By Carlo Mazza
The idea 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 steered through Grothendieck in 1964 because the underlying constitution at the back of the myriad cohomology theories in Algebraic Geometry. We now recognize that there's a triangulated thought of factors, stumbled on by means of Vladimir Voevodsky, which suffices for the improvement of a passable Motivic Cohomology concept. even though, the lifestyles of factors themselves continues to be conjectural.
The lecture notes structure is designed for the ebook to be learn via a sophisticated graduate pupil or knowledgeable in a comparable box. The lectures approximately correspond to one-hour lectures given by way of Voevodsky through the direction he gave on the Institute for complex research in Princeton in this topic in 1999-2000. moreover, a few of the unique proofs were simplified and enhanced in order that this publication can be a useful gizmo for examine mathematicians.
This ebook presents an account of the triangulated thought of reasons. Its objective is to introduce Motivic Cohomology, to advance its major homes, and at last to narrate it to different identified invariants of algebraic types and earrings corresponding to Milnor K-theory, étale cohomology, and Chow teams. The e-book is split into lectures, grouped in six components. the 1st half offers the definition of Motivic Cohomology, established upon the inspiration of presheaves with transfers. a few straight forward 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 publication is the fourth half, proposing the triangulated type of factors. ultimately, 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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Extra info for Lecture notes on motivic cohomology
Example text
By convention Ztr ((X, x)∧0 ) = Z and Ztr ((X, x)∧q ) = 0 when q < 0. 13. The presheaf Ztr ((X1 , x1 ) ∧ · · · ∧ (Xn , xn )) is a direct summand of Ztr (X1 × · · · × Xn ). In particular, it is a projective object of PST. Moreover, the following sequence of presheaves with transfers is split-exact: {xi } 0 → Z → ⊕i Ztr (Xi ) → ⊕i, j Ztr (Xi × X j ) → · · · · · · → ⊕i, j Ztr (X1 × · · · Xˆi · · · Xˆ j · · · × Xn ) → ⊕i Ztr (X × · · · Xˆi · · · × Xn ) → → Ztr (X1 × · · · × Xn ) → Ztr (X1 ∧ · · · ∧ Xn ) → 0.
11). 2. For any scheme T over k, Ztr (T ) is an e´ tale sheaf. P ROOF. Since PST(k) is an additive category, we have the required decomposition of Ztr (T )(X Y ) = HomCork (X Y, T ). 2. As U × T → X × T is flat, the pullback of cycles is well-defined and is an injection. Hence the subgroup Ztr (T )(X) = Cork (X, T ) of cycles on X × T injects into the subgroup Ztr (T )(U) = Cork (U, T ) of cycles on U × T . 1 (1) is exact at Ztr (T )(U), take ZU in Cork (U, T ) whose images in Cork (U ×X U, T ) coincide.
It will consist of three steps: (1) Construction of θ : H n,n (Spec F, Z) → KnM (F). 5. (2) Construction of λF : KnM (F) → H n,n (Spec F, Z). 6. 8. 9) ⇒ ∃ λF (3) Proof that these two maps are inverse to each other. 11). Before starting the proof of the theorem we need some additional properties of motivic cohomology and Milnor K-theory. 10 is the group of zero cycles of (A − 0) . 2. We have H p,q (Spec F, Z) = Hq−p C∗ Ztr (G∧q m )(Spec F) for all p and q. In particular we have H n,n (Spec F, Z) =H0 C∗ Ztr (G∧n m )(Spec F) 1 = coker Ztr (G∧n m )(A ) ∂0 − ∂1 ✲ Ztr (G∧n )(Spec F) .