Let $R$ be a commutative ring with identity. Assume that $X = Spec(R)$ with the Zariski topology.
When is this space path connected? And also we want to know the topological fundamental group of the space $X$. How can we think about these questions?
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8$\begingroup$ Unfortunately, any topological space with a "generic point" (whose closure is the whole space) is contractible. This covers any integral domains. $\endgroup$– Tyler LawsonFeb 24, 2014 at 7:02
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1$\begingroup$ For questions like these, I find it helpful to spend some time getting familiar with the homotopy theory of finite topological spaces. The underlying topological space of the spectrum of a noetherian ring is very similar to a finite topological space (although there are a few minor differences). $\endgroup$– zebFeb 24, 2014 at 7:38
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4$\begingroup$ Are you sure this is the question you want to ask? These kinds of spaces are very poorly behaved from the point of view of homotopy theory. There are more interesting ways to apply homotopy-theoretic methods in algebraic geometry. $\endgroup$– Qiaochu YuanFeb 24, 2014 at 8:13
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2$\begingroup$ Every finite $T_0$ space is the spectrum of some ring, so Spec(R) can have the weak homotopy type of any finite CW-complex. It seems plausible to me that Spec(R) is always weak equivalent to the nerve of its specialization poset (this is true for any finite space, and seems not hard to prove by an induction argument if $R$ is Noetherian). $\endgroup$– Eric WofseyFeb 24, 2014 at 10:17
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2$\begingroup$ In particular, the example of $C(X)$ suggests that for completely arbitrary rings, there is no easy way to tell whether the spectrum is path-connected. $\endgroup$– Eric WofseyFeb 24, 2014 at 10:45
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1 Answer
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The point is that you may be asking the 'wrong' question. The use of paths is inappropriate in this algebraic geometric context, and hence also ideas such as 'path connected' and 'fundmental group(oid)'although there are analogues.
The classical fundamental group classifies covering spaces of $X$, and that is a useful property to generalise. Look at Grothendieck's SGA1 (if you can read French) for the original material on this, but there are lots of more recent sources 'out there' (including some well written surveys, some done by various Masters students, that are a good read and get you to the point fairly quickly without a lot of generality! One such that I have used is 'M. A. D. Robalo, 2009, Galois Theory towards Dessins d’Enfants, Master’s thesis, Instituto Superior Technico, Lisboa'. Another very good source is by Dubuc and de la Vega, (and which can be found on the ArXiv as math.CT/0009145.) There are analogues of the fundamental groupoid and of covering spaces and then you can ask if the fundamental groupoid is connected (corresponding to path connectedness in the classical topological case) That fundemantal groupoid is usually thought of as a profinite groupoid, and that may encode the topological information that you are hoping for in the second part of the question. (I should add that it is best to consider general schemes with the étale topology, but to start with that may not be necessary.)
I hope this helps.
answered Feb 24, 2014 at 12:16