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ETALE SHEAVES WITH TRANSFERS As U × T → X × T is flat, the pullback of cycles is well-defined and is an injection. Hence the subgroup Ztr (T )(X) = Cork (X, T ) of cycles on X × T injects into the subgroup Ztr (T )(U ) = Cork (U, T ) of cycles on U × T . 1 is exact at Ztr (T )(U ), take ZU in Cork (U, T ) whose images in Cork (U ×X U, T ) coincide. We may assume that X and U are integral, and that the ´etale map U → X is finite; let F and L be their respective generic points. 11, because if L lies in a Galois extension L and G = Gal(L /F ), then ZL lies in CorF (L , TF )G = CorF (L, TF ).

If M is in She´t (Et/k), then M → π∗ π ∗ M is an isomorphism by Ex. 8. Thus π ∗ is faithful. By category theory, π ∗ π∗ π ∗ ∼ = π ∗ , so for F locally constant we have a natural isomorphism π ∗ π∗ F ∼ = F. 10. Let L be a Galois extension of k, and let G = Gal(L/k). Show that Ztr (L) is the locally constant ´etale sheaf corresponding to the module Z[G]. 11. Any locally constant ´etale sheaf has a unique underlying ´etale sheaf with transfers. ´ LECTURE 6. ETALE SHEAVES WITH TRANSFERS 54 Proof. Let Z ⊂ X × Y be an elementary correspondence and let Z be the normalization of Z in a normal field extension L of F = k(X) containing K = k(Z ).

Z is a preimage of the pair. Now whenever F is a sheaf and X is smooth, each presheaf U → F (U ×X) is also a sheaf for the Zariski topology. In particular each Cn F is a sheaf and C∗ F is a complex of sheaves. Thus C∗ Ztr (Y ) is a complex of Zariski sheaves. 26 above) that the complex C∗ Ztr (Y ) is not exact. There we showed that the last map may not be surjective, because its cokernel H0 C∗ Ztr (Y )(S) = Cor(S, Y )/A1 -homotopy can be non-zero. 3 below. Recall that the (small) Zariski site Xzar over a scheme X is the category of open subschemes of X, equipped with the Zariski topology.