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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.

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btw, the title of your question is not formatted correctly. <sub> appears as plaintext. probably you can fix the title via tex $F_1$. – joro Oct 8 '11 at 16:56
I think I understand what you're asking. I don't know if there is any standard terminology, but Tate's name often gets attached to this sort of thing, as in "its motive or whatever is Tate". I recently went to a talk by Dimca on some related phenomena. So you might look at some of his papers on the arxiv. – Donu Arapura Oct 8 '11 at 17:37
Presumably the $q^2\mapsto q$ behavior is just that for many purposes, $\mathbb C$ already behaves like it has one element --- at least, its Euler characteristic is $1$ --- whereas $\mathbb R$ behaves for things like Euler characteristic as if it had $-1$ elements. So this really supports the "is the same" behavior. – Theo Johnson-Freyd Oct 8 '11 at 18:46
Greg -- the differentials in the chain complex of the Schubert cell decomposition are certainly non-zero in general (take projective spaces for example). They are however zero mod 2. – algori Oct 8 '11 at 20:22
I think what you are describing is that the cohomology of $X$ is pure Tate. Another example would be toric varieties and $\overline M_{0,n}$. – Dan Petersen Oct 8 '11 at 20:59
up vote 13 down vote accepted

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:

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

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Good point! But it would be slightly better news if there were only one definition, up to equivalence, rather than several. – Greg Kuperberg Oct 9 '11 at 1:54
I'm not sure whether the different definitions are known to be different.... – JSE Oct 9 '11 at 3:09
Pena and Lorscheid detail known implications between the definitions here: – Michael Joyce Oct 9 '11 at 3:19
Peter Arndt and I thought about grassmannians over Durov's F_1. It turns out that one needs to glue along morphisms more general than any of the topologies Durov considers (in particular, morphisms that are not flat but which become flat after base change to any semiring or ring). – Jeffrey Giansiracusa May 19 '14 at 18:31

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.

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My impression (which could be fatuous for all I know) is that --- except for the fact that $\mathbb{C}$ has valuation Euler characteristic 1 --- the field with one element can be viewed as a benevolent exercise in self-deception. In the sense that there isn't really any such field and things will go wrong eventually. But maybe my wording was too strong. – Greg Kuperberg Oct 8 '11 at 21:22
The field of one element is more about the number field/function field analogy, which I guess will go wrong eventually but has been very fruitful in number theory and is far from exhausted. – Felipe Voloch Oct 8 '11 at 21:34
BTW, I don't know if Soulé's notes were "published" either, but they are in the arXiv. Three cheers for the arXiv; it turned 20 recently. – Greg Kuperberg Oct 8 '11 at 22:05
So suppose that $X$ is a polynomial-count scheme over $\mathbb{Z}$ with respect to finite-field specializations. Is it necessarily also polynomial count in the valuation Euler characteristic sense for $\mathbb{C}$ and $\mathbb{R}$? If in addition, $X(\mathbb{C})$ and/or $X(\mathbb{R})$ is compact, is the polynomial also the Poincaré-Hilbert polynomial of the cohomology as stated? – Greg Kuperberg Oct 9 '11 at 0:17
@Greg: If $X$ is smooth and proper over the integers, this is the main theorem of the paper of van den Bogaart and Edixhoven. – Dan Petersen Oct 9 '11 at 6:22

The following paper of Toen and Vaquie may be interesting.

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