Compact and quasi-compact - MathOverflow

Compact and quasi-compact

2010-03-03T14:37:28Z

Why do algebraic geometers still use the term "quasi-compact" when they almost never deal with Hausdorff spaces? They certainly use "local" rather than "quasi-local" (local = quasi-local + noetherian), so is there any reason other than historical contingency?

Do algebraic geometers who do work in other fields still follow this convention when they write other papers? If they do, do they write at the beginning of the paper something along the lines of "by compact, we mean quasi-compact and Hausdorff"?

Answer by BCnrd for Compact and quasi-compact

2010-03-03T15:23:02Z

It is a useful technique in algebraic geometry to work over the complex numbers with the analytic topology...

Answer by YBL for Compact and quasi-compact

2010-03-03T15:25:33Z

I think it comes from Bourbaki. In french, "compact" is what you call "quasi-compact and Hausdorff". Calling a non-Hausdorff space compact would make no sense hence "quasi-compact".

I imagine it is the same thing for "local = quasi-local + noetherian". For me, a commutative ring is local if it has only one maximal ideal. There is no noetherian hypothesis in the definition.

Answer by Donu Arapura for Compact and quasi-compact

2010-03-03T16:01:56Z

Personally, I was never a big fan of the Bourbaki convention of including the Hausdorff axiom into compactness. I would be happy if we dropped the "quasi", but perhaps it would lead to too much confusion at this point in time. On the other hand, schemes were once called preschemes, so algebraic geometers are capable of change.

(This topic is a bit a diversion I think, and I'm not seriously suggesting that the terminology should be changed in this instance. In general, however, I think it is OK to occasionally break with tradition and modify terminology when it's genuinely ungainly. Of course mathematics, like any human endeavor, is full of choices which in hindsight may seem a little odd and perhaps inconsistent. Most of us can live with that. As a complex geometer, I am happy to use "Riemann surface" and "elliptic curve" in the same sentence.)

Answer by Pete L. Clark for Compact and quasi-compact

2010-03-03T16:12:47Z

The condition of quasi-compactness in the Zariski topology bears little resemblance to the condition of compactness in the classical analytic topology: e.g. any variety over a field is quasi-compact in the Zariski topology, but a complex variety is compact in the analytic topology iff it is complete, or better, proper over $\operatorname{Spec} \mathbb{C}$.

I think many algebraic geometers think to themselves that a variety is "compact" if it is proper over the spectrum of a field. I have heard this terminology used and occasionally it shows up in (somewhat informal) writing.

So a perhaps more accurate brief answer is that in algebraic geometry the distinction between quasi-compact and quasi-compact Hausdorff is very important, whereas in other branches of geometry non-Hausdorff spaces turn up more rarely.

Anyway, many mathematicians have been happy with the quasi-compact / compact distinction for about 50 years, so I don't think this usage is going away anytime soon.

To address the last question: when writing for a general mathematical audience, it is a good idea to give an unobtrusive heads up as to your stance on the quasi-compact / compact convention. (The same probably goes for other non-universal conventions in mathematics.) If I were speaking about profinite groups, I would say something like:

"A profinite group is a topological group which can be expressed as an inverse limit of finite discrete groups. Equivalently, a topological group is profinite if it is compact (Hausdorff!) and totally disconnected."

This should let people know what side I'm on, and thus be able to understand me. When writing for students, I might take pains to be more explicit, using a "By compact I mean..." construction as you have indicated above.

Answer by Charles Staats for Compact and quasi-compact

2010-09-02T13:27:01Z

This is far from standard, but in my mind, when I'm doing algebraic geometry, "quasicompact" means "compact in the Zariski topology," and "compact" means "compact in the analytic topology" (or is not used at all if I'm not working over a topological field). Thus, in principle, having explained that the terms were being used this way, one could write statements like

"$\mathbb{A}^n_{\mathbb{C}}$ is quasicompact but not compact"
and
"Projective varieties are both compact and quasicompact."

But I would be very hesitant to do so, since I've never seen the terminology used this way so explicitly.