The Stacks project

Lemma 110.47.2. There exists an affine scheme $X = \mathop{\mathrm{Spec}}(A)$ whose underlying topological space is Noetherian and an injective $A$-module $I$ such that $\widetilde{I}$ has nonvanishing $H^1$ on some quasi-compact open $U$ of $X$.

Proof. See above. Note that $\mathop{\mathrm{Spec}}(A) = \mathop{\mathrm{Spec}}(k[x, y])$ as topological spaces. $\square$

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