Lecture 1

Motivation 1. material of these lectures originated from attempts to prove duality for the generalized motivic homotopy theories on general noetherian schemes

2. I used as a model the approach which was succesful in the ladic cohomology: there the theorems about cohomology theories (i.e. l-adic cohomology with coefficients in sheaves or complexes of sheaves) are obtained as corollaries of theorems about the functorial behavior of the derived categories of l-adic sheaves.

3. A great summary of fundamental theorems of this type belongs to Deligne. See ...

4. unfortunately some of the very basic items of Deligne’s summary are out of reach in the motivic setting

5. Deligne works with the derived categories of complexes of l-adic sheaves with constructible cohomology sheaves. The direct analog of these categories in the context of the motivic stable homotopy are the motivic analogs of the Spanier-Whitehead categories. For these categories we do not even known how to construct the direct image functor for a morphism X → Spec(k) unless X is proper or k has resolution of singularities

6. to have right adjoints f_{∗} and f^{!} we have to work with larger motivic stable homotopy categories (over a base) which are analogs of the derived categories of complexes with locally constructible cohomology

7. The behavior of internal Hom is unknown in these larger categories. In particular, it is not known that the motivic homotopy category over an algebraically closed field has “enough” reflexive objects (i.e. objects X such that X → DD(X) is an isomorphism) unless the field has resolution of singularities.

8. Because of thsi problem we have to reformulate Deligne’s summary without D. Since same problems occur with other properties of ⊗ and internal Hom we have to avoid them as well.

9. This has an important advantage: things get simplier.

10. Let us see what we are left with:

(a) (f_{∗}, f^{∗}, f_{!}, f^{!}), “functoriality”, adjunctions, base change

(b) for any X autoequivalence Σ : K ↦ K(1)[2], identifications f_{∗} = f_{!}, f^{!} = Σ^{d}f^{∗}

(c) triangulated structure on D(X)’s and the gluing distinguished

triangles

11. Let us show how these data can be used to prove two facts ...