# Quail Entry: Hurwitz Thm

Around 12/21/09 I looked into how to prove Hurwitz thm (but I didn’t write down some of the details until 1/1/10). I more or less followed Hartshorne, but the proof comes as the corollary of a number of a number of other results whose full strength is not needed for just Hurwitz thm. Below is roughly how I unfolded the proof of Hurwitz’s thm…

Hurwitz: curves (complete over alg. cl. field (also nonsingular!)) and is finite. Then

(1)

Where is the ramification divisor defined as .

The first step is to interpret Hurwitz as a statement about sheaves of differentials, this also leads to a better understanding of ramification divisor. (this can be proved via Riemann Roch), in any case Hurwitz now can be written as

(2)

Any by definition is closely related to . These three sheaves of differentials fit into a s.e.s

(3)

pf: general theory of differentials associated to ring maps says exactness holds everywhere except possibly the injection

*Note the exact seq. of differentials associated to is in some ways analogous to the l.e.s of a triple in homology.

to show the first map is injection the key is to show the maps is nonzero at the generic point; indeed is the map is not injective, the something nonzero maps to zero, but at the generic point everything nonzero is invertible to this would say a unit mapped to zero, hence the map would be zero. Now the result follows because

is 0 at the generic point, means the first map is surjective so certainly not zero. For the generic point, . An algebra thm says that the dimension of this space is , also is separable, and in the separable case the inequality is an equality, so the space in equation is zero.

Now describe the sheaves of differentials and maps in terms of local parameters. Let be a loc. parm at and a loc. par. for . Now

. The fact that its free of rank 1 follows from an alg. result of the form is a free module of rank for a loc. ring whose residue field is isomorphic to ; also require to be perfect and to be a localization of a finitely generated algebra. That is a generator follows from Nakayama’s lemma and that becomes generator after modding out by max ideal.

Get something similar for . Localize (3) at . Have and its local map, means for some unit ; the natural number is called the ramification index. If the characteristic doesn’t divide then the ramification is tame; otherwise its wild.

When all the ramifications are tame, : first map in localized version of (3) is . If ramification tame, then image of injection is hence quotient has length given by . But if the ramification is wild, then and for ; e.g. our unit could be then . So in general the length is .

The upshot is considering as a closed subscheme, get

Now use (3) and state via divisors. The canonical divisors are defined such that So tensoring (3) with the inverse of this sheaf we get

Would like to say and hence conclude and be done by taking degrees, but for this need iso

Compatibility (ex. II.6.8): above and then .

Recall on divisors is first defined on a point via and then extended by linearity. Note the curves being nonsingular implies they are locally factorial so can use Weil = Cartier = Pic. The problem essentially asks if the following two maps agree

(4)

Can reduce to the case

If it holds for a point, then for general have .

It holds very generally (for sheaves of modules) that . Enough to show iso on the level of presheaves because then the left side is the sheafifacation of the presheaf on the right which is the right side. And on the level of presheaves it boils down to the following isomorphism

and this can be proved by showing the left side has the universal property of the right side.

the upshot

(point to justify!; made some remarks in ogus excerpts) is flat, this implies is exact (note is always exact). So for both of the following two sequences are exact

Remains to show note this makes sense because is effective; the subschemes are differentiated in that the associated ideal sheafs are

Clear to see only supported on . Say , . Now take stalks of the s.e.s (4). Seems like a general fact that this arises essentially because

parametrize the same sets. It follows that , now more or less because is the ideal sheaf that cuts out the subscheme ; using the local parameters above, i.e. that it follows that

So localizing (4) at each produces

hence which also shows

This gives all the pieces of Hurwitz thm.

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- Published:
- January 1, 2010 / 11:58 pm

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- Quail, wall scribble

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