1
$\begingroup$

In all the literature I have read, etale cohomology is defined for smooth varieties. Suppose $X/\mathbb{Q}$ is a singular variety over $\mathbb{Q}$, does there exist etale cohomology $H_{\text{et}}^*(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})$ defined for it? If $H_{\text{et}}^*(X_{\bar{\mathbb{Q}}},\mathbb{Q}_{\ell})$ is defined, is it still a representation of the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$? Does the characteristic polynomial of the geometric Frobenius still counts the rational points of $X$?

$\endgroup$
3
  • 2
    $\begingroup$ What literature have you read? I don't know any serious introduction to etale cohomology that doesn't cover the case of singular varieties. $\endgroup$
    – Will Sawin
    Oct 25, 2017 at 20:21
  • 1
    $\begingroup$ Étale cohomology is defined for any scheme (or stack, or ...). I don't know a single reference that insists on sticking to smooth varieties. And you cannot use Frobenius to count rational points if your base field is $\mathbb Q$. $\endgroup$ Oct 25, 2017 at 20:21
  • $\begingroup$ @WillSawin I learned etale cohomology from some research papers and online short note, I will check some serious books! $\endgroup$
    – Wenzhe
    Oct 25, 2017 at 20:28

1 Answer 1

12
$\begingroup$

Just to answer the last question, the key statement is that for a proper variety over $\mathbb F_q$, the number of $\mathbb F_{q^n}$ points is equal to the alternating sum of the traces of $\operatorname{Frob}_{q^n} = \operatorname{Frob}_q^n$ on the etale cohomology. (Or for a general variety, if you take etale cohomology with compact supports).

If you have a variety $X$ over $\mathbb Q$, take a model over $\mathbb Z$, and look at the $\mathbb F_{p^n}$-points, then you can get this statement in terms of the trace of the Frobenius element $\operatorname{Gal}(\overline{\mathbb Q}|\mathbb Q)$ acting on $H^*_{et}(X_{\overline{\mathbb Q}} , \mathbb Q_\ell)$ if you can show that the cohomology is ``the same" in characteristic zero and characteristic $p$. More precisely, you want to show that the cohomology sheaves on $\operatorname {Spec} \mathbb Z$ are lisse. One case you can do this is when the integral model you have chosen is smooth and proper over $\mathbb Z$. This is where a smoothness assumption is helpful.

However, if you compute directly the etale cohomology in characteristic $p$, the smoothness is not needed for the Lefschetz fixed point formula.

Also, if you have a variety over $\mathbb Q$ and are comparing its reductions to different primes $p$, for any given integral model this lisseness assumption will hold for all but finitely many primes $p$, so if you are willing to throw away finitely many primes you don't need any additional assumptions..

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.