By Carlo Mazza
The thought of a reason is an elusive one, like its namesake "the motif" of Cezanne's impressionist approach to portray. Its lifestyles used to be first prompt via Grothendieck in 1964 because the underlying constitution at the back of the myriad cohomology theories in Algebraic Geometry. We now understand that there's a triangulated thought of factors, came upon via Vladimir Voevodsky, which suffices for the advance of a passable Motivic Cohomology thought. notwithstanding, the life of explanations themselves continues to be conjectural.
The lecture notes structure is designed for the publication to be learn by means of a sophisticated graduate scholar or knowledgeable in a comparable box. The lectures approximately correspond to one-hour lectures given by way of Voevodsky in the course of the direction he gave on the Institute for complicated research in Princeton in this topic in 1999-2000. additionally, some of the unique proofs were simplified and enhanced in order that this e-book can be a great tool for study mathematicians.
This e-book presents an account of the triangulated thought of causes. Its goal is to introduce Motivic Cohomology, to increase its major homes, and eventually to narrate it to different recognized invariants of algebraic forms and jewelry corresponding to Milnor K-theory, étale cohomology, and Chow teams. The publication is split into lectures, grouped in six elements. the 1st half provides the definition of Motivic Cohomology, dependent upon the concept of presheaves with transfers. a few trouble-free 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 center of the booklet is the fourth half, featuring the triangulated classification of causes. ultimately, the comparability with greater Chow teams is built partly five.
Titles during this sequence are copublished with the Clay arithmetic Institute (Cambridge, MA).
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Extra resources for Notes on Motivic Cohomology
ETALE SHEAVES WITH TRANSFERS 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 is exact at Ztr (T )(U ), take ZU in Cork (U, T ) whose images in Cork (U ×X U, T ) coincide. We may assume that X and U are integral, and that the ´etale map U → X is finite; let F and L be their respective generic points. 11, because if L lies in a Galois extension L and G = Gal(L /F ), then ZL lies in CorF (L , TF )G = CorF (L, TF ).
If M is in She´t (Et/k), then M → π∗ π ∗ M is an isomorphism by Ex. 8. Thus π ∗ is faithful. By category theory, π ∗ π∗ π ∗ ∼ = π ∗ , so for F locally constant we have a natural isomorphism π ∗ π∗ F ∼ = F. 10. Let L be a Galois extension of k, and let G = Gal(L/k). Show that Ztr (L) is the locally constant ´etale sheaf corresponding to the module Z[G]. 11. Any locally constant ´etale sheaf has a unique underlying ´etale sheaf with transfers. ´ LECTURE 6. ETALE SHEAVES WITH TRANSFERS 54 Proof. Let Z ⊂ X × Y be an elementary correspondence and let Z be the normalization of Z in a normal field extension L of F = k(X) containing K = k(Z ).
Z is a preimage of the pair. Now whenever F is a sheaf and X is smooth, each presheaf U → F (U ×X) is also a sheaf for the Zariski topology. In particular each Cn F is a sheaf and C∗ F is a complex of sheaves. Thus C∗ Ztr (Y ) is a complex of Zariski sheaves. 26 above) that the complex C∗ Ztr (Y ) is not exact. There we showed that the last map may not be surjective, because its cokernel H0 C∗ Ztr (Y )(S) = Cor(S, Y )/A1 -homotopy can be non-zero. 3 below. Recall that the (small) Zariski site Xzar over a scheme X is the category of open subschemes of X, equipped with the Zariski topology.