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We conclude with an application of these ideas to homological algebra. 19. The abelian category She´t (Cork ) has enough injectives. Proof. The category S = She´t (Cork ) has products and filtered direct limits are exact, because this is separately true for presheaves with transfers and for ´etale sheaves. That is, S satisfies axioms AB5 and AB3∗ . 3, the family of sheaves Ztr (X) is a family of generators of S. 1]) that this implies that S has enough injectives. 20. Let F be an ´etale sheaf with transfers.

13 shows that it does not work in the Zariski topology, because the transfer E(X) → E(S) need not factor through the sum of the E(Ui ). 21. If F is any ´etale sheaf with transfers, then its cohomology presheaves He´nt (−, F ) are presheaves with transfers. Proof. 20 is a resolution of sheaves with transfers. Since the forgetful functor from PST(k) to presheaves is exact, and H n (−, F ) is the cohomology E ∗ (F ) as a presheaf, we see that H n (−, F ) is also the cohomology of E ∗ (F ) in the abelian category PST(k).

32 LECTURE 2. 27. 12 implies that there are adjoint functors i∗ : PST(k) → PST(F ), i∗ : PST(F ) → PST(k). Show that there is a natural transformation π : i∗ i∗ M → M whose composition πη with the adjunction map η : M → i∗ i∗ M is multiplication by [F : k] on M . Lecture 3 Motivic cohomology Using the tools developed in the last lecture, we will define motivic cohomology. It will be hypercohomology with coefficients in the special cochain complexes Z(q), called motivic complexes. 1. For every integer q ≥ 0 the motivic complex Z(q) is defined as the following complex of presheaves with transfers: Z(q) = C∗ (Ztr (G∧q m ))[−q].