[Math] the Zariski topology good/bad for

ag.algebraic-geometrysheaf-theory

In a comment to this question the quotation "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry" appears.

Well, so some spontaneous questions arise:

1) What is Zariski topology good for in algebraic geometry? In other words, what can you do with it, without referring directly to some finer Grothendieck topologies?

2) On the other hand, which concepts which are analogs of concepts in -say- analytic geometry or topology, really need finer Grothendieck topologies to be generalized to the algebro geometric setting?

I think one could mention vector bundles for the 1), and principal bundles and projective bundles for 2). And also cohomology for 2), since it seems that the étale topology is best suited for reproducing a cohomology that resembles the singular one in the analytic setting. And also sheaf cohomology for 1).

Edit: I think it would also be nice if in the answers there were some brief comments about the sufficiency (or not) of Zariski topology to capture the essential geometric picture borrowed from analytic geometry and/or topology about (some of) the following contexts:

  • Patching of morphisms
  • Inverse function theorem & implicit function theorem
  • Existence of tubular neighbourhoods (?)
  • Local reducubility vs. global reducibility (analytic branches of a variety at a point..)
  • "Infinitesimal" properties given by the local ring at a point (Zariski vs. Hensel)
  • Vector bundles
  • Coherent sheaves (and patching thereof…)
  • Principal bundles (local triviality…)
  • Projective bundles (as above…)
  • In general, "locally trivial" fiber bundles (isotriviality vs. triviality…)
  • Sheaf cohomology with constant coefficients vs. of coherent sheaves
  • Cech cohomology (as above…)
  • Covering spaces and fundamental group
  • Extracting topological/homotopical information via coverings
  • (any other topic you find interesting)

Best Answer

While the Zariski topology has its limitations, it amazes me how well it does work. A few brief points in its defense:

  1. It's easy to define. In the classical case of affine spaces over a field, it's the weakest topology for which points are closed and polynomials are continuous.

  2. It can be used to give precise meaning to the word "generic" or "general", as in "a general matrix is diagonalizable, therefore to prove Cayley-Hamilton it suffices..."

  3. For coherent sheaves, it's the right topology; cohomology works as expected. At a more sophisticated level, cohomology is upper semicontinuous in the Zariski topology, and this is very important for many arguments.

So for the younger generation out there who are thinking of doing away with it: please don't!

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