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 $X$?
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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 SawinCommented Oct 25, 2017 at 20:21
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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 $\mathbb Q$. $\endgroup$– R. van Dobben de BruynCommented Oct 25, 2017 at 20:21
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$\begingroup$ @WillSawin I learned etale cohomology from some research papers and online short note, I will check some serious books! $\endgroup$– WenzheCommented Oct 25, 2017 at 20:28
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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..
answered Oct 25, 2017 at 20:37