An algebraic geometry trivia (Math Puzzle #3?)

mieda

United States85 Posts

September 10 2010 07:16 GMT

Take any scheme X and consider its Zariski topology with non-closed points so we're looking at all of the prime spectra, not just the max spectra. Prove that X is simply connected. In fact, pi_n(X) is trivial for all n >= 1. (it's not hard once you see the trick!

)

If you just look at the max spectra for the points and complex manifold topology on varieties over C it really isn't true at all. For example, take any complex torus, which has nontrivial fundamental groups and is a complex curve (in fact an elliptic curve).

Which explains why topologically Zariski space isn't so interesting from topological point of view. This is why we use etale fundamental groups instead.

Edit: Probably should assume that X is irreducible also ;p

incnone

17 Posts

September 10 2010 08:44 GMT

I don't think I buy into this example that a complex torus is not simply connected in the classical (mSpec) Zariski topology. I'm going to attempt to prove that the torus (or any one-dimensional complex variety) is simply connected:

+ Show Spoiler +

The mSpec-Zariski topology on the torus T has it that a set is open iff it is cofinite (or empty). So if X is a topological space, then f: X -> T is continuous iff the preimage of every point is closed.

So suppose I have a continuous map f from [0,1] into T with f(0) = f(1). I'm going to extend it to a homotopy F : [0,1] x [0,1] -> T with F( *, 1) = pt. Choose any injection G of [0,1] x (0, 1) into T, and define F to be equal to G on [0,1] x (0, 1), and to be some point p on [0, 1] x {1}. Then the preimage of any point q of T is the union of some closed subset f^{-1}(q) of [0,1] x {0} along with possibly a single point of [0,1] x (0,1), and also (if q = p) the line [0,1] x {1} and is therefore certainly closed. So F is continuous and f is nulhomotopic.

mieda

United States85 Posts

September 10 2010 09:15 GMT

I should've been careful there. I meant for the topology of complex manifold structure on torus. When I said max spectra I should've said just the points, and the topology was of complex manifold.

Edit: OP is Edited to clarify this.

spoolinoveryou

United States503 Posts

September 10 2010 10:33 GMT

holy crap i think my head just exploded... i wish i could understand what you guys were talking about hahahaha.

Terranlisk

Singapore1404 Posts

September 10 2010 12:39 GMT

Is this something they asked you at your interview too?

mieda

United States85 Posts

September 10 2010 15:20 GMT

On September 10 2010 21:39 MyHeroNoob wrote:
Is this something they asked you at your interview too?

No, it's more something you might try on the side as you read SGA 1 .

The question is just for trivia interest, since no one considers traditional fundamental groups on algebraic varieties anyway (especially for positive characteristic case).

mieda

United States85 Posts

September 10 2010 21:22 GMT

I'm going to throw this away.

The solution is quite simple / silly in a way: Suppose S^n -> X is any continuous map. Send the interior, D^{n+1} (without the boundary), to the generic point of X. This is continuous and extends the map S^n -> X to D^{n+1} -> X and the latter clearly homotopic to a constant map.

Muirhead

United States557 Posts

September 10 2010 22:16 GMT

Where do you go to school mieda? I think the fact that Zariski topology isn't Hausdorff is enough to show you shouldn't be applying methods better suited to CW complexes

mieda

United States85 Posts

September 10 2010 23:22 GMT

On September 11 2010 07:16 Muirhead wrote:
Where do you go to school mieda? I think the fact that Zariski topology isn't Hausdorff is enough to show you shouldn't be applying methods better suited to CW complexes

Sure, as I keep repeating, no one considers these fundamental groups on these spaces anyway. That's why it's a "trivia" :p , just to see what happens when one does :p

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