An effective Cartier divisor on a scheme $X$ (a closed subscheme of $X$ that is locally cut out by one equation that is not a zero-divisor) is always a regular embedding of codimension 1 (in the stalks, it is cut out by one equation that is not a zero-divisor). The converse is true if $X$ is locally Noetherian, using Nakayama's lemma to "lift" from the stalk to an "honest neighborhood".

Is there a (necessarily non-Noetherian) example of a codimension 1 regular embedding that is not an effective Cartier divisor?

(See section 8.4.7 of the March 23, 2013 version of the notes **here** for some discussion, if you care.)

I am asking this out of idle curiosity rather than any need for it, but I am genuinely curious.