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algebraic geometry

Keywords: 
variety,ring,polynomial
Type of Math Object: 
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Mathematics Subject Classification

14-00 no label found

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The first sentence says: "Algebraic geometry is the study of algebraic objects using geometrical tools." Surely this is back to front: isn't it the study of geometry (curves, surfaces, etc) using algebraic tools?

- Michael (not an algebraic geometer)

Hmm. No, not really. The point is that those things we call "curves" and "surfaces" are really algebraic objects: a curve over a finite field has no natural differentiable structure, and its topology doesn't tell you much of anything about it. So to be able to use all the geometric wizardry, you have to do some magic. That's where schemes come in.

In fact, when looking at truly geometric objects (complex manifolds, say), you get almost nothing new from algebraic geometry. At least, nothing new that I know of. Generally the algebra makes your proofs harder: the general outline is the same as the topological proof but you have to replace trivial facts about one-point topological spaces with difficult algebraic facts.

Now I just have to make the entry say all this.

The study of geometry using algebraic tools is essentially the subject of non-commutative geometry.

Did you mean this the other way around, i.e.,

The study of algebra using geometric tools is essentially the subject of non-commutative geometry.

Either way, *that* I'd like to see explained. I have only the very vaguest idea what non-commutative geometry (and finite geometry, etc.) is.

No, I mean geometry using algebraic tools. It is algebraic geometry from the other direction, so to speak. As I see it, algebraic geometry gives you:
geometry -> commutative algebra (geometry is algebra)
So, you could take the opposite view that algebra is/defines geometry,
commutative algebra -> geometry (commutative algebra is geometry)
This naturally leads to regarding noncommutative algebras as defining some sort of "noncommutative" geometry. You can then start building an algebra<->geometry dictionary along the lines of:

ring = ring of functions on a "space"
projective modules over a ring = vector bundles over a "space" (K-theory comes in here)
Hopf algebra (quantum group) = algebra of functions on a "group"
etc

Ah, very interesting. Would you like to add some brief comment to the entry to the effect that noncommutative geometry tries to generalize algebraic geometry to handle noncommutative rings? I don't know if we actually have anything on noncommutative geometry yet, but it might be nice to draw some attention to it.

Equally, Riemann and others resolves the problem. I feel that, actually, the algebraic geometry is a powerfull discipline.

Algebraic geometry is certainly one influence, but not the only one. It depends on personal taste for the dictionary line
ring = ring of functions on a space
If you like your spaces to be manifolds, then a ring won't convey enough structure. You will need something like
C*-algebras = algebra of continuous functions
This branch of non-commutative geometry draws heavily on operator theory/functional analysis.

Noncommutative geometry may actually be considered as an area in, or subfield of, Non-Abelian Algebraic Topology; for further details please see also $http://planetmath.org/encyclopedia/AlgebraicTopology.html$, and perhaps also contribute to this topic...
bci1

One of my favorite quotes (translated from the French, and not by me) is from Sophie Germain:

Algebra is but written geometry; geometry is but drawn algebra.

Roger (also not an algebraic geometer)

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