J. Milne. Lectures on  Etale Cohomology. 1998. http://www.jmilne.org/math.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ae Spec(A) of the image of w so that the induced homomorphism hG : R Gamma G factors through a ring homomorphism R g Gamma C f Gamma G where: i. C = R[t 1 ; t k ] and g : R Gamma C is the natural ring homomorphism. ii. f : C Gamma G is an etale ring homomorphism. cf. [16]) Theorem 4.1 and an inductive argument show that, RC = C ; and by Theorem 3.2 C G = G : Therefore there exists an open affine covering of Spec(D) fSpec(D i )g i=1; n ; and open affine subsets of Spec(A) fSpec(A i )g i=1; n ; so that A i D i = D i for i = 1; n. ....

J.S. Milne, " ' Etale cohomology", Princeton University Press, 33 (1980).

.... on Q with rational coefficients (this is a slight abuse of notation) let h 2 CH 1 (Q) be the Chern class of the tautological invertible sheaf OQ (1) Let CH num (Q) be the quotient of CH (Q) modulo numerical equivalence; this is a finite dimensional graded Q algebra (see [3] Chapter 19, and [11], Chapter 6, Theorem 11.7) By a result of Kurano [7] the (rational) Chow group CH (Spec Rm ) of the local ring Rm is identified (up to reindexing by codimension) with the quotient graded ring CH (Q) hCH (Q) Hence there is a natural graded surjection from CH (Spec Rm ) onto CH num (Q) hCH ....

J. S. Milne, ' Etale Cohomology, Princeton Math. series No. 33, Princeton Univ. Press (1980).

....: HE=K;N (S) HE=K;N (S 0 ) and so we obtain a contravariant functor H E=K;N : Sch =K Sets; called the Hurwitz functor of genus 2 covers of E=K. Remark. Part (b) of the above Proposition 3. 3 shows that if N 3, then the functor HE=K;N is a separated presheaf in the fpqc topology (cf. [Mi1], p. 49 and or [BLR] p. 199) which is a necessary condition for the functor to be (finely) representable; cf. BLR] Proposition 8.1 1. In fact, if N is invertible in K, then we shall see later (cf. Theorem 5.18) that H E=K;N is representable by a smooth affine curve H E=K;N which is an open ....

....[N ] Theta E 0 [N ] N denote the e N pairings of E and E 0 . Proof. Viewing E[N ] as a (locally constant) etale sheaf of Z=NZ modules on S et , let 2 A E[N ] be its second exterior power with respect to A = Z=NZ (which is formed analogous to the tensor product of etale sheaves; cf. [Mi1], p. 79) thus, 2 A E[N ] is a (locally constant) etale sheaf of A modules. Since e N is an alternating pairing (cf. KM] p. 90, 505) we have an induced A homomorphism of etale sheaves e N : 2 E[N ] N because for any etale 22 sheaves M ,N of A modules we have the canonical ....

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J. Milne, ' Etale Cohomology. Princeton U. Press, Princeton, 1980. 54

....is similar. Note rst that the presheaves Cycl(X=S; r) Q are separated with respect to the h topology, i.e. the canonical morphisms of presheaves Cycl(X=S; r) Q (Cycl(X=S; r) Q ) h are monomorphisms. Therefore according to the standard construction of the sheaf associated with a presheaf (see [11], 1] it is sucient to show that for any co nial class of h coverings fU i Sg i=1; n of S the following sequence of abelian groups is exact: Cycl(X=S; r) Q (S) M i Cycl(X=S; r) Q (U i ) M i;j Cycl(X=S; r) Q (U i S U j ) We may obviously replace the covering fU i Sg by the ....

....; k xn . Then there is a nite surjective morphism S 0 S such that for any i = 1; n and any point y over x i the eld extension k y =k i contains E i . Proof: Replacing S by the disjoint union of its irreducible components we may assume that S is integral. Using Zariski s main theorem [11] one can reduce the problem to the case when S is a semi local scheme and x i are the closed points of S. We may obviously assume that E i are normal extensions of k x i and since E i are nitely generated over k x i using induction on the minimal number of generators we may further assume that E ....

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J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980.

....on A and D : f Gamma1 (supp C) the preimage of the curve C identified with its reduced inverse image. Then D is an intersecting elliptic divisor on B. If C is proportional, then also D is. Proof. Let E be an elliptic curve on B. By the base change property for etal morphisms (see e.g. [Mil], I, Prop. 3.3) the restriction f Gamma1 (E) Gamma E of f is etal, too. Especially, f Gamma1 (E) is smooth, hence this preimage is a disjoint finite union of smooth irreducible curves. These curves have to be elliptic because this is the only possibility of unramified covers of ....

Milne, J.S.: ' Etale cohomology, Princeton Univ. Press, 1980

.... [ a) Gamma Gamma Gamma H n F i L=F Gamma Gamma Gamma H n L n L=F Gamma Gamma Gamma H n F : Let X be an algebraic variety defined over F . The group Z i n (X) a x2X (i) H n F (x) is called the group of H n cycles of codimension i on X. There is the complex [3] [12] (char F 6= 2) Delta Delta Delta Z i Gamma1 n 1 (X) Z i n (X) Z i 1 n Gamma1 (X) The cohomology groups of this complex in Z i n (X) we denote by H i (X; H n i ) If X = A 1 F , then H 0 (X; H n ) H n F and H i (X; H n ) 0 for i 1. In other words, ....

.... E nE=F y y nE=F k n F hn;F Gamma Gamma Gamma Gamma H n F commutes [17] If F is a finite, local or global field, then h n;F is an isomorphism [13] 18] If X is an algebraic variety over F , then for an etale sheaf M on X et by H (X; M) we denote etale cohomology groups [12]. The group H (X; Z=2Z) we shall denote by H X. The degree deg x of a closed point x 2 X is the degree of the extension F (x) F . By [1; n] we denote the set f1; 2; ng. In the section 2 we prove certain properties of the Bockstein map. The next section is devoted to the study of ....

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J. S. Milne, ' Etale Cohomology, Princeton Math. Series, 1980.

....with the short exact sequence of Z p group schemes 1 Gamma G der Gamma G c Gamma T Gamma 1 THE CONJECTURE OF LANGLANDS AND RAPOPORT 9 shows that c(K p ) T (Z p ) This is because H 1 et (Z p ; G der ) H 1 (F p ; G der ) 0, where the first equality is 4. 5 in Ch.III of [Mi. 80] and the second equality is Lang s Theorem. B) The argument used at the beginning of the proof of (1) shows that T (Q) is dense in T (R) Theta T (Q p ) Let K = K p K p where K p is open compact in G(A p f ) From A) and B) we obtain the identity T (Q)nT (A f ) T 0 (R)c(K) T (Z (p) ....

J.S. Milne, ' Etale Cohomology, Princeton Univ. Press, 1980.

.... CAlg(F ) taking a Gamma set P to the separable F algebra A(P ) def = Map Gamma (P; F sep ) with the multiplication given by the rule (fg) p) f(p) Delta g(p) There is also a functor m : CAlg(F ) Gamma Sets; A 7 Hom F Gammaalg (A; F sep ) The following statement is well known ([11]) Proposition 1.5 The functors l and m are mutually inverse equivalences of categories. 12 Consider the functor j : Alg(F ) C given on objects by the rule j(A) Spec F; A) For an F algebra homomorphism f : A B, viewing B as a left A op Omega F B module as follows: a op Omega b) ....

J.S.Milne, ' Etale cohomology, Princeton Univ. Press, Princeton, N.J., 1980.

....the sum resp. union of the translates of X P under the automorphisms of Gamma. With the reduced scheme structure X is a disjoint union of closed subvarieties resp. a closed subvariety of G j n 1 m . In both cases X is affine and n dimensional. In particular we have H n 1 (X) 0 in MM c.f. [M] VI Th. 7.2. For any object H with a Gamma action in a Q linear abelian category we write H( eH for the isotypical component of H where e 2 Q[ Gamma] is the idempotent corresponding to : e = j Gammaj Gamma1 X fl2 Gamma (fl) Delta fl Gamma1 : Then we have: H n (G j n 1 m ) ....

J.S. Milne, ' Etale cohomology, Princeton 1980

....mod l cohomology. Proof. Both pro spaces in 14 being connected, the isomorphism on the zero dimensional cohomology is trivial. Also, recall that for any noetherian ring R we have a natural isomorphism ( 7] p. 49) H ( R) et ) H et (Spec(R) F l ) and the following short exact sequence ([12] 4.11) 1 R Theta = R Theta ) l H 1 et (Spec(R) F l ) l torsion of P ic(R) 0 where F l is the constant sheaf and H et is the etale cohomology (see [12] p. 84) Because in our case, P ic(R) does not have l torsion for R = A or A p with p any prime ideal of A, by the ....

....have a natural isomorphism ( 7] p. 49) H ( R) et ) H et (Spec(R) F l ) and the following short exact sequence ( 12] 4. 11) 1 R Theta = R Theta ) l H 1 et (Spec(R) F l ) l torsion of P ic(R) 0 where F l is the constant sheaf and H et is the etale cohomology (see [12], p. 84) Because in our case, P ic(R) does not have l torsion for R = A or A p with p any prime ideal of A, by the assumptions made about A (see also [13] p. 74) it follows that H 1 ( A) et ) A Theta = A Theta ) l F r2 1 l (16) 10 MARIAN FLORIN ANTON and H 1 ( r i=1 ....

Milne, J.S.: Etale Cohomology, Princeton Univ. Press, Princeton, 1980

....) et (f 2 ) et : A p1 ) et ( A p2 ) et (A) et given by 2.6 induces an isomorphism on mod 3 cohomology. Proof. For any noetherian ring B we have a natural isomorphism H ( B) et ) H et (Spec(B) k) 9] p. 49) where H et is the etale cohomology (defined as in [16], p. 84) and k is the constant sheaf. Because H q et (Spec(B) k) 0 if B = A or A p and q 2 [15] we can use the following short exact sequence ( 16] 4.11) 1 B Theta = B Theta ) 3 H 1 et (Spec(B) k) 3 torsion of P ic(B) 0 5 to conclude the proof. Indeed, P ic(B) ....

....a natural isomorphism H ( B) et ) H et (Spec(B) k) 9] p. 49) where H et is the etale cohomology (defined as in [16] p. 84) and k is the constant sheaf. Because H q et (Spec(B) k) 0 if B = A or A p and q 2 [15] we can use the following short exact sequence ([16], 4.11) 1 B Theta = B Theta ) 3 H 1 et (Spec(B) k) 3 torsion of P ic(B) 0 5 to conclude the proof. Indeed, P ic(B) 0 if B = A or A p (being principal ideal domains) and B Theta = B Theta ) 3 is generated by i and 1 Gamma i if B = A, and by 3 mod 7 if B = A p ....

Milne, J.S.: Etale Cohomology, Princeton Univ. Press, Princeton, 1980

....Any relative linear bundle (L; OE) is locally trivial in the Zariski topology and the automorphism group of the trivial relative bundle (O U ; Id : O U ) jY = O Y U ) is canonically isomorphic to Gamma(U; GX;Y ) This implies the first formula. The second follows from the Hilbert theorem 90 ([15][ch.3] and the five lemma. Corollary 2.2 Assume that n is invertible on X, then P ic(X; Y ) nP ic(X; Y ) ae H 2 et (X; j ( n ) here j is the extension by zero functor see [15] Proof: This follows from the lemma 2.1 in view of the exact sequence of etale sheaves: 0 Gamma j ( n ....

....to Gamma(U; GX;Y ) This implies the first formula. The second follows from the Hilbert theorem 90 ( 15] ch.3] and the five lemma. Corollary 2. 2 Assume that n is invertible on X, then P ic(X; Y ) nP ic(X; Y ) ae H 2 et (X; j ( n ) here j is the extension by zero functor see [15]) Proof: This follows from the lemma 2.1 in view of the exact sequence of etale sheaves: 0 Gamma j ( n ) Gamma GX;Y n Gamma GX;Y Gamma 0: Assume that X is integral and denote by K the field of rational functions on X. The relative Cartier divisor on X is a Cartier divisor D such, ....

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J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980.

....a condition almost always fulfilled in practice. The construction of etale cohomology is quite involved. The original work of Grothendieck and his collaborators [D4, G4, G4a, G5] is still the only place where a detailed treatment of its technical properties can be found. The monographs [FK] and [Mi] are easier to approach, but deal mainly with the case of smooth varieties. We will make a thumbnail sketch of how the adic etale cohomology groups H i et (X; Q ) that we shall use are constructed. The analogy between the etale topos and the category of sheaves on the space underlying a ....

J. S. Milne, ' Etale cohomology, Princeton Univ. Press, 1980.

.... length(C ffl ) such that (D ffl ; fL 0 i g i ; f 0 i g i ) D ffl 2 ; fL 00 i g i ; f 00 i g i ) Gamma1) n Gamma1 Delta [F 0 3 ] 2:32) and (D ffl 1 ; fL 0 i g i ; f 0 i g i ) D ffl 2 ; fL 00 i g i ; f 00 i g i ) Gamma1) n Gamma1 Delta [F 0 3 M F 2 ]: 2:33) where F 0 3 = D n Gamma1 is projective if length(C ffl ) 1, and F 0 3 = F 3 if length(C ffl ) 1. Here we may use Hypothesis 2.13 to assert that (D ffl 2 ; fL 00 i g i ; f 00 i g i ) has the same value on the right sides of (2.32) and (2.33) It follows easily from ....

....Proposition 2.15 shows part (a) of Theorem 2.3. We now show parts (b) and (c) of Theorem 2.3 by induction on length(C ffl ) n Gamma m 1. If length(C ffl ) 0 then C ffl is the zero complex and (C ffl ; fL i g i ; f i g i ) 0 by (2.19) so we are done. From diagram (2. 3) we have [M n ] = HomZ (L n ; Z) F n ] Gamma [H n (C ffl ) codiv ] in G 0 (ZG) 2:36) Suppose length(C ffl ) 1. By Corollary 2.8, C i = 0 if i 6= n, and H n (C ffl ) C n 6= 0. Furthermore, D n Gamma1 = M n is a projective Z[G] module, and (C ffl ; fL i g i ; f i g i ) ....

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Milne, J.: ' Etale Cohomology, Princeton Univ. Press, Princeton, N. J., 1980.

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J. Milne. Lectures on  Etale Cohomology. 1998. http://www.jmilne.org/math.

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J.S. Milne, " Etale Cohomology", Princeton Mathematical Series 33, Princeton, New Jersey: Princeton University Press. XIII, 1980.

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J. S. Milne, " Etale Cohomology", Princeton Mathematical Series 33, Princeton University Press, Princeton (1980).

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J. Milne, Etale cohomology, 33, Princeton Math., 1980.

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J. Milne, Etale cohomology, 33, Princeton Math., 1980.

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J. Milne,  Etale cohomology, Princeton Univ. Press, Princeton, N.J., 1980.

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J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980.

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J. S. Milne, ' Etale cohomology, Princeton Univ. Press 1980

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J. Milne, ' Etale Cohomology, Princeton Univ. Press, 1980.

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J.S.Milne, ' Etale cohomology, Princeton Univ. Press, Princeton, N.J., 1980.

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J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980. 55

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