The etale fundamental group and etale cohomology with compact support

Before me, the following was asked: etale fundamental group and etale cohomology of curves

However, that question dealt only with projective curves.

Question

Let $$X$$ be any scheme (or if you prefer something more concrete, a variety over some field), and let $$l$$ be some prime different from the characteristics of the residue fields of $$X$$ (respectively, the characteristic of the field over which the variety is defined), then is there an isomorphism $$Hom_{cont}(\pi_1^{et}(X),\mathbb{Q}_l)\cong H^1_c(X,\mathbb{Q}_l)$$?

• Why $H^1_c$ instead of $H^1?$ Oct 25, 2011 at 3:39

In general, it's always true (for a connected scheme) that $H^1_{et}(X, \mathbb{Z}/l \mathbb{Z}) = \hom(\pi_1^{et}(X), \mathbb{Z}/l\mathbb{Z})$ (not compactly supported). Taking inverse limits over $l$ then gives the claim.
The reason this is true is that $H^1_{et}(X, \mathbb{Z}/l\mathbb{Z}$) can be computed by Cech cocycles, and from this it follows that elements of this group classify torsors over $\mathbb{Z}/l\mathbb{Z}$ (i.e. sheaves with a $\mathbb{Z}/l\mathbb{Z}$-action which are locally the constant $\mathbb{Z}/l \mathbb{Z}$ (in the etale topology)). But these, by descent theory, are the same as Galois covers of $X$ with Galois group $\mathbb{Z}/l\mathbb{Z}$, and (by Galois theory) classified by maps from the etale fundamental group into $\mathbb{Z}/l\mathbb{Z}$.
• What do you mean by the first statement? That $H^1_{et}(X,\mathbb{Z}/l\mathbb{Z})=hom(\pi_1^{et}(X),\mathbb{Z}/l\mathbb{Z})$? This would be false in algebraic topology. Think of $hom(\pi_1^{et}(X),\mathbb{Z}/l\mathbb{Z})$ as really being $(\pi_1^{et}(X))^{ab}\otimes \mathbb{Z}/l\mathbb{Z}$... Oct 24, 2011 at 19:38
• @Ariyan: The statement about $H^1$ should always be true (on sites in general). @Matt: Thanks, and whoops! Oct 24, 2011 at 23:57