I am just starting to learn about schemes and algebraic geometry in general, but I am finding it very hard to visualize things. For example, affine schemes that look like varieties are easily visualized. But how about infinite dimensional or nonproper things? Or fibre products of schemes? So just throwing out this question to all algebraic geometers: When you are doing your research, how much of your results come from the geometric intuition? If one were to start the research in algebraic geometry, what would you say the most important things is?
[Math] How much of scheme theory can you visualize
ag.algebraic-geometryschemessoft-question
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Best Answer
Well, you asked 10 different questions, and I am not sure what you mean by "nonproper" ($Spec A$ is not proper). But let's see.
A scheme is a very geometric object, with practice - or maybe just habit - one learns to visualize it quite well. If you already see geometrically $Spec$ of a finitely generated algebra over a field $k$ (including algebras with nilpotents which you visualize as "thickenings", including $k$ not algebraically closed which you visualize as Galois orbits; you looked at these, right? these are important steps) then you are almost there. Add some other standard examples such as $Spec(\mathbb Z)$, DVR, a double-headed snake (the first nonseparated scheme), and you already know plenty do start doing research.
Infinite-dimensional algebras? Well, I suppose it is just as hard or easy to imagine them as infinite-dimensional spaces.
The fiber product is a perfectly geometric notion as well, and fairly easy to visualize. You begin by looking at fiber products of sets and you progress from there through some standard examples. Isolating a fiber of a morphism is an important case. And then look at some examples where the residue fields of the scheme points change. Learn the simple way to compute the tensor product $A\otimes_R B$ by using generators and relations of $A$, and you will be up and running in no time.
As far as the balance of geometry vs algebra, I suppose that depends on a person and everybody is different. My advisor used to say that geometry comes first and then later algebra follows, and I tend to agree. I think you get nowhere without geometric intuition.
But if you are serious, at some point you will need a solid commutative algebra foundation. Fortunately these days there are plenty of nice books, starting with the very nice and elementary "Undergraduate commutative algebra" by Miles Reid.