0.1.1 Algebraically closed fields and Vector Spaces

Steinitz theorem that the isomorphism class of an
algebraically closed field is given by its
characteristic and its transcendence degree, is thus a
very important theorem for us, and---for us---it is
the raison d'etre of algebraically closed fields, and
it is an explanation of why they occur in mathematics
everywhere. (In model theory terms, he theorem actually
states the (uncountable) categoricity of the theory of
an algebraically closed field of certain
characteristic in the language of rings. )

Algebraically closed fields appearing everywhere---from
our point of view, an algebraic variety is just the
same as a field; indeed, a natural language for an
algebraic variety consists of all its subvarieties (and
subvarieties of its Cartesian powers). And it is a
theorem of Model Theory that, for any algebraic
variety, this language is able to define the field, and
the operations on it...And such a field, obviously,
recovefunctions on the variety, the ring of regular
functions, etc....

A vector space is also determined uniquely by its
dimension; the proper language for vector spaces is
when each linear (or rather, affine) function has its
own name; the base field is thus ingrained in the
language itself. And indeed, we want to speak about the
vector space, not the field!

The Slogan above naturally and naively leads to a
conjecture that

any categorical theory is essentially equivalent either
to a vector field or an algebraically closed field.

The conjecture was proven false, but it has been
recovered in the following way.


0.2 Zariski Geometries

With each variety X, one might associate the geometry of
its closed subvarieties. Essentially, it is the Zariski
topology on X. One might formulate some properties of
the Zariski topology in a formal way, like the
following ones:

* it is Notherian,

* the image of projection is constructible

One might also speak of dimension, which satisfies

* formulae for dimension of union

* a formulae connecting the dimension of kernel and the
dimension of image i

* semi-continuity of dimensions of fibres

* (if X is smooth) a formula for dimension of intersection

A Zariski geometry is a topology satisfying there
abstract properties (except the formula for
intersections) of a Zariski topology of a variety. It is
called pre-smooth if if it also satisfies the formula
for intersections.

In fact, those properties are very strong. In dimension
1 (that is, when the whole set has dimension 1), a
Zariski geometry is always "almost" comes from Zariski
topology on a curve; "almost" means that it it a finite
cover of such. In general, one might reasonably recover
a notion of a ring of "regular" (definable) functions
associated to such a geometry, multiplicities of
intersections, and other.

This fits in the ideology described above; the fact
that we can say something useful (speak of dimension)
already tell us a lot.

However, as mentioned above, not all examples of
Zariski geometries come from Zariski topologies of
varieties (over an algebraically closed field.) In
fact, we believe that non-trivial examples might be
quantum manifolds, supermanifold, i.e. come from
non-commutative geometry.

One could hope also for other results of this sort; for
example, one might consider a universal cover of a
toric variety or an abelian variety, and then consider
the associated geometry of analytic sets. This will be
a purely algebraic notion which can be axiomatised and
studied. For such covers, one may actually prove
uniqueness (categoricity) results; in a way, such
results show what is the relation between complex
topology and the abstract, discrete automorphism group
of the universal cover (that is closely related to
Galois group.) That is what I try to do.