Re: The division lattice as a category: is 0 prime?

[Vaughan Pratt <pratt@cs.stanford.edu>]

Jeff Egger wrote:
> And I thought that every generation since Dedekind, Krull and Noether
> knew that divisibility lattices are (in the general case) a red herring
> and that it is the lattice of ideals of a ring (or its opposite, if you
> prefer) which is really important.
 > ...
 > although I don't really understand your motivation.

Right, I should have been clearer about the motivation.  I wanted to
construct the division lattice abstractly from the primes in some
finitary way, analogously to how one can construct the power set 2^X as
the free upper semilattice generated by the singletons of X.  Putting
that in terms of ideals, I'd like to be able to form all the ideals of Z
from just the prime ideals.  I don't know much about ring theory so I
could be confused about this, but I would have thought intersecting them
could only get you the square-free ideals.  Starting from the prime
power ideals takes care of that but what's the trick for getting all the
ideals from just the prime ideals?  The category Div was my suggestion
for that, but if there's a more standard approach in ring theory I'd be
happy to use that instead (or at least be aware of it---Div is starting
to grow on me).

Now that I think of it, I suppose the standard completion must be the
formation of finite subdirect products (aka sums?) of the quotients Z_p
= Z/pZ over the prime ideals pZ.  By including Z along with the Z's,
that way you reconstruct Div with the lower part consisting of Z_n =
Z/nZ and the upper part n.Z (if I understand the notation).  That puts
the ring structure of Z back into play however, which doesn't feel quite
as "pure" as simply closing a flat inverted CPO under finite coproducts.

>  Perhaps the answer to your original
> question is to take (finite-valued) sheaves on this space of primes,

Right, that (by Yoneda) was the completion under finite colimits
approach at the end of my 10:40 am message this morning, which didn't
"work" in the sense of not being the minimal solution and not having an
obviously pleasing structure either.  Completion under finite coproducts
was as small as I could make it, and initially I was miffed that there
was still this junk above the division lattice that I was hoping would
go away.  But then I decided that rather than complicate the completion
process to prevent 0 from sprouting sow's ears above it, I'd try to make
a silk purse out of the ears.  This ended up being the two-part
Fundamental Theorem of Arithmetic via the single construction.

With coproducts instead of colimits it's still sheaves but with the
condition that if the stalk at * is nonempty then all the other stalks
must be empty.  I don't know what the abstract-nonsense name for that is.

Vaughan