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Note that if $T$ is reduced, this morphism has dense image. If $T$ is a locally compact totally disconnected hausdorff space, it can be given explicitely a reduced scheme structure, which is affine iff $T$ is compact. Besides $T \to Spec \Gamma(T)$ is an dense open immersion. Thus it is not closed as long $T$ is not compact. I've proven this here. Thus a counterexample would be $T = \mathbb{Q}_p$. I expect that others will show you more "natural" examples. |
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If $T$ is a locally compact totally disconnected hausdorff space, it can be given explicitely a reduced scheme structure, which is affine iff $T$ is compact. Besides $T \to Spec \Gamma(T)$ is an dense open immersion. Thus it is not closed as long $T$ is not compact. I've proven this here. Thus a counterexample would be $T = \mathbb{Q}_p$. I expect that others will show you more "natural" examples. |
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