Skip to main content
MathOverflow

Return to Question

added 32 characters in body
Source Link
user19475
user19475

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{Br}(X)$$\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".)

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 $\mathrm{H}^2_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{Br}(X)$, 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".)

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".)

added 570 characters in body
Source Link
user19475
user19475

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 $\mathrm{H}^2_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{Br}(X)$, 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".)

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?

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 $\mathrm{H}^2_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{Br}(X)$, 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".)

Source Link
user19475
user19475

representability of étale and motivic cohomology by schemes

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?