I'm trying to read chapter XIII of SGA1, and I'd appreciate some help about a few issues I'm having.

  1. Definition 2.1.1. is of tamely ramified sheaves. The definition is as such: if $U$ is an open subscheme of $X$ (which is a scheme over $S$), $F$ a sheaf of sets on $U$, and $Y$ the reduced induced closed subscheme $X - U$, which we assume to be a normal crossings divisor. $F$ is said to be tamely ramified over $X$ along $Y$ relatively to $S$, if for every geometric point $\bar s$ of $S$, and for every closed point $y$ of $Y_{\bar s}$, the restriction of $F$ to the field of fractions $K$ of $\mathcal{O}_ {X_{\bar s}}$ is representable by the spec of an etale $K$-algebra $L$, tamely ramified over $\mathcal{O}_ {X_ {\bar s, y}}$.

    I can't say that I fully understand this. I can think of several interpretations of this, but I'd feel more comfortable if someone will corroborate one of them. I'm guessing $\mathcal{O}_ {X_{\bar s}}$ means the stalk of the generic point of $X_{\bar s}$ in $X$, rather than the structure sheaf of $X_{\bar s}$. Still, I feel this definition is lacking. For example, $L$ is an etale $K$-algebra, but it makes sense to ask that it be tamely ramified over $\mathcal{O}_ {X_ {\bar s, y}}$? Also, by sheaf in sets I'm guessing they mean a stack in the small etale site fibered in sets such that it's a functor. I guess what I'm really asking is how to think of this. How does this definition relate to tamely ramified covers? How do these "tamely ramified" sheaves have to do with the tame $\pi_1$'s?

  2. This one is mostly me not knowing French well enough: In the second paragraph of 2.1.5. it says "Si maintenant $F$ est un faisceau en groupes sur $U$, l'image inverse sur $S'$ d'un torseur sous $F$ moderement ramifie sur $X$ relativement a $S$ est un torseur sous $F'$ moderement ramifie sur $X'$ relativement a $S'$."

    I don't understand this grammatically. Is it "If now $F$ is a sheaf of groups of $U$, the inverse image of $S'$ of a sub-torsor of $F$ ..." or "of $F$-torsors", or "sub-sheaf of $F$ that is made up of torsors". None of these makes much sense to me. Do you understand the statement and how it would make sense?

  • $\begingroup$ The French you ask about reads in English as: "If now $F$ is a sheaf of groups on $U$, the inverse image on $S'$ of a torsor under $F$ tamely ramified on $X$ relative to $S$ is a torsor under $F'$ tamely ramified on $X'$ relative to $S'$". No surprises there, I think. (Note: "sur" means "on", not "of"; "de" means "of". "faisceau en groupes" literally means "sheaf in groups", but of course that is not how we say it in English.) $\endgroup$
    – Emerton
    Apr 28 '10 at 5:21

If you look in the text, $y$ is not a closed point but a maximal point, meaning a maximally generic point. Then $O_{X_{\bar{s}y}}$ is a discrete valuation ring, and so it makes sense to talk about tameness of extensions. Also, surely she's working in the etale topology, otherwise it would be kind of silly to try and represent functors by etale covers, but I didn't actually see where she says that. It's got to be written somewhere, though!

I suppose you're right about a sheaf of sets being a stack with no non-identity maps, but taking this as a definition is exactly as sensible as defining a set to be a category with no non-identity maps.

"torseur sous F" = "torsor under F" = "F-torsor"

  • 1
    $\begingroup$ "maximally generic" = "generic". The "maximal point" terminology (at least in the context of algebraic geometry) is really confusing and should not be used. In section 0 it is said that she is working in the etale topology throughout. $\endgroup$
    – BCnrd
    Apr 27 '10 at 23:35
  • $\begingroup$ Aha! I was not aware of this usage of "maximal point". Okay, this should make things make sense. $\endgroup$
    – H. Hasson
    Apr 28 '10 at 1:43
  • $\begingroup$ Actually, neither was I, but I know that to get a discrete valuation ring, you have to localize at a point of codimension 1, so it seemed reasonable to infer that maximal meant generic. If I may, it's usually a good idea to keep some simple examples in mind, especially when reading things written in the formal style. It can save you a lot of confusion. $\endgroup$
    – JBorger
    Apr 28 '10 at 7:01

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