# Injective map on spectra [closed]

I know that if we have a surjection $$f:B\rightarrow A$$, this induces an injection on the spectra $$f^* \colon \operatorname{Spec} A\hookrightarrow \operatorname{Spec} B.$$

What about the opposite? Does an injection in the spectra of affine schemes induce a surjection in rings?

I would assume if such a nice property held I would have found it somewhere online, so can you provide a counterexample? Also, is there a natural set of conditions such that if $$f^*$$ satisfies them, then we can conclude that $$f$$ is a surjection?

## closed as off-topic by Pace Nielsen, Qiaochu Yuan, abx, Sean Lawton, user44191Mar 15 at 1:41

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• Is $A\rightarrow S^{-1}A$ surjective?. – Einfacher Schreiberling Mar 14 at 17:49
• Nope. Think of a localization $A\to A[f^{-1}]$ – Qfwfq Mar 14 at 17:49
• Ok.. almost at the same time! :D – Qfwfq Mar 14 at 17:49
• Oh of course you are right! Thank you both very much. – Ioannis Zolas Mar 14 at 18:16
• Or think about $\mathbb{Q} \subseteq \mathbb{R}$. But you do want to think about closed embeddings of (affine) schemes, those come from surjective ring maps. See also II, Exercise 3.11 in Hartshorne as well. – Karl Schwede Mar 14 at 18:30