Re: differential/algebraic geometry

On 21.06.2005 05:43, hale@xxxxxxxxxx wrote:
>
> Orion wrote:
>
>>Is differential/algebraic geometry and multilinear algebra basically
>>one and the same?
>
>
> I'm currently reading about algebraic geometry, so I am just a
> beginner in that area.
>
> Dieudonne has written a brief article (but very technical) on
> the history and development of algebraic geometry which is
> worth reading. I don't have the reference at hand though.
> It might give some answers to your question.
>
> Of course, differential geometry and algebraic geometry are
> not basically the same since the former deals with smooth
> functions on R^n while the latter deals with polynomials
> on k^n (wher R is reals and k is a field).

I would like to go a step further: Algebraic geometry is a geometrical
extension of commutative algebra where local objects are associated to
commutative rings and not only (finitely generated) algebras over a
field k. The latter is certainly one of the classical examples of the
beginning point of the algebraic theory.

>
> However, I get the feeling that they (or their underlying ideas)
> are related or are similar. For example, both have a version of
> the Riemann-Roch theorem that relates the genus g with algebraic
> properties (in the algebraic geometry case).

But this goes certainly back to the existence of a (co-)homology theory
on the objects in differential and algebraic geometry, resp. In my
opinion, this does not necessarily imply to some kind of similarity.

>
> Also, both consider the germs of functions at a point. Both
> consider the tangent space at a point.

This is a common conceptual approach on rather different objects which
appear to be similar in very special cases. Here I am thinking of
Serre's GAGA (géométrie algébrique et géométrie analytique): an
equivalence of certain sub-categories of analytical and algebraic
varieties of the field of complex numbers. It basically refers to
compact varieties which implies that some cohomological vector spaces
are finite dimensional.

>
> One major feature of differential geometry is that the
> geometry is considered to be locally Euclidean at a point
> and that charts are used to globally patch sections together.
> This part I am not sure about: it also appears that algebraic
> geometry is considered to be locally affine and that charts are
> used to globally patch secitons together. If this is true, I
> think this would provide reasons to consider them both to
> be related or similar.
>
> -- Bill Hale
>

Basically, the two theories share a common principle which is widely
spread in mathematics. If a theory is wanted to be 'geometrized' then
you look for standard objects translated to a topological space with
some additional structure (e.g., the sheaf of continuous functions, of
differentiable functions). Then a geometrical object is a topological
space with additional structures which locally look like the standard
objects.

In differential geometry the standard objects are open sets in some real
vector space (not necessarily finite dimensional) and in algebraic
geometry affine schemes stemming from a commutative algebra.

Because of the way of construction of geometric objects, the machinery
of category theory, universal algebra and (co)homology theory apply. So
far I agree that the theories appear to be similar. But this argument
seems to valid for all objects constructed as briefly described above.

But the crucial question is what intrinsic properties of the objects can
be revealed. If you get down to answer this, you get assertions
distinguishing geometric theories from each other.

Similarities occur due to equivalences of categories. Mostly they are
provided by huge theorem which are not obvious at all.

I named Serre with one of his famous theorems (GAGA), so I should forget
to name one of the 'fathers' of algebraic geometry: Alexandre Grothendieck.


Cheers,
J.
.

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