Zero is not a local ring

Let R be a ring such that for every x in R either x or 1 – x is invertible. Then I claim that R is a local ring. Take some time to think this through…

Brian Conrad complained here that the statement above is not true because the zero ring is not a local ring. I agree with him. The same mistake was made in the stacks project! Argh!

Fixing it led me to review the definition of a locally ringed topos. I want the definition of a locally ringed topos (see Definition Tag 04EU) when applied to a ringed space to produce a locally ringed space. Hence I decided to add a condition that guarantees that 1 is “nowhere” 0 on a locally ringed topos. Any complaints?

Note that Exercise 13.9 of Exposee IV in SGA IV suffers from the same confusion too (although, of course, I may be misreading it). I also haven’t read Hakim’s thesis which SGA tells you to do (my bad). Have you?

2 thoughts on “Zero is not a local ring

  1. Johan, if you want a reader to think through the 1st paragraph before getting to the 2nd paragraph and to possibly fall into the trap of overlooking the zero ring, perhaps change the name of the blog post to something else (such as “What is…a local ring?”). Feel free to then delete this comment.

    • Haha! Yes, well I want people to feel really dumb (like I did) when they read the second paragraph, especially if they make the same mistake after having read the title of the blog post. So I’ll leave it like this.

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