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From: peno@enea.UUCP (Pekka Nousiainen)
Newsgroups: net.math
Subject: Re: Euler's formula and arithmetic dimensions
Message-ID: <1100@enea.UUCP>
Date: Wed, 27-Nov-85 23:42:17 EST
Article-I.D.: enea.1100
Posted: Wed Nov 27 23:42:17 1985
Date-Received: Sat, 30-Nov-85 14:16:37 EST
References: <1095@enea.UUCP> <64@brl-tgr.ARPA>
Reply-To: peno@enea.UUCP (Pekka Nousiainen)
Organization: Enea Data, Sweden
Lines: 54

>This formula doesn't even talk about volume, except to the extent
>that every number can be thought of as the volume of SOMEthing.

You're right.  All I can offer is "proof by authority":  Manin is one
of the best living mathematicians.

>Tell us more about Spec(Z), whose elements are apparently just the
>positive prime integers.  What is this "algebro-geometric spectrum"?
>I have to object to calling every factor algebra a "dimension".

Sorry about using fancy words.  Your objection though isn't valid.
A factor algebra, or just any commutative ring A, corresponds to a
geometrical object with properties such as dimension and connectivity.
Here's an explanation via an [ inaccurate ] example:

--begin--technical--

Consider a geometrical object S such as a sphere or the 3-dimensional
space.  To measure things in S one introduces coordinate functions,
that is, continuous functions f: S -> Reals.  Such functions can be
added and multiplied (just add or multiply the values of the functions)
so they form a ring A.  One can recover the set S (but not its geometry)
from A: a point x in S corresponds to the set of functions f such that
f(x) = 0.  These sets are exactly the "prime ideals" of A.

In a general ring A, the prime ideals play the same role as prime
numbers do in integers.  The points of Spec(A) are the prime ideals of A.
Returning to the example, we have S = Spec(A) as abstract sets.  To get
further, assume S is the 3-dim space of points (x,y,z).  Instead of
all continuous functions, take only polynomial functions, such as
f(x,y,z) = 3 + x^2*y + z^6.  These form another ring B.  Now Spec(B)
is much more interesting.  As in the case of A, prime ideals of B have
"zero sets" in S.  Any maximal chain of primes in B corresponds to some
chain of zero sets looking like ( "-" means inclusion)

	(point)  -  (curve)  -  (surface)  -  (whole S).

You can read off the correct dimension dim(E) = 3 from the structure
of B alone (length of longest prime chain).

To come to the point, what is the dimension of the integers Z ?  The
prime ideals are the set consisting of 0, and for each prime number p 
the set of multiples of p.  A longest chain is any 0 - p.  The integers 
are one-dimensional.  In math, Z is just another a "curve" (granted,
a weird curve).

--end--technical--

The above is hardly enough to convince, but the ideas are at the very
center of pure mathematics.  As for Manin's claim, I suspect Spec(Z)
is much too simple (there are others).

--
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