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Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow 0$, it is enough to show that $H^1_{et}(X,\mathbb{Q})$ vanishes. It is known, for instance by 2.1 of Deninger's 1988 JPAA paper, that $H^1_{et}(X,\mathbb{Q})$ vanishes when $X$ is normal.

Note: there are two arguments I think are incorrect that claim to show $H^1_{et}(X,\mathbb{Z})$ always vanishes. The first is that $\mathbb{Z}$ is flasque in the etale topology. This is false. For instance, over the function field $\mathbb{C}(x,y)$, the long exact sequence in cohomology for $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}/n\rightarrow 0$ shows that $H^2(\mathbb{C}(x,y),\mathbb{Z})$ is non-zero. So, $\mathbb{Z}$ cannot be flasque. The second argument is that $H^1_{et}(X,\mathbb{Z})=Hom_{cont}(\pi_1^{et}(X),\mathbb{Z})$, where $\pi_1^{et}(X)$ is the etale fundamental group of $X$, which is a profinite group. Since it is profinite, the $Hom$ group above vanishes. But, the claimed equality between $H^1_{et}(X,-)$ and $Hom_{cont}(\pi_1^{et}(X),-)$ only holds for torsion sheaves, as far as I have been able to determine.

I am in fact interested in several things. First, either an example of $X$ such that $H^1_{et}(X,\mathbb{Z})$ is non-zero, or a proof that this always vanishes. Second, the same thing but where we only look at affine $X$. In particular, if it exists, I would love to see an example of a commutative ring $R$ where $H^1_{et}(Spec R,\mathbb{Z})$ is non-zero, if this is possible.

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We always have the formula $H^1(X,\mathbb{Z})=Hom_{cont}(\pi^{et}_1(X),\mathbb{Z}$ (where the progroup $\pi^{et}_1(X)$ is defined as the fundamental group of the small etale topos of $X$, say). But $\pi_1(X)$ is known to be profinite only for $X$ noetherian and normal, so that this does not contradicts the counter example of Vistoli below. –  Denis-Charles Cisinski Dec 27 '11 at 23:49
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As Denis-Charles Cisinski mentions, you have to be careful with your definition of $\pi_1^{et}$ - the version defined in SGA1 only uses finite etale covers, while the "gros" version in SGA3 uses all etale covers. One has examples of gros etale fundamental groups that are infinite and discrete, coming from maximally degenerate curves of positive arithmetic genus. –  S. Carnahan Dec 28 '11 at 10:48
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Thanks for the clarification about the etale fundamental group. –  Benjamin Antieau Dec 29 '11 at 22:33
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Dear Benjamin, If you are still interested in this kind of thing, this discussion in comments at the Secret Blogging Seminar is highly relevant: sbseminar.wordpress.com/2009/04/20/… Regards, –  Emerton Mar 21 '13 at 12:17
    
@Emerton: thanks! Definitely looks useful. –  Benjamin Antieau Mar 22 '13 at 5:26
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2 Answers

up vote 15 down vote accepted

The standard example is a copy of $\mathbb A^1_k$, where $k$ is an algebraically closed field, with two points glued. In algebraic terms, $X = \mathop{\rm Spec}R$, where $R := k[x,y]/(y^2 - x^3 + x^2)$. Consider the finite morphism $\pi\colon \mathbb A^1 \to X$, which yields an exact sequence $$ 0 \to \mathbb Z_X \to \pi_*\mathbb Z_{\mathbb A^1} \to i_*\mathbb Z_p \to 0, $$ where $p$ is the singular point of $X$. Since both $\pi_*\mathbb Z_{\mathbb A^1}$ and $i_*\mathbb Z_p$ have trivial étale cohomology , by taking global sections we see that $\mathrm H^1(X, \mathbb Z) = \mathbb Z$.

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More generally, if $X$ is proper over an algebraically closed field, then $H^1(X,\mathbb Z)$ is isomorphic to the cocharacter module of the maximal torus of the Picard variety $Hom(\mathbb G_m,Pic^0)$.

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Dear Thomas, can you give a reference/proof for this nice result? –  Timo Keller Sep 30 '12 at 9:33
    
It is the content of my paper: The affine part of the Picard scheme. –  Thomas Geisser Sep 30 '12 at 16:38
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