What is the Zariski topology good/bad for? - MathOverflow

What is the Zariski topology good/bad for?

2010-04-15T20:32:47Z

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)

Answer by Kevin Lin for What is the Zariski topology good/bad for?

2010-04-15T22:56:58Z

(I guess this is an elaboration on one of your comments in your last paragraph.)

One reason why the Zariski topology is "bad" is that the higher cohomologies ($H^i$ for $i > 0$) of a constant sheaf on an irreducible space (in particular irreducible varieties with the Zariski topology) are zero, because such a sheaf will be flasque (exercise!).

I guess this is one of the motivations for the etale topology and for etale cohomology. The etale cohomology of (certain) constant sheaves will be nontrivial, indeed, it will coincide with the singular cohomology of the associated analytic variety. This is the "Comparison Theorem", see for example Milne's book on etale cohomology for the details.

Answer by Alberto García-Raboso for What is the Zariski topology good/bad for?

2010-04-16T01:07:25Z

As Kevin said, the higher cohomology groups of constant sheaves on irreducible varieties are zero when working with the Zariski topology. Also, "fibre bundles aren't locally trivial" and "the inverse mapping theorem fails". Instead of writing about the last two myself, let me refer you to the wonderful notes of fellow MOer JS Milne on étale cohomology (page 10, The inadequacy of the Zariski topology)

Answer by Emerton for What is the Zariski topology good/bad for?

2010-04-16T02:29:11Z

The Zariski topology is part of the basic structure of varieties and schemes. Unlike other, fancier Grothendieck topologies, it is actually a topology, defined by subsets of the variety/scheme, and so gives rise to the notion of closed, as well as of open, subset. The closed subsets are the algebraic subsets (in the variety case) or (the spaces underlying) the closed subschemes (in the scheme case), which are what the study of algebraic geometry is to a large extent about.

If you look at Hartshorne, Chapters IV and V, you will find a lot of geometry of curves and surfaces (as well as some geometry of higher dimensional varieties too), all of it treated without recourse to any topology other than the Zariski topology.

Added: A good example, in my opinion, is the (direct) proof that the addition law on a smooth plane cubic curve is associative. (By direct, I mean the proof working from the definition of the group law in terms of collinear points, not the more involved proof that proceeds by identifying the curve with its Picard variety.) It is not hard to see that associativity holds when the three points being added are in suitably general position. To conclude, one can either make complicated special cases arguments in the various non-generic situations, or, one can make a continuity argument in the Zariski topology. The latter argument is simple (in that it uses the most basic kinds of arguments about continuous maps in general topology) and decisive. It illustrates perfectly the role of the Zariski topology in geometric arguments.

One could generalize from this example as follows: two of the most fundamental notions of algebraic geometry are the complementary concepts of general and special position, and these notions are precisely what the Zariski topology captures (just as the topology on a metric space captures the notion of closeness in the metric).

Answer by Donu Arapura for What is the Zariski topology good/bad for?

2010-04-16T12:37:56Z

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

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.
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..."
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!

Answer by Sean Tilson for What is the Zariski topology good/bad for?

2010-04-18T00:23:10Z

algebraic geometry is not my field at all, but it seems like this is a bit of an answer, and i don't really know the details but maybe someone can fill them in. The Zariski topology is not good for doing homotopy theory. I have heard this at many seminars, specifically from someone who does motivic homotopy theory. So from that perspective it is not the right topology, but i can only say that i have heard this not that i understand why the Zariski topology is bad. I guess just naively it seems like it would be pretty hard doing homotopy theory on "any" line where your only open sets are finite complements.

Answer by Kevin Lin for What is the Zariski topology good/bad for?

2010-04-18T00:33:11Z

In light of your edit, Remark 2.1.2 of these notes of Christoph Sorger might be useful.