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 $W$?


  • 2
    $\begingroup$ (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. $\endgroup$ – Qing Liu Nov 4 '11 at 7:59

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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