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I have two questions.

  1. What class of schemes are the Weil conjectures true for? Projective schemes? Do I need additional hypotheses? I know from my question (The Weil conjecture involving Betti numbers.) that we need a scheme over the Spec of some ring of integers.
  2. Can you provide an example of a scheme as in 1. that is not a projective variety?

Thank you in advance.

Joseph Harrison
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    You may be conflating two things. 1."Usual" Weil conjectures is stated for smooth projective varieties over $\mathbb{F}_q$. Reduction isn't really involved. 2. When the variety over $\mathbb{F}_q$ comes from good reductions of varieties over $Spec(O_K)$ though, there may be comparison theorems + base change theorems that establishes isomorphism between classical singular cohomology and etale cohomology. The latter seems to be what you are asking for, but is not part of Weil conjectures. See http://www.math.lsa.umich.edu/~mmustata/lecture5.pdf p.16 for some sample ... – dummy Feb 04 '23 at 22:23
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    comparison theorems result. – dummy Feb 04 '23 at 22:24
  • @dummy thanks for the comment and the reference. 1. I see that the conjectures are stated for smooth varieties, but I can still define the zeta function for a scheme (although some hypotheses may be needed to make the sum/product converge) and so I'm wondering if the conjectures are still true here. 2. What do you mean by "good reduction"? PS: I see the equality in your reference between the singular cohomology with $\ell$-adic coefficients and the étale cohomology. Unfortunately, I am not advanced enough to understand such statements yet. Also I am used to seeing singular homology... – Joseph Harrison Feb 05 '23 at 00:51
  • ... I suppose that smoothness allows us to use Poincaré duality. – Joseph Harrison Feb 05 '23 at 00:52
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  • I am not sure what happens once you leave varieties, but see https://mathoverflow.net/questions/293812/which-part-of-weils-conjecture-fails-for-a-nice-singular-variety for what works/fails for singular variety.
    • – dummy Feb 05 '23 at 05:19
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  • A priori you can define varieties directly over $\mathbb{F}q$. But of course, if you have a variety over $\mathbb{Z}$, you can "take the equations mod $p$ to talk about reduction mod $p$ (or in scheme theoretic language, this is the base change $X \times{Spec\mathbb{Z}} Spec (\mathbb{F}_p)$). Of course, smoothness is important in Weil conjectures (per the MO post above), so you care about when the mod $p$ reduction is smooth. When that is the case, we say $X$ has good reduction mod $p$.
    • – dummy Feb 05 '23 at 05:23
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    Also just as a remark, I mentioned comparison theorems because in your linked question, you are trying to compare Betti numbers in singular homology vs what you get from zeta function over $\mathbb{F}_q$ (hence etale cohomology). For that reason there had to be some pathway between $\mathbb{F}_q$ to $\mathbb{C}$. I just wanted to point out that this path way is not part of Weil conjectures, and that Weil conjectures can be purely a statement of varieties over $\mathbb{F}_q$ alone, and does not need to involve reduction. – dummy Feb 05 '23 at 05:30
  • @dummy 1. thank you for linking that question and 2. thank you for the explanation of good reduction as requiring that the base change be smooth. I guess, more generally, we can also have the base change $X \times_{\textrm{Spec} ; \mathscr{O}_L} \textrm{Spec} ; \mathscr{O}_L/\mathfrak{p}$ when $X \to \textrm{Spec} ; \mathscr{O}_L$ is a scheme over the ring of integers of a number field $L$ and $\mathfrak{p}$ is a prime of $\mathscr{O}_L$. 3. I'm afraid I do not understand where you say that the conjectures do not need to involve reduction. How would we obtain the finite field variety... – Joseph Harrison Feb 05 '23 at 18:59
  • ... without reduction? – Joseph Harrison Feb 05 '23 at 18:59
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    $\mathbb{F}_p$ is a field. You can write down polynomial equations with coefficients in $\mathbb{F}_p$ directly. So you can directly write down varieties over $\mathbb{F}_p$. Smoothness is also defined over $\mathbb{F}_p$. Basically none of varieties/their smoothness requires a lift to $\mathbb{Z}$ or other ring of integers to define/think about. – dummy Feb 05 '23 at 19:13
  • @dummy so you're saying that by making this pathway from $\mathbf{F}_q$ to $\mathbf{C}$ that you talk about (we do this by connecting the singular homology to étale cohomology I suppose) we do not need to talk about the topology on a complex variety at all? PS: can you write an answer so that I can accept it? I think this question can be called solved now :) thank you again for your comments – Joseph Harrison Feb 05 '23 at 19:24
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    A big motivation of scheme theory is to define etale cohomology so that one can make sense of "geometry over $\mathbb{F}_q$. So yes, in particular we don't need to talk about topology on complex variety at all - this is not part of Weil conjectures.

    However, one (e.g. Weil) does believe that the numbers coming out from "geometry over $\mathbb{F}_q$" match the numbers from classical complex varieties; making sense of this is where comparison theorems come in.

     – dummy Feb 05 '23 at 19:29