}$. Then $V(I) = V(J)$ implies we have $I^ c \subset J$ and $J^ d \subset I$ for some $c, d \geq 1$ by elementary properties of the Zariski topology (see Algebra, Section 10.17 and Lemma 10.32.5). Hence the $I$-adic and $J$-adic completions of $A$ agree, see Algebra, Lemma 10.96.9. Thus the equivalence follows from Lemma 30.23.1 in this case.

In general, using what we said above and the fact that $X$ is quasi-compact, to choose $c, d \geq 1$ such that $\mathcal{I}^ c \subset \mathcal{J}$ and $\mathcal{J}^ d \subset \mathcal{I}$. Then given an object $(\mathcal{F}_ n)$ in $\textit{Coh}(X, \mathcal{I})$ we claim that the inverse system

is in $\textit{Coh}(X, \mathcal{J})$. This may be checked on the members of an affine covering; we omit the details. In the same manner we can construct an object of $\textit{Coh}(X, \mathcal{I})$ starting with an object of $\textit{Coh}(X, \mathcal{J})$. We omit the verification that these constructions define mutually quasi-inverse functors. $\square$

## 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Your email address will not be published.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)