Changes for Etale Cohomology Theory by Lei FU

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1 Changes for Etale Cohomology Theory by Lei FU Page v, line 4: Change During the preparation of this book to I would like to thank Jiangxue Fang, Enlin Yang, Takeshi Saito and Hao Zhang for pointing out errors, misprints, and improvement of an earlier edition of this book. During the preparation of the book,. Page 1, line 6: Change pf to of. Page 25, line 2: Change h : S S S S to h : S S S S. Page 33, line 3: Change from to by. Page 49, lines 6 to 5: Delete We leave the proof of the following proposition to the reader. Page 78, line 3: Replace free by free of finite rank. Page 81, line 1: Replace A[t 1,..., t n ] by I. Page 86, line 7 to page 87, line 10: Replace the text by the following one: is surjective. So there exists a B-homomorphism θ : B[t 1,..., t n ]/J B[t 1,..., t n ]/J 2 which is a section of the projection p : B[t 1,..., t n ]/J 2 B[t 1,..., t n ]/J. Let φ : B[t 1,..., t n ] B[t 1,..., t n ]/J 2 be the canonical homomorphism and let φ = θpφ. We have pφ = pφ. By the proof of Lemma 2.6.1, φ φ : B[t 1,..., t n ] J/J 2 is a B-derivation. It corresponds to a (B[t 1,..., t n ]/J)-module homomorphism ψ : Ω B[t1,...,t n]/b B[t1,...,t n] (B[t 1,..., t n ]/J) J/J 2. 1

2 We claim that ψ is a left inverse of the canonical homomorphism δ : J/J 2 Ω B[t1,...,t n]/b B[t1,...,t n] (B[t 1,..., t n ]/J). Indeed, for any x J, we have ψδ(x + J 2 ) = ψ(dx 1) = (φ φ )(x) = φ(x) θpφ(x) = x + J 2 since pφ J = 0. This proves our claim. So the sequence Page 96, line 6: Insert before It follows the following: One can verify that Spec A n is the smallest open neighborhood of n in Spec A and hence O Spec A (Spec A n ) = A n. Page 135, lines 20: Delete the up-side-down question mark. Page 172, line 9: Switch F and F on the righthand side of the equation, that is, change the righthand side to Hom(F, f P F ). Page 172, line 9: Change n+1 i=1 to n+1 i=0. Page 175, line 10: Change δ 0 to δ 1. Page 175, line 9: Change morphism to U-morphism. Page 179, line 11: Delete the upside down question mark before From now on. Page 180, line 11: Replace leaf by left. Page 209, line 7: Change f! F f F to f! F f F. Page 212, line 2: Change Ȟ1 (U, F ) = 0 to Ȟi (U, F ) = 0 for all i 1. page 212, line 2: Change Ȟ1 (f U, F ) = 0 to Ȟi (U, f F ) = 0 for all i 1. Change Ȟ1 (U, f F ) = 0 to Ȟi (U, f F ) = 0 for all i 1. Page 213, line 2: Change... for all i 1. to... for all i 1, that is, F is flasque. Page 213, line 1: Delete this line. 2

3 Page 214, lines 20, 21: Delete these two lines. Page 231, line 9: Change the last 0 to 1. Page 244, line 6: Insert connected after For any. Page 245, line 18: Insert before We thus prove the following the following: The two sets G with G acting by right multiplication and with π 1 (X, γ) acting through two homomorphisms φ i : π 1 (X, γ) G (i = 1, 2) are isomorphic if and only if there exists g G such that g 1 φ 1 (σ)g = φ 2 (σ) for any σ π 1 (X, γ). Page 245, line 19-23: Proposition is not precise. Replace it by the following: Proposition Let X be a connected noetherian scheme, γ a geometric point in X, G a finite group, and cont.hom(π 1 (X, γ), G) the set of continuous homomorphisms from π 1 (X, γ) to G. Define an equivalence relation on cont.hom(π 1 (X, γ), G) so that two continuous homomorphisms φ i : π 1 (X, γ) G (i = 1, 2) are equivalent if there exists g G such that g 1 φ 1 (σ)g = φ 2 (σ) for any σ π 1 (X, γ). Then Ȟ1 (X, G X ) is isomorphic to the set of equivalent classes of cont.hom(π 1 (X, γ), G). Page 246, line 1: Insert noetherian before scheme. Page 246, lines 6-7: Delete and X has finitely many irreducible components. Page 255, line 10: Change a to the. Page 257, line 7: Change u λ to u λµ. Page 260, lines 11: Change v : X X λ to v λ : X X λ. Page 274, line 1: Change For any s S, we have T (s) S to We have s S if and only if T (s) S. Page 300, line 8: Change G to G. Page 300, line 10: Change τ n C (F ) to τ n C (G ). Page 304, line 5: Change im d n+1 X to im dn X. 3

4 Page 319, line 3: Replace the two O Xet by O Xet. Page 323, line 3: Change principle to principal. Page 323, line 2 and line 1: Change O Xet to O X et. Page 332, line 2: Change close to closed. Page 342, line 15: Change O Xn to O Yn. Page 347, line 9 to the last line of page 349 Replace the proof of Proposition by the following more simple proof: Proof. The canonical isomorphisms Hom(f F, f g g F ) = Hom(f f F, g g F ), Hom(g f f F, g F ) = Hom(f f F, g g F ) map f (adj) Hom(f F, f g g F ) and g (adj) Hom(g f f F, g F ) to the same element in Hom(f f F, g g F ) defined by the composite f f F adj F adj g g F. Let h = gf = fg. Our assertion follows from the commutativity of the following diagram: Hom(f F, g f g F ) = Hom(g f F, f g F ) = Hom(f g f F, g F ) = = Hom(f F, h g F ) = Hom(h f F, g F ) = = Hom(f F, f g g F ) = Hom(f f F, g g F ) = Hom(g f f F, g F ) Page 352, line 15: Change Y Y to Y Y. Page 370, line 5: Change q 0 to q 1. Page 370, line 2: Change point to points. Page 376, line 9: Change X-scheme to Y -scheme. 4

5 Page 376, line 4: Change S f(x) to S f( x). Page 381, line 15: Change surjective to faithfully flat. Page 382, line 7: Insert before Our assertion follows the following: If g is flat, then the morphism S s S s is surjective. Page 382, line 14: Change surjective to faithfully flat. Page 397, line 10: Change Z/n to 0. of Y. Page 408, line 3: Change the function field of Y to the perfect closure of the function field Page 408, line 7: Change and to , and Page 408, lines 7-10: Change there exists an open subset W... of f W : f 1 (W ) W to there exists an open subset V 0 of Y, a finite surjective radiciel morphism W V 0, and a compactification U W = W Y U U W g W of the base change f W : U W W of f : U Y Page 408, lines 14 to 13: Change So there exists an open subset V of W to By and 5.8.3, there exists an open subset V of V 0. Page 421, line 12: Change Y to Y. Page 422, line 16: Change f to f. Page 422, line 18: Change f to f. Page 437, line 7: Change H 0 (X, i Z/n) to H 0 (X, i Z/n). Page 437, line 10: Change H 1 (X, i Z/n) to H 1 (X, i Z/n). Page 453, line 3: Change L D + (Y, A) to L ob D + (Y, A) 5

6 Page 453, lines 4 to 3: Change noetherian ring to noetherian torsion ring. Page 454, line 1-2: Change M D (X, A) to M ob D (X, A). Page 463, line 14: Change homomorphism to morphism. Page 463, line 15: Change Hom to Hom. So this line becomes Hom (j! Z/n( d)[ 2d], K) Hom (f! k! Z/n, K). Page 463, line 4: Change S to Y. Page 466, line 11: Change i < dimx 1...i = dimx 1 to q < dimx 1...q = dimx 1. Page 472, line 9: Change S s to Ỹȳ. Page 473, line 17: Change ker (Tr f ) Ff 2 W and coker (Tr f ) Ff 2 W to ker (Tr f F 2 ) and fw coker (Tr f F 2 ). fw Page 473, line 15: Change (ker (Tr f ) F 2 f W ) O and (coker (Tr f ) F 2 f W ) O to (ker (Tr f F 2 fw )) O and (coker (Tr f F 2 )) O. fw Page 473, line 10: Change (ker (Tr f ) F 2 f W ) A and (coker (Tr f ) F 2 f W ) A to (ker (Tr f F 2 fw )) A and (coker (Tr f F 2 fw )) A. Page 475, line 9: Change Tr h 1 to Tr 1 i h i. Page 477, line 14: Change Y Y W to Y Y W. Page 486, lines 20-22: Change the four Hom in the diagram (4) to Hom. So the diagram becomes Hom (j! Z/n( N)[ 2N], h K) Hom (g! k!z/n, h K) (4) Hom (j! Z/n( N)[ 2N], K) Hom (g!k! Z/n, K), Page 490, lines 11 to 9: Delete We have Z/n(d) = Z/n L Z/n Z/n(d) = C (Z/n) Z/n I in D(X, Z/n). Page 500, line 3: Change the two S to X. 6

7 Page 500, line 5: Change S to X. Bottom of page 500 to top of page 501: Delete Let 0 I 0 I 1 be an injective resolution... So we have R q f G (f G ) = { G if q = 0, 0 if q 1. and replace it by the following: We have a biregular spectral sequence E pq 2 = H p (G, R q f f G ) R p+q f G (f G ). As R q f f G = 0 for q 1, the spectral sequence degenerates and we have H p (G, f f G ) = R p f G (f G ). Each stalk of f f G is an induced G-module. So we have H p (G, f f G ) = { (f f G ) G if p = 0, 0 if p 1. If follows that R q f G (f G ) = { G if q = 0, 0 if q 1. Page 501, the last paragraph: Change Let X be a scheme on which G acts on the right by Let X be a scheme with the trivial G-action. Delete G acts trivially on X on line 16. Delete of A-modules on line 15. Change A on lines 14, 13, 12, 9 to Z. Move this paragraph to the place on page 500 before the paragraph Let X be a scheme on which G.... Page 509, line 6: Change S to S. Pgae 510, line 5: Add on X after modules. Page 512, line 15: Change (X Y i ) iη to (X Y i ) η. Page 521, line 13: change = to. Page 521, line 5: Change i R u+v j F to i R u+v j F η. Page 524, line 13: change < n to < d. 7

8 Page 526, line 5: change H om (H om (K, A)) to H om (H om (K, A), A). Page 535, line 8: Change F to F n. Page 540, line 5: Change (ii) to 5.8.9, Page 551, line 7: Change f n : K n K n to f n : K n K n. Page 555, after line 22: Insert the following paragraph: Suppose X is a compactifiable k-scheme for some separably closed field k, and let K ob D b c(x, R). Then (H i (X, K n )) and (H i c(x, K n )) are A-R λ-adic. We define H i (X, K) = lim n H i (X, K n ), H i c(x, K) = lim n H i c(x, K n ). If F = (F n ) is a λ-adic sheaf, then (H i (X, F n )) and (H i c(x, F n )) are A-R λ-adic. We define H i (X, F ) = lim n H i (X, F n ), H i c(x, F ) = lim n H i c(x, F n ). Page 569, line 4: Change (A, F ) to (A, F A). Page 570, line 4: Insert finite before k-morphism. Page 574, line 11: Change Z l to Z l. Page 574, line 14: Change G x to G x. Page 575, line 1: Change the two j! to j!. Page 576, line 2: Change j! to j!. Page 577, line 9: After Proposition 4, insert and III 7. Page 577, lines 10, 12, 14: Replace degd K /K, by x X length(ω X /X) x. Page 577, line 11: Delete where D K /K is the different of K /K. Bottom of page 577 to top of page 578: Delete The different D K /K is an... We thus have 2 2g = (2 2g)[K : K] degd K /K., and replace it by the following paragraph: Hence χ(x, Ω X /k) = χ(x, p Ω X/k ) + χ(x, Ω X /X). 8

9 By the Riemann-Roch formula, we have χ(x, Ω X /k) = 1 g + deg(ω X /k) = g 1, χ(x, p Ω X/k ) = 1 g + deg(p Ω X/k ) = 1 g + (2g 2)[K : K]. Moreover, since Ω X /X is a sky-scrapper sheaf, we have χ(x, Ω X /X) = length(ω X /X) x. x X We thus get g 1 = 1 g + (2g 2)[K : K] + length(ω X /X) x. Our assertion follows. x X Page 580, line 2: Delete the sign up-side-down?. (I have no idea how it appears in the printed book. There is nothing like this in the latex file.) Page 582, lines 7 and 8: Change X to U. (So the equation looks like [RΓ c (U, K)] = [RΓ(U, K)].) Page 584, line 11: Replace the equation α x (K) = i ( 1) i dim Hom G (Q l [G] Zl [G] Sw x, H i (K) η ) by α x (K) = i ( 1) i Sw x (H i (K) η ) Page 584, line 9: Before Then we have, insert See [Laumon(1987)] and for the definition of the Swan conductor Sw x (H i (K) η ) at x of the Q l -representation H i (K) η of Gal( η/η). Page 585, line 9: Change A scheme to A scheme X. Page 591, line 9: Change to Page 603, line 1: Change deg to det. 9