Re: 'analytic' vs. 'algebraic'

On 28.01.2006 09:53, john_ramsden@xxxxxxxxxxxxxx wrote:

> Generally, "analytic" refers to topics and methods that
> involve explicit use of limits in metric spaces, whereas
> "algebraic" is used for arguments involving the symmetry
> and combinatorics etc of algebraic systems such as groups
> and fields, although limits crop up even in the latter
> such as limits of groups and field completions.
>
> For example, there are two strands of modern algebraic
> geometry these days (ignoring the word "algebraic" in
> that phrase!): the algebraic approach developed by
> Zariski and Grothendieck et al of schemes, versus a
> more analytic approach that uses complex variable
> theory, and there is a large amount of overlap
> between them.
>
> Often, the contrasting methods give information about
> "local" properties of an object such as a curve or
> surface as opposed to "global" properties. I think
> it's reasonable to say that analytic methods tend
> to be more useful for the first whereas algebraic
> methods such as cohomology shed more light on the
> second, although no doubt both contribute to each.
>

Two comments from my perspective:

(1) I could imagine that the famous article from J.P.Serre, GAGA,
Géométrie Algébrique et Géometrie Analytique was one source of
inspiration for the terms discussed here - or the understanding in those
days just gave Serre the idea for the title.

The essence of the article is that every projective variety over the
complex numbers considered as an analytic space (which is something like
a complex manifold with singularities allowed) can be defined by a
finite set of polynomials. Furthermore, analytic sheaves on these spaces
come from algebraic sheaves and the analytic and algebraic cohomologies
coincide.


(2) One of the technical terms in German for calculus (in several real
variables) is /Analysis/. I follow the idea that analytic methods apply
transcendental techniques like continuity and differentiability etc. in
appropriate spaces. Analytic objects can be defined both explicitly and
implicitly. For the latter there are solutions of differential equations
which cannot be expressed in a 'closed form'.

So I would focus on the methods applied to distinguish the terms
'analytic' and 'algebraic'.

J.
.