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I have read several articles which use a version of the Weil decent criterion for covers, but the reference is always to Weil's original paper (1956 - The field of definition of a variety). I would like to know how one makes the transition. Note that when I say covers I mean a morphism between covers $(f:V\rightarrow W), (f':V'\rightarrow W)$ is a $g:V\rightarrow V'$ commuting with the covering maps.

For reference, here is the theorem from Weil's paper, which I simplified assuming Galois:

Let $k/k_0$ be a finite, Galois extension, $H=Gal(k/k_0)$. Let $V$ be a projective variety defined over $k$. Elements of $H$ have a natural action on varieties and morphisms defined over $k$ (for example by acting on the coefficients if we embed into projective space). Suppose for each $\sigma, \tau \in H$, we have an isomorphism $f_{\tau,\sigma}:\sigma(V)\rightarrow \tau(V)$. Then we have a model $V_0$ over $k_0$ for $V$ if the following are satisfied:

(i) $f_{\tau,\rho}=f_{\tau,\sigma}\circ f_{\sigma,\rho}$ for all $\sigma,\tau,\rho\in H$.

(ii) $f_{\tau \omega, \sigma \omega}=\omega(f_{\tau,\sigma})$ for all $\sigma,\tau\in H$, $\omega \in Gal(k_0^{sep}/k_0)$.

Now I suspect that to make the translation, one takes $f:V\rightarrow W$ and looks at the graph $\Gamma_f\subseteq V\times W$. Let's suppose we can satisfy the criteria above (with the $f_{\tau,\sigma}$ morphisms of covers) for $\Gamma_f$. Then we get some $\Gamma_0$ defined over $k_0$ and an isomorphism $\varphi:\Gamma_0\times k \rightarrow \Gamma_f$. There are two points I can't resolve:

(1) How do we know we have $\Gamma_0\subseteq V_0\times W_0$ for some models $V_0,W_0$ of $V,W$ (we can get the models of $V$ and $W$ over $k_0$ from the covering data). And further that it is the graph of a morphism $V_0\rightarrow W_0$.

(2) How do we know that the map $\varphi$ corresponds to a morphism of covers of $W$?

Thanks

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(1) You are descending a closed subvariety of $(V_0\times W_0)_k$ to $k_0$, so you get a closed subvariety of $V_0\times W_0$. That $\Gamma_0$ is the graph of a morphism is equivalent to the first projection $\Gamma_0\to V_0$ being an isomorphism. As this is true after extension to $k$, it is already true over $k_0$ because $k$ is faithfully flat over $k_0$. (2) The condition of being a morphism of covers is the equality of two morphisms. The latter can be checked over any field extension. –  Qing Liu Nov 4 '11 at 7:59

1 Answer 1

As I understand it, question 1 is about whether closed immersions descend along Galois covers, and question 2 is about whether being an open immersion or a surjection is a property that descends along Galois covers.

You can find the answer to the first question in SGA 1 Exp 8, section 5, where the Galois cover condition is replaced by a much weaker condition of fpqc cover.

You can find the answer to the second question in EGA IV vol. 2 2.7.1, or SGA 1 Exp 8, section 4, again in somewhat more general language than what you strictly need.

There are also proofs in Vistoli's notes. I'm afraid I don't know about any proofs that use Weil's old terminology, and I couldn't think of a good reason to look for one.

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