@ZhenLin It's cool. It also gives me a guiding intuition about how to move through the space. Much like how when you turn your monitor pi/2 radians the mouse starts moving to the wrong directions.
When you know where you have to go, then you know where you can pee and how to get there.
@Asaf: There isn't really any geometry in the Zariski topology. It's like asking what the geometry of the space that is Stone-dual to a Boolean algebra.
@QED: There are easier ones, like elliptic curves. But Fermat's equation is more famous.
@ZhenLin You know, "geometry" means the measurement of land. Of course in mathematics we change that to "measurement of space". The fact that you are unable to comprehend the Zariski space does not mean it cannot be measured.
@Zhen: I'm sure that you understand it much much better than I do, or will ever do. I just don't see why you geometers are so adamant not to try and interpret it visually.
Just because it doesn't make sense doesn't mean that it doesn't make sense. Much like this very sentence.
It's quite easy to interpret geometrically, in the case where A is a finitely-generated integral algebra over the complex numbers. Then in that case we do have a genuine geometrical object which is amenable to geometric intuition.
For example, if A = C[x], then Spec A is essentially just the complex plane.
The trouble is that the Zariski topology is essentially the cofinite topology on the complex plane.
I suddenly understand. You are just insisting that everything that has the name geometry will look, at least somehow, at least locally enough, as real vector space.
It is interesting, I am thinking about it and I try to move my hands around to somehow make shapes and movements as in my mind... but there simply too many dimensions. Perhaps if string theory works...
How many dimensions are they claiming now? 26 or 12?
It's funny, you know. My geometrical intuition is crap. However I can imagine incredibly complicated structures so vividly that most of my intuition in set theory comes from actually seeing in my mind how the set I am working with is ordered.
The first time I learned that about myself was when I was reading about Suslin trees, the book kept saying that the underlying set is omega_1, and I kept thinking it means with the usual ordering and how could that be a tree.
Then I realized that it is actually just the set, and the order of the tree is different.
At the same moment, I will never forget this, the line I had in my mind for omega_1 just splitting into a huge and complicated tree.
For passing the course in commutative and homological algebra, so I won't have to take any course work during the next semester and I could actually work on my thesis.
