# Lecture notes on motivic cohomology by Carlo Mazza

By Carlo Mazza

The suggestion of a reason is an elusive one, like its namesake "the motif" of Cezanne's impressionist approach to portray. Its life used to be first prompt through Grothendieck in 1964 because the underlying constitution in the back of the myriad cohomology theories in Algebraic Geometry. We now comprehend that there's a triangulated idea of factors, chanced on by means of Vladimir Voevodsky, which suffices for the improvement of a passable Motivic Cohomology idea. even though, the lifestyles of causes themselves is still conjectural.

The lecture notes layout is designed for the booklet to be learn via a sophisticated graduate scholar or a professional in a comparable box. The lectures approximately correspond to one-hour lectures given by way of Voevodsky through the path he gave on the Institute for complex examine in Princeton in this topic in 1999-2000. additionally, the various unique proofs were simplified and stronger in order that this booklet can also be a great tool for learn mathematicians.

This booklet presents an account of the triangulated thought of reasons. Its objective is to introduce Motivic Cohomology, to improve its major homes, and at last to narrate it to different recognized invariants of algebraic types and earrings similar to Milnor K-theory, étale cohomology, and Chow teams. The ebook is split into lectures, grouped in six components. the 1st half provides the definition of Motivic Cohomology, established upon the idea of presheaves with transfers. a few simple comparability theorems are given during this half. the speculation of (étale, Nisnevich, and Zariski) sheaves with transfers is built in elements , 3, and 6, respectively. The theoretical middle of the ebook is the fourth half, providing the triangulated classification of factors. eventually, the comparability with greater Chow teams is constructed partly 5.

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

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**Example text**

An ) ∈ (A1 − 0)n (E). [x] = NE/F ([x]) ˜ Since x˜ is a rational point of XE , [x] ˜ is the image under λE of its coordinates. So [x] = NE/F λE {a1 , . . , an }. 1) commutes. 1) λE✲ H n,n (Spec E, Z) NE/F NE/F ❄ KnM (F) ❄ ✲ H n,n (Spec F, Z). 11. 1) commutes. P ROOF. 3 (3) we may assume that [E : F] = l for some prime number l. Assume first that F has no extensions of degree prime to l and [E : F] = l. 3) in [BT73] states that in this case KnM (E) is generated by the symbols a = {a1 , . .

18 2. 1. Simplicial decomposition of ∆n × A1 P ROOF. 17 induce maps hi = F(1X × θi ) : Cn F(X × A1 ) → Cn+1 F(X). 11]) from i∗1 = ∂0 h0 to i∗0 = ∂n+1 hn . 13], the alternating sum sn = ∑(−1)i hi is a chain homotopy from i∗1 to i∗0 . 19. If F is a presheaf then the homology presheaves HnC∗ F : X → HnC∗ F(X) are homotopy invariant for all n. 20. 2]) The surjection F → H0C∗ F is the universal morphism from F to a homotopy invariant presheaf. 21. Set H0sing (X/k) = H0C∗ Ztr (X)(Spec k). 10). If X is projective, H0sing (X/k) ∼ = sing 1 1 CH0 (X).

17] for details. , the inclusion is an exact functor. 18), we need two preliminary results. ˇ ˇ to be the Cech complex If p : U → X is an e´ tale cover, we define Ztr (U) ··· p0 −p1 +p2 ✲ Ztr (U ×X U) p0 −p ✲1 Ztr (U) ✲ 0. 12. Let p : U → X be an e´ tale covering of a scheme X. , the following complex is exact Ztr (U) as a complex of e´ tale sheaves. ··· p0 −p1 +p2 ✲ Ztr (U ×X U) p0 −p ✲1 Ztr (U) p ✲ Ztr (X) → 0 40 ´ 6. ETALE SHEAVES WITH TRANSFERS P ROOF. As this is a complex of sheaves it suffices to verify the exactness of the sequence at every e´ tale point.