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59.6 A computation

How do we compute the cohomology of $\mathbf{P}^1_\mathbf {C}$ with coefficients $\Lambda = \mathbf{Z}/n\mathbf{Z}$? We use Čech cohomology. A covering of $\mathbf{P}^1_\mathbf {C}$ is given by the two standard opens $U_0, U_1$, which are both isomorphic to $\mathbf{A}^1_\mathbf {C}$, and whose intersection is isomorphic to $\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} = \mathbf{G}_{m, \mathbf{C}}$. It turns out that the Mayer-Vietoris sequence holds in étale cohomology. This gives an exact sequence

\[ H_{\acute{e}tale}^{i-1}(U_0\cap U_1, \Lambda ) \to H_{\acute{e}tale}^ i(\mathbf{P}^1_ C, \Lambda ) \to H_{\acute{e}tale}^ i(U_0, \Lambda ) \oplus H_{\acute{e}tale}^ i(U_1, \Lambda ) \to H_{\acute{e}tale}^ i(U_0\cap U_1, \Lambda ). \]

To get the answer we expect, we would need to show that the direct sum in the third term vanishes. In fact, it is true that, as for the usual topology,

\[ H_{\acute{e}tale}^ q (\mathbf{A}^1_\mathbf {C}, \Lambda ) = 0 \quad \text{ for } q \geq 1, \]


\[ H_{\acute{e}tale}^ q (\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} , \Lambda ) = \left\{ \begin{matrix} \Lambda & \text{ if }q = 1\text{, and} \\ 0 & \text{ for }q \geq 2. \end{matrix} \right. \]

These results are already quite hard (what is an elementary proof?). Let us explain how we would compute this once the machinery of étale cohomology is at our disposal.

Higher cohomology. This is taken care of by the following general fact: if $X$ is an affine curve over $\mathbf{C}$, then

\[ H_{\acute{e}tale}^ q (X, \mathbf{Z}/n\mathbf{Z}) = 0 \quad \text{ for } q \geq 2. \]

This is proved by considering the generic point of the curve and doing some Galois cohomology. So we only have to worry about the cohomology in degree 1.

Cohomology in degree 1. We use the following identifications:

\begin{eqnarray*} H_{\acute{e}tale}^1 (X, \mathbf{Z}/n\mathbf{Z}) = \left\{ \begin{matrix} \text{sheaves of sets }\mathcal{F}\text{ on the étale site }X_{\acute{e}tale}\text{ endowed with an} \\ \text{action }\mathbf{Z}/n\mathbf{Z} \times \mathcal{F} \to \mathcal{F} \text{ such that }\mathcal{F}\text{ is a }\mathbf{Z}/n\mathbf{Z}\text{-torsor.} \end{matrix} \right\} \Big/ \cong \\ = \left\{ \begin{matrix} \text{morphisms }Y \to X\text{ which are finite étale together} \\ \text{ with a free }\mathbf{Z}/n\mathbf{Z}\text{ action such that } X = Y/(\mathbf{Z}/n\mathbf{Z}). \end{matrix} \right\} \Big/ \cong . \end{eqnarray*}

The first identification is very general (it is true for any cohomology theory on a site) and has nothing to do with the étale topology. The second identification is a consequence of descent theory. The last set describes a collection of geometric objects on which we can get our hands.

The curve $\mathbf{A}^1_\mathbf {C}$ has no nontrivial finite étale covering and hence $H_{\acute{e}tale}^1 (\mathbf{A}^1_\mathbf {C}, \mathbf{Z}/n\mathbf{Z}) = 0$. This can be seen either topologically or by using the argument in the next paragraph.

Let us describe the finite étale coverings $\varphi : Y \to \mathbf{A}^1_\mathbf {C} \setminus \{ 0\} $. It suffices to consider the case where $Y$ is connected, which we assume. We are going to find out what $Y$ can be by applying the Riemann-Hurwitz formula (of course this is a bit silly, and you can go ahead and skip the next section if you like). Say that this morphism is $n$ to 1, and consider a projective compactification

\[ \xymatrix{ {Y\ } \ar@{^{(}->}[r] \ar[d]^\varphi & {\bar Y} \ar[d]^{\bar\varphi } \\ {\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} } \ar@{^{(}->}[r] & {\mathbf{P}^1_\mathbf {C}} } \]

Even though $\varphi $ is étale and does not ramify, $\bar{\varphi }$ may ramify at 0 and $\infty $. Say that the preimages of 0 are the points $y_1, \ldots , y_ r$ with indices of ramification $e_1, \ldots e_ r$, and that the preimages of $\infty $ are the points $y_1', \ldots , y_ s'$ with indices of ramification $d_1, \ldots d_ s$. In particular, $\sum e_ i = n = \sum d_ j$. Applying the Riemann-Hurwitz formula, we get

\[ 2 g_ Y - 2 = -2n + \sum (e_ i - 1) + \sum (d_ j - 1) \]

and therefore $g_ Y = 0$, $r = s = 1$ and $e_1 = d_1 = n$. Hence $Y \cong {\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} }$, and it is easy to see that $\varphi (z) = \lambda z^ n$ for some $\lambda \in \mathbf{C}^*$. After reparametrizing $Y$ we may assume $\lambda = 1$. Thus our covering is given by taking the $n$th root of the coordinate on $\mathbf{A}^1_{\mathbf{C}} \setminus \{ 0\} $.

Remember that we need to classify the coverings of ${\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} }$ together with free $\mathbf{Z}/n\mathbf{Z}$-actions on them. In our case any such action corresponds to an automorphism of $Y$ sending $z$ to $\zeta _ n z$, where $\zeta _ n$ is a primitive $n$th root of unity. There are $\phi (n)$ such actions (here $\phi (n)$ means the Euler function). Thus there are exactly $\phi (n)$ connected finite étale coverings with a given free $\mathbf{Z}/n\mathbf{Z}$-action, each corresponding to a primitive $n$th root of unity. We leave it to the reader to see that the disconnected finite étale degree $n$ coverings of $\mathbf{A}^1_{\mathbf{C}} \setminus \{ 0\} $ with a given free $\mathbf{Z}/n\mathbf{Z}$-action correspond one-to-one with $n$th roots of $1$ which are not primitive. In other words, this computation shows that

\[ H_{\acute{e}tale}^1 (\mathbf{A}^1_\mathbf {C} \setminus \{ 0\} , \mathbf{Z}/n\mathbf{Z}) = \mathop{\mathrm{Hom}}\nolimits (\mu _ n(\mathbf{C}), \mathbf{Z}/n\mathbf{Z}) \cong \mathbf{Z}/n\mathbf{Z}. \]

The first identification is canonical, the second isn't, see Remark 59.69.5. Since the proof of Riemann-Hurwitz does not use the computation of cohomology, the above actually constitutes a proof (provided we fill in the details on vanishing, etc).

Comments (4)

Comment #564 by Dustin on

Hey, first of all thanks for this resource.

Now that those niceties are out of the way, of course, I'm here to nitpick. I think there's an error in the discussion of the H^1 of A^1-0. The answer should be dual to what's written: it should be Hom(\mu_n,Z/nZ). The error is in the "We leave it to the reader to see...".

Comment #565 by on

Thanks for your encouragement. As to your nitpick: I was thinking of (some of) these coverings as given by with action of in given by where is a primitive th root of . But I guess your point is that one can equally well think of this the other way round, i.e., as the map sending to . To check which one is right, you'd have to compute the addition of two of these torsors (addition of torsors is given by taking the product of the torsors over the base and then dividing by the anti-diagonal action of the group). This is annoying, but I did it to make sure that you were right.

Anyhoo, maybe a better way of getting at it is that there is an absolutely canonical -torsor over , namely, given by the th power map. This gives a canonical element whose image over the complex numbers tells us you are correct.

Fixed here. Thanks!

Comment #1701 by Yogesh More on

Regarding the opening line "How do we compute the cohomology of with ?", which cohomology are you referring to - etale or singular (of course the answer turns out to be the same, and perhaps the same method works for both - cech cohomology). From the next sentence involving Cech cohomology I thought you were referring to singular cohomology, and being a beginner I don't know yet if Cech cohomology computes etale cohomology for abelian sheaves (as opposed to quasi-coherent ones) - if so can you link to it (I searched but couldn't find it)? But then the remaining sentences convinced me you were referring to etale cohomology of .

minor grammatical fix: in line 3 "and which intersection is isomorphic" I think should be whose intersection is isomorphic

Comment #1747 by on

OK, I fixed the which and whose confusion, see here. About the other stuff. There is a bit of information about Cech cohomology later in the chapter. But the discussion is a bit loose, because it comes from a lecture I gave a long time ago. If you have a precise suggestion of how to change the text please feel free to put a further comment here.

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