Schemes over ℤ with a “graded existence over 𝔽₁” - MathOverflow

Schemes over ℤ with a “graded existence over 𝔽₁”

2011-10-08T16:48:53Z

Let $X$ be a (separated, and with whatever other tameness conditions are appropriate) scheme over the integers $\mathbb{Z}$. (If you don't like schemes much, imagine that $X$ is described by algebraic equations over the integers, so that it is possible to make the algebraic variety $X(k)$ for any field $k$.) I am interested in the case in which $X(k)$ is "the same" for all fields $k$ so that it is possible to discuss its properties over the "field with one element $\mathbb{F}_1$". (Actually I think that $\mathbb{F}_1$ is a scam to some extent, but I wanted to make the question catchy.) What I mean is examples such as when $X$ is a Grassmannian. In this case the Poincaré-Hilbert polynomial of the topological cohomology $H^*(X(\mathbb{C}),\mathbb{Q})$ is the same up to the change of variables $q^2 \mapsto q$ as that of the cohomology $H^*(X(\mathbb{R}),\mathbb{Z}/2)$ , and that's the same as the polynomial expression of the number of $\mathbb{F}_q$-rational points of $X(\mathbb{F}_q)$, and probably a lot of other things are the same. They're all the same because the Schubert cell decomposition is perfect in the sense that it has a vanishing differential over either $\mathbb{C}$ or $\mathbb{R}$, and it has equally nice properties over any other field.

Is there a name for a scheme with this type of good behavior? The question is not entirely rigorous, except in the primitive form of reducing only to the fields listed above. Even so, most schemes clearly don't share these stability properties possessed by Grassmannians. For instance, the picture fails badly for an elliptic curve.

I got two good answers to the question (and a third one that was not bad). In the interest of wrapping things up, I would like to accept one of them. The answers are basically tied in mathematical value, but I particularly like Jordan Ellenberg's prose. The picture from both answers taken together is that people have worked both on upper bounds --- all schemes that could possibly qualify for the question --- and lower bounds --- schemes that by design must be allowed as examples. The former are called "polynomial count" varieties; the latter are sometimes offered as varieties defined over $\mathbb{F}_1$. Pure Tate cohomology is another type of upper bound. It seems that the discrepancies between the upper and lower bounds are not well understood.

Answer by Felipe Voloch for Schemes over ℤ with a “graded existence over 𝔽₁”

2011-10-08T21:02:25Z

See Katz's appendix to the paper "Mixed Hodge polynomials of character varieties", by Hausel and Villegas which appeared in Inventiones in the last year or two. The varieties you are looking for are called polynomial count varieties. There has been a few other papers on this recently, e.g. one by van der Bogaart and Edixhoven. There is also an older (2003?) preprint of Soulé (Les variétés sur le corps à un élément) which I don't know if ever got published.

BTW Why do you think the field of one element is a scam? To me, the word scam evokes willful intent to deceive.

Answer by JSE for Schemes over ℤ with a “graded existence over 𝔽₁”

2011-10-09T01:49:54Z

I think if you want a good answer to your question you will have to consult papers by people who don't think F_1 is a scam! To be more precise: since F_1 doesn't actually exist, what do people think it is? And I think most people would say words like: there should be some category of schemes over Z which are somehow "isotrivial" or "the same over every prime," and we call this category the category of schemes over F_1, and this serves as a "definition" of F_1. And, in addition, its supplies the name you requested for schemes with this good behavior.

I think it is stronger than having all cohomology of Tate type, though I'm no expert on this. Grassmannianns are schemes over F_1 (I'm pretty sure) but elliptic curves never are.

Now this is all just philosophy unless without an actual definition somewhere in the picture. The good news is, there is one! Actually, several! The one I like best is due to Jim Borger:

http://arxiv.org/abs/0906.3146

So have a look at this and see if "schemes with Lambda-structure" are what you're looking for.

Answer by Sasha for Schemes over ℤ with a “graded existence over 𝔽₁”

2011-10-09T04:01:39Z

The following paper http://ens.math.univ-montp2.fr/~toen/souz.pdf of Toen and Vaquie may be interesting.