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Can one develop a theory of étale homology via étale cosheaves? The hope is that this would, for example, return the Tate module (and not its dual) for an elliptic curve, and it would return group homology for $\rm Spec$ of a field.

The first step in this direction is to notice that you can define an étale cosheaf on $\rm Spec$ of a field, in which the sections are coinvariants, rather than invariants, of the associated Galois module. In this case I believe we would recover group (Galois) homology.

(See Cosheaf homology and a theorem of Beilinson (in a paper on Mixed Tate Motives) if you don't know what cosheaves are.)

I'm not expecting this to prove anything new, but it would make certain formulations nicer (for some definition of "nice"). E.g., I'm hoping we would get theorems that look similar to the relations between singular homology and cohomology, and we would get a comparison with singular homology for finite coefficients.

I feel like this should work, though I'm not entirely sure about the ability to cosheafify.

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Sure thing! There's an equivalence of categories between cosheaves valued in profinite abelian groups and sheaves valued in torsion abelian groups, by Pontryagin duality one could say. So you can directly define the first etale homology of Z_ell and it will give you the l-adic Tate module. But it's formally the same as defining the l-adic Tate module to be Pontryagin dual to H^1 with Z/l^\infty-coefficients. I agree with you, though, that the cosheaf perspective -- though equivalent -- seems more natural in many cases.

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I think you mean Poincare? –  David Corwin Feb 14 '13 at 8:19
No, I literally mean Pontryagin dual. Think of it this way: etale cohomology with Z/l^infty coefficients should be equal to continuous homomorphisms from \pi_1(X) to Z/l^infty, or to continuous homomorphisms from H_1(X;Z_l) to Z/l^infty. This just means that H^1(X;Z/l^infty) is Pontryagin dual to H_1(X;Z_l). –  Dustin Clausen Feb 14 '13 at 15:53
(for a nice reference from the Cech perspective, see Mitchell's paper math.uiuc.edu/K-theory/0346/topk.pdf in section 3.) –  Dustin Clausen Feb 14 '13 at 18:16
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