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?