# generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question.

Let $p : X \rightarrow Y$ be a finite morphism of algebraic varieties over $\mathbb{C}$ with $Y$ smooth and $X$ normal. By Abhyankar's theorem, we know that if the ramification locus of $p$ is a divisor with normal crossings, then $X$ has quotient singularities (and in particular has rational singularities).

I would like to know if there are weaker hypotheses on the ramification locus which would imply that $X$ is Cohen-Macaulay? One important thing is that I don't want to assume that $p$ is flat (in fact I want to prove that $p$ is flat).

Thanks!

• Could you please give a reference for Abhyankar's result? – Francesco Polizzi Jul 17 '14 at 14:41
• @FrancescoPolizzi There is this paper by Viehweg : uni-due.de/~mat903/preprints/ec/rational_sing.pdf (lemma 2). But he does not prove it, he gives a reference to a paper in German, which I am not able to read. – Libli Jul 17 '14 at 14:53
• @FrancoPolizzi It's in SGA 1 XII.5.2. – Will Sawin Jul 17 '14 at 15:32
• @WillSawin: thanks. Maybe you meant SGA 1 XIII.5.2? – Francesco Polizzi Jul 18 '14 at 8:45
• @FrancescoPolizzi yes, sorry for my mistake. – Will Sawin Jul 18 '14 at 15:14