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
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