2
$\begingroup$

Let $\pi:X\to\mathcal X$ be a presentation of an Artin stack $\mathcal X$ of finite type over a field $k,$ and let $f:Y\to X$ be a finite \'etale covering. Does there exist a finite \'etale covering $Y'\to X$ factoring through $Y,$ such that $Y'$ can be given descent structure, i.e. there exists an isomorphism $pr_1^*Y'\cong pr_2^*Y'$ over $X\times_{\mathcal X}X$ satisfying cocycle condition, so that $Y'\to X$ descend to a finite \'etale covering $\mathcal Y\to\mathcal X$?

$\endgroup$

1 Answer 1

4
$\begingroup$

I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $\mathcal{X}$ is a point, and $X$ is a smooth variety with non-trivial fundamental group, say, an elliptic curve (or $\mathbb{A}^1-\{0\}$). Then $\pi$ is a presentation. Let $f:Y\to X$ be a non-trivial finite etale cover, say, the cover of the elliptic curve by an isogeneous elliptic curve. Then your question becomes: `is there a trivial (i.e., lifted from $\mathcal{X}$) cover $Y'$ of $X$ with a map to $Y$? This is of course not true.

$\endgroup$
2
  • $\begingroup$ Thanks. I wanted to know if any locally constant constructible sheaf on the stack $\mathcal X$ can be trivialized by a finite etale cover (which is true for schemes). That's why I asked that descent question. I was told that the answer to this lcc sheaf question is Yes, because one can take the espace etale of the sheaf... $\endgroup$
    – shenghao
    Dec 19, 2009 at 6:22
  • $\begingroup$ I see. Well, the original question is curious by itself, while, as you explain, the answer to the lcc sheaf question is essentially tautological. $\endgroup$
    – t3suji
    Dec 19, 2009 at 14:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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