# Notes on Motivic Cohomology by Carlo Mazza

By Carlo Mazza

The suggestion of a intent is an elusive one, like its namesake "the motif" of Cezanne's impressionist approach to portray. Its lifestyles used to be first advised by means of Grothendieck in 1964 because the underlying constitution in the back of the myriad cohomology theories in Algebraic Geometry. We now recognize that there's a triangulated thought of explanations, stumbled on by way of Vladimir Voevodsky, which suffices for the advance of a passable Motivic Cohomology concept. despite the fact that, the life of causes themselves continues to be conjectural.

The lecture notes layout is designed for the booklet to be learn through a complicated graduate scholar or a professional in a comparable box. The lectures approximately correspond to one-hour lectures given by means of Voevodsky through the direction he gave on the Institute for complicated examine in Princeton in this topic in 1999-2000. furthermore, a few of the unique proofs were simplified and stronger in order that this ebook can be a useful gizmo for study mathematicians.

This booklet offers an account of the triangulated concept of reasons. Its objective is to introduce Motivic Cohomology, to enhance its major homes, and at last to narrate it to different recognized invariants of algebraic forms and jewelry comparable to Milnor K-theory, étale cohomology, and Chow teams. The ebook is split into lectures, grouped in six elements. the 1st half offers the definition of Motivic Cohomology, dependent upon the suggestion of presheaves with transfers. a few hassle-free comparability theorems are given during this half. the idea of (étale, Nisnevich, and Zariski) sheaves with transfers is constructed in components , 3, and 6, respectively. The theoretical center of the booklet is the fourth half, providing the triangulated class of factors. eventually, the comparability with larger Chow teams is constructed partially five.

Titles during this sequence are copublished with the Clay arithmetic Institute (Cambridge, MA).

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**Example text**

14. Let U = {Ui → X} be a Zariski open covering of X. , the following sequence is exact as a complex of ´etale sheaves: 0 → Ztr (U1 ∩ . . ∩ Un ) → . . → ⊕i Ztr (Ui ) → Ztr (X) → 0. Proof. 12 to U = U1 U2 . 12. It follows that Ztr (Uˇ ) → Ztr (X) is exact for n = 2. For n > 2, the exactness follows by induction on n. 15. If U is the cover of P1 by A1 = Spec k[t] and Spec k[t−1 ], and we mod out by the basepoint t = 1, we obtain the exact sequence 0 → Ztr (Gm ) → 2Ztr (A1 , 1) → Ztr (P1 , 1) → 0.

An ´etale sheaf is locally constant if π ∗ π∗ F → F is an isomorphism. We will write Shelc ´t for the full subcategory of She´t (Sm/k) consisting of all locally constant sheaves. 8. Let F be the locally constant sheaf π ∗ M corresponding to the G-module M . If X is connected, and l is the separable closure of k in H 0 (X, OX ), show that F (X) = M H where H = Gal(ksep /l). Conclude that π∗ F is the Galois module M . Note that F (X) = M H is also defined if X is normal. 9. The functors π ∗ and π∗ induce an equivalence between the category Shelc ´t and the category of discrete modules over the profinite group Gal(ksep /k).

Similarly, we write (Am • ⊗ A • )∗ n for the chain complex associated to diag(Am • ⊗A• ). 1]) yields a quasi-isomorphism ∇ : Am ∗ ⊗A∗ → (A• ⊗A• )∗ . m+n n we have Therefore if we find a simplicial map m : diagAm • ⊗ A• → A• m+n n m which, composed with the previous one, also a map (A• ⊗ A• )∗ → A∗ gives the multiplicative structure. Unfolding the definitions again, we have: i Ani = Ztr (G∧n m )(U × ∆ ). 9: i ∧n i ∧(m+n) Ztr (G∧m )(U × ∆i ). 4]). 10 is homotopy associative. 11. For each smooth X, there are pairings: H p,q (X, Z) ⊗ H p ,q (X, Z) → H p+p ,q+q (X, Z).