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

[Bill Lawvere <wlawvere@buffalo.edu>]

Vaughn remarks
...........I would have thought intersecting them could only get you the
square-free ideals........

Indeed, as Jeff points out, we learned from Kummer and Dedekind to replace
elements by ideals, but we categorists have been late in providing a clear
account of this transition and, in particular, of the reason why the
result is  not primarily a lattice, but a monoidal closed category with
colimits.

Below I will elaborate on the following three points:

(1) The actual "ideal number" functor itself is clear enough (though never
made explicit), but why should it exist?

(2) The standard account of "why" is very categorical, but does not
directly address the algebraic category of rings nor the geometry of
intersection theory.

(3) The universal algebraists have developed a tool that might be applied
to the "why", but for some reason the universality is not often applied to
algebra or to geometry.

In more detail:

(1) The Kummer functor I goes from rigs (or K-rigs, where K is given, e.g.
Z or Q) to 2-rigs, where 2 is the 2-element rig in which 1+1=1. (Yes, the
rig that launched ring theory is not itself a ring). The functorality, as
well as the multiplication itself, depends on the set-theoretic operation
of image. The principal ideal concept is a natural transformation from M
to MI where M is the underlying multiplication.

(2) A rig can serve (not only as functions on a scheme but) as an abstract
general whose semantically corresponding concrete general is its category
of modules, which is a monoidal closed category with colimits. This
2-functor can be composed with the functor to posets that extracts from
the big category of modules just the submodules of the unit object. Again,
the image operation must in general be applied to the result of tensoring
two submodules (because of the lack of flatness). A monoidal poset with
colimits is also a 2-rig.

(3) Intersecting closed subspaces of a space may give only a shadow of a
description of their clash (e.g. the clash of Africa & Europe produces a
bulge i.e. the Alps). Although geometric figures are in general singular,
a notion of closed subspace which refines the notion of mere subset
provides a useful partial record.
In terms of the rigs of variable quantity on the spaces there is a
corresponding refinement:
The distributive lattice of radical ideals is refined to the monoidal
poset of all ideals. The ideal product under discussion is a key
ingredient in a construction of unions of subspaces that takes into
account the clashes. As it would be desirable geometrically to see even
nonsingular figures as images of maps in the category of spaces itself, it
would dually be desirable to see R/ab as the result of a construction on
R/a and R/b within the category of K-rigs itself, without the detour (2)
through modules. At least the case where K is a ring is indeed covered in
principle by a construction called the "commutator" (a misleadingly
particular "general terminology", ....groups are apparently not a typical
algebraic category). That this construction does reduce to a certain
concatenation of limits and colimits has been shown by categorists in
terms of congruence relations.

But the application to rings and ideals still remains to be done.

Bill


************************************************************
F. William Lawvere, Professor emeritus
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************



On Thu, 27 Sep 2007, Vaughan Pratt wrote:

> 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
>
>
>
>