Lemma 109.44.1. There exists an affine scheme $X = \mathop{\mathrm{Spec}}(A)$ and a closed subscheme $T \subset X$ such that $T$ is Zariski locally on $X$ cut out by ideals generated by idempotents, but $T$ is not cut out by an ideal generated by idempotents.

Proof. See above. $\square$

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