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There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27) But under some very reasonable additional conditions - I think quasi-compactness will be sufficient, if you are happy with using Zorn's lemma - the result holds. Use/prove the existence of a closed point, and the fact that localizing a reduced ring still gives you a reduced ring. Brunoh, you don't seem to believe me, would you like me to write it out in detail? |
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There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27) But under some very reasonable additional conditions - I think quasi-compactness will be sufficient, if you are happy with using Zorn's lemma - the result holds. Use/prove the existence of a closed point, and the fact that localizing a reduced ring still gives you a reduced ring. Brunoh, you don't seem to believe me, would you like me to write it out in detail? |
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There do exist schemes without a closed point, yes. (Liu, exercises 3.3.26/27) But under some very reasonable additional conditions - I think quasi-compactness will be sufficient - the result holds. Use/prove the existence of a closed point, and the fact that localizing a reduced ring still gives you a reduced ring. |
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