The Stacks project

Lemma 110.66.1. Non-finite $H^0$.

  1. There exists a ring $R$ and a projective scheme $X$ over $R$ such that $H^0(X, \mathcal{O}_ X)$ is not a finite $R$-module.

  2. There exists a ring $R$ and a finite type quasi-coherent module $\mathcal{F}$ on $\mathbf{P}^1_ R$ such that $H^0(\mathbf{P}^1_ R, \mathcal{F})$ is not a finite $R$-module.

Proof. See discussion above. $\square$

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