One has $\mathrm{H}^1_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^2(X,\mathbf{Z}(1)) = \mathrm{Pic}(X)$ (étale and motivic cohomology).

Is étale or motivic cohomology in other dimensions also representable by a scheme?

For the Brauer group $\mathrm{H}^2_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^3(X,\mathbf{Z}(1)) = \mathrm{Br}(X)$, this is false: Why is there no Brauer scheme?

But what about other dimensions and other coefficients than $\mathbf{G}_m$? For example, one has for $f: X \to S$ and $\ell$-adic cohomology $\mathrm{R}^1f_*\mathbf{Z}_\ell(1) = T_\ell\mathbf{Pic}_{X/S}$.

Also, the formal Brauer group is pro-representable: http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf p. 389, Chapter 18, Corollary 1.13.

(I assume that $X/S$ is "nice".)

  • 3
    $\begingroup$ If $X$ is a smooth complex projective variety, $H^2_{\rm ét}(X,\mathbb{G}_m)$, the Brauer group of $X$, is isomorphic to $(\mathbb{Q}/\mathbb{Z})^{n}$ for some $n$ ($=b_2-\rho $ for the experts). It is certainly not "representable by a scheme". $\endgroup$ – abx Jun 24 '17 at 14:01
  • $\begingroup$ mathoverflow.net/questions/185630/why-is-there-no-brauer-scheme $\endgroup$ – Daniel Loughran Jun 24 '17 at 15:42
  • $\begingroup$ And in other dimensions? $\endgroup$ – user19475 Jun 24 '17 at 16:10

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