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 Jan25 comment Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused) Yes, I just wanted to be sure. Thank you for this. It completely cleared up everything. Jan25 comment Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused) I think I understand what I'm doing wrong now. Starting with $Y\to X$ as in the question, the base change gives $Y\times_X X^\prime \to X^\prime$. This is not necessarily etale, but once you normalize you obtain $Y^\prime\to X^\prime$ and this morphism is etale by Abhyankar. Now, if by some chance $Y^\prime = Y\times_X X^\prime$ (i.e., the base change is normal) then faithfully flat descent implies that the morphism we started out with was already etale. Thank you very much QiL!! Jan25 comment Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused) By "some maximal ideal" do you mean "any maximal ideal", or does Abhyankar really mean that there is maybe just "one" maximal ideal? Jan24 asked Is there a construction known for associating a K3 surface to a curve or cover of curves Jan24 asked Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused) Jan24 accepted Is a group with only finitely many subgroups of index n (for all n) finitely generated Jan24 comment Is a group with only finitely many subgroups of index n (for all n) finitely generated Great! I guess one can also use this to obtain a non-abelian example by taking the product of $\mathbf Q/\mathbf Z$ with a finite non-abelian group $G$. Then $\mathbf Q/\mathbf Z \times G$ is not finitely generated, but it also has only finitely many subgroups of index $n$, right? Jan24 asked Is a group with only finitely many subgroups of index n (for all n) finitely generated Jan24 accepted Is a finite field extension of a imperfect field imperfect Jan23 asked Is a finite field extension of a imperfect field imperfect Jan20 comment Is $M_g$ a subvariety of $M_{h}$ for some $h>g$ No, but maybe something like Knudsen clutching can be used? Jan19 asked Is $M_g$ a subvariety of $M_{h}$ for some $h>g$ Jan19 asked Are there generalizations of Prym varieties to higher dimensions Jan19 comment Is the relative rank function with respect to an ample line bundle non-decreasing Ok. I should have thought of that...I can still save the situation. In my case, $\mathcal L = \Omega^1_{X/K}$. What can we do in this case? Of course, if you take a rational variety this is not going to work, but let's assume $X$ is such that $h^0(\Omega_{X/K}) >0$. Jan19 comment Is the relative rank function with respect to an ample line bundle non-decreasing I fixed some typos and inaccuracies. The answer is still incomplete though. I need to figure out why $\mathcal L \cong \mathcal O_X(D)$ with $D$ effective. Do you know if this is true? Jan19 revised Is the relative rank function with respect to an ample line bundle non-decreasing deleted 46 characters in body Jan18 answered Is the relative rank function with respect to an ample line bundle non-decreasing Jan18 revised Is the relative rank function with respect to an ample line bundle non-decreasing added 516 characters in body Jan18 asked Is the relative rank function with respect to an ample line bundle non-decreasing Jan13 comment Are there moduli spaces of higher-dimensional varieties When I say moduli stack I mean a Deligne-Mumford stack solving the moduli problem. Sorry for not saying that explicitly.