# Advances in Discrete and Computational Geometry by Chazelle B., Goodman J.E., Pollack R. (eds.)

By Chazelle B., Goodman J.E., Pollack R. (eds.)

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18, are described by the following diagram, where the ϕ are (exact) forgetful functors and both functors aet´ are exact. 17. Uniqueness. Suppose that two e´ tale sheaves with transfers F1 and F2 satisfy the conditions of the theorem. We already know that F1 (X) = F2 (X) = Fet´ (X) for all X and we just need to check that F1 ( f ) = F2 ( f ) holds when f : X → Y is a morphism in Cork . This is given if f comes from Sm/k. Let y ∈ F1 (Y ) = F2 (Y ) = Fet´ (Y ). Choose an e´ tale covering p : U → Y so that y|U ∈ Fet´ (U) is the image of some u ∈ F(U).

28. 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. Hint: XF → X is finite. 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].

Xn } = 0 in K∗M (E), this induces a well-defined map f : M Ztr (G∧n m )(Spec F) → Kn (F). 5 the composition of f with the face operators is zero. We define θ to be the map induced on the cokernel. If x is an F-point of (A1F − 0)n then its coordinates x1 , . . , xn are nonzero elements of F. We shall write [x1 : · · · : xn ] for the class of x in H n,n (Spec F, Z). The map θ is obviously surjective since θ ([x1 : · · · : xn ]) = {x1 , . . , xn } for x1 , . . , xn in F. Now let us build the opposite map, λF .