# Regularization methods for ill-posed problems by V A Morozov; Michael I Stessin By V A Morozov; Michael I Stessin

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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.