I'm a student meeting commutative algebra and algebraic geometry for the first time. The idea of studying every (commutative) ring geometrically via its spectrum (as a locally ringed space) is amazing. The techniques of homological algebra appear very quickly - already in dimension theory. Sheaf cohomology comes up later in algebraic geometry too (I'm not there yet).

However, the basic ideas of algebraic topology kind of seem like they're missing (at least at the basic level): we have this topological space - the spectrum, but no books seem to play with it in the sense of deformation retracts, fundamental groups, etc.

This MO question starts with:

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology.

So I'm guessing the answer to the following question will be a one-liner, but still: Why?

A comment by the user Anonymous on his answer to the linked question mentions the maximal spectrum is a deformation retract of the prime one, so it looks at least like basic homotopical concepts are not completely useless.

What are some examples of these?

  • $\begingroup$ Doing homotopy theory to the prime spectrum is completely the wrong thing (basically you're completely ignoring the structure sheaf and that's the most important piece of the data; in some sense the underlying space exists only so that a structure sheaf can live on it). But there are various ways that ideas from algebraic topology show up; look up, for example, the etale fundamental group, etale cohomology, etc. $\endgroup$ – Qiaochu Yuan Dec 22 '15 at 19:44
  • $\begingroup$ @QiaochuYuan ah, right - the spectrum itself misses loads of information. Thanks! $\endgroup$ – Arrow Dec 22 '15 at 22:50

Spectra of rings are not the right kind of spaces to understand via fundamental groups, higher homotopy groups, usual (co)homology groups, etc. This is easy to see already in the case of the Zariski affine line over a field. The only closed sets are finite...so what are the continuous maps from the circle, en route to computing the fundamental group? Well, there are tons-in particular, any function with finite fibers (no point has infinite inverse image) is continuous, so e.g. any permutation of the circle is a continuous map from the circle to the affine line over $\mathbb{C}$. This is clearly totally ridiculous and can't give us any useful information.

The problem can be understood less concretely: schemes are not really topological spaces. In other words, it's of no interest, in general, to study continuous maps between the raw topological spaces of schemes. For instance, there is the theorem (maybe someone can remind me of the name attached to it) that every so-called spectral space, i.e. every sober $T_0$ quasicompact space with a basis of quasicompact opens closed under finite intersections, is homeomorhic to the spectrum of a ring. But all quasicompact schemes are spectral spaces, so for instance projective schemes are equivalent, as topological spaces, to affine ones-this approach is telling us absolutely nothing about the geometry.

Since the late '60s, considerations like these, as well as many others, have led geometers from Grothendieck on down the generations to be somewhat skeptical of the conceptual value of the locally ringed approach to schemes. It's arguably clearer to take the "functor-of-points" approach, which has the advantage of avoiding inappropriate questions about topological spaces, since the latter no longer explicitly appear. This is also the only way to study algebraic spaces and stacks, useful higher abstractions in modern geometry.

In any case, one does want analogues in geometry for the useful tools of topology-the problems in the first paragraphs just tell us that we need a less naive generalization. There is, for instance, an algebraic ("etale") fundamental group which uses the covering space perspective on the topological fundamental group as its starting point. So one tries to define a group of covering schemes over a given scheme, and in particular to directy generalize covering spaces. This works pretty well, and is related to the first algebraic analogue of singular cohomology (which has the same problems as the fundamental group, since it eventually depends on continuous maps from subspaces of real vector spaces,) namely etale cohomology. There are now many other cohomology theories for varieties and schemes, which have many of the same properties as topological cohomology theories. Homotopy theory is harder to generalize, beyond the fundamental group, but it is possible, to some extent, by studying something like a whole complex (simplicial set) of schemes, and using the abstract homotopy theory of such complexes to understand how the schemes fit together. This is tied to the theory of motives, which has long been seen as one of the most ambitious goals of the whole modern approach to algebraic geometry, and which is still very much an active area of research.

  • $\begingroup$ Wonderful answer! Could you elaborate on when the maximal spectrum is a deformation retract of the prime spectrum? $\endgroup$ – Arrow Dec 22 '15 at 22:50
  • 3
    $\begingroup$ Nope, no idea, and I doubt it's of interest. But thanks very much! $\endgroup$ – Kevin Carlson Dec 23 '15 at 0:05

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