Lefschetz theorem for abelian fundamental group with modulus
aa r X i v : . [ m a t h . AG ] D ec LEFSCHETZ THEOREM FOR ABELIAN FUNDAMENTALGROUP WITH MODULUS
MORITZ KERZ AND SHUJI SAITO
Abstract.
We prove a Lefschetz hypersurface theorem for abelian fundamen-tal groups allowing wild ramification along some divisor. In fact, we show thatisomorphism holds if the degree of the hypersurface is large relative to the rami-fication along the divisor. Statement of main results
Lefschetz hyperplane theorems represent an important technique in the study ofGrothendieck’s fundamental group π ( X ) of an algebraic varieties X (we omit basepoints for simplicity). Roughly speaking one gets an isomorphism of the form ι Y/X : π ( Y ) ∼ −→ π ( X )for a suitable hypersurface section Y → X if dim( X ) ≥
3. Purely algebraic Lefschetztheorems for projective varieties satisfying certain regularity assumptions were de-veloped in [SGA2]. The case of non-proper varieties
X, Y is more intricate becauseone needs a precise control of the ramification at the infinite locus. We show inthe present note that for the abelian quotient of the fundamental group a Lefschetzhyperplane theorem does in fact hold. Our basic technical ingredient is the higherdimensional ramification theory of Brylinski, Kato and Matsuda which is recalledin Section 2. We expect that there is a non-commutative analog of our Lefschetztheorem, which should have applications to ℓ -adic representations of fundamentalgroups, especially over finite fields as studied in [EK].To formulate our main result, let X be a normal variety over a perfect field k and let U ⊂ X be an open subset such that X \ U is the support of an effectiveCartier divisor on X . Let D be an effective Cartier divisor on X with support in X \ U . We introduce the abelian fundamental group π ab ( X, D ) as a quotient of π ab ( U ) classifying abelian ´etale coverings of U with ramification bounded by D .More precisely, for an integral curve Z ⊂ U , let Z N be the normalization of theclosure of Z in X with φ Z : Z N → X , the natural map. Let Z ∞ ⊂ Z N be the finiteset of points x such that φ Z ( x ) U . Then π ab ( X, D ) is defined as the Pontryagindual of the group fil D H ( U ) of continuous characters χ : π ab ( U ) → Q / Z such thatfor any integral curve Z ⊂ U , its restriction χ | Z : π ab ( Z ) → Q / Z satisfies thefollowing inequality of Cartier divisors on Z N : X y ∈ Z ∞ art y ( χ | Z )[ y ] ≤ φ ∗ Z D, where art y ( χ | Z ) ∈ Z ≥ is the Artin conductor of χ | Z at y ∈ Z ∞ and φ ∗ Z D is thepullback of D by the natural map φ Z : Z N → X .Such a global measure of ramification in terms of curves has been first consideredby Deligne and Laumon, see [La].Now assume that X is smooth projective over k (we fix a projective embedding)and that C = X \ U is a simple normal crossing divisor. Let Y be a smoothhypersurface section such that Y × X C is a reduced simple normal crossing divisor on Y and write deg( Y ) for the degree of Y with respect to the fixed projectiveembedding of X . Set E = Y × X D . Then one sees from the definition that the map Y ∩ U → U induces a natural map ι Y/X : π ab ( Y, E ) → π ab ( X, D ) . Our main theorem says:
Theorem 1.1.
Assume that Y is sufficiently ample with respect ( X, D ) (see Def-inition 3.1). If d := dim( X ) ≥ , ι Y/X is an isomorphism. If d = 2 , ι Y/X issurjective.
The prime-to- p part of the theorem is due to Schmidt and Spiess [SS], where p = ch( k ). Below we see that Y is sufficiently ample if deg( Y ) ≫ Corollary 1.2.
Let X be a normal proper variety over a finite field k . Then π ab ( X, D ) is finite, where π ab ( X, D ) = Ker (cid:0) π ab ( X, D ) → π ab (Spec( k )) (cid:1) . Proof.
In case X and X \ U satisfy the assumption of Theorem 1.1, the corollaryfollows from the corresponding statement for curves. The finiteness in the curvescase is a consequence of class field theory. For the general case, one can take by [dJ]an alteration f : X ′ → X such that X ′ and X ′ \ U ′ with U ′ = f − ( U ) satisfy theassumption of Theorem 1.1. Then the assertion follows from the fact that the map f ∗ : π ab ( U ′ ) → π ab ( U ) has a finite cokernel. (cid:3) Corollary 1.2 can also be deduced from [Ra, Thm. 6.2]. It has recently beengeneralized to the non-commutative setting by Deligne, see [EK].Theorem 1.1 is a central ingredient in our paper [KeS]. There we use it to constructa reciprocity isomorphism between a Chow group of zero cycles with modulus andthe abelian fundamental group with bounded ramification. In fact Theorem 1.1allows us to restrict to surfaces in this construction.2.
Review of ramification theory
First we review local ramification theory. Let K denote a henselian discretevaluation field of ch( K ) = p > O K of integers and residue field κ .Let π be a prime element of O K and m K = ( π ) ⊂ O K the maximal ideal. By theArtin-Schreier-Witt theory, we have a natural isomorphism for s ∈ Z ≥ ,(2.1) δ s : W s ( K ) / (1 − F ) W s ( K ) ∼ = −→ H ( K, Z /p s Z ) , where W s ( K ) is the ring of Witt vectors of length s and F is the Frobenius. Wehave the Brylinski-Kato filtration indexed by integers m ≥ log m W s ( K ) = { ( a s − , . . . , a , a ) ∈ W s ( K ) | p i v K ( a i ) ≥ − m } , where v K is the normalized valuation of K . In this paper we use its non-log versionintroduced by Matsuda [Ma]:fil m W s ( K ) = fil log m − W s ( K ) + V s − s ′ fil log m W s ′ ( K ) , where s ′ = min { s, ord p ( m ) } . We define ramification filtrations on H ( K ) := H ( K, Q / Z )as fil log m H ( K ) = H ( K ) { p ′ } ⊕ ∪ s ≥ δ s (fil log m W s ( K )) ( m ≥ , fil m H ( K ) = H ( K ) { p ′ } ⊕ ∪ s ≥ δ s (fil m W s ( K )) ( m ≥ , EFSCHETZ THEOREM FOR ABELIAN FUNDAMENTAL GROUP WITH MODULUS 3 where H ( K ) { p ′ } is the prime-to- p part of H ( K ). We note that this filtration isshifted by one from Matsuda’s filtration [Ma, Def.3.1.1]. We also let fil H ( K ) bethe subgroup of all unramified characters. Definition 2.1.
For χ ∈ H ( K ) we denote the minimal m with χ ∈ fil m H ( K ) byart K ( χ ) and call it the Artin conductor of χ .We have the following fact (cf. [Ka] and [Ma]). Lemma 2.2. (1) fil H ( K ) is the subgroup of tamely ramified characters. (2) fil m H ( K ) ⊂ fil log m H ( K ) ⊂ fil m +1 H ( K ) . (3) fil m H ( K ) = fil log m − H ( K ) if ( m, p ) = 1 . The structure of graded quotients:gr m H ( K ) = fil m H ( K ) / fil m − H ( K ) ( m > K be the absolute K¨ahler differential module and putfil m Ω K = m − mK ⊗ O K Ω O K . We have an isomorphism(2.2) gr m Ω K = fil m Ω K / fil m − Ω K ≃ m − mK Ω O K ⊗ O K κ. We have the maps F s d : W s ( K ) → Ω K ; ( a s − , . . . , a , a ) → s − X i =0 a p i − i da i . and one can check F s d (fil n W s ( K )) ⊂ fil n Ω K . Theorem 2.3. ( [Ma] ) The maps F s d factor through δ s and induce a natural map fil n H ( K ) → fil n Ω K which induces for m > an injective map (called the refined Artin conductor for K ) (2.3) art K : gr n H ( K ) ֒ → gr n Ω K . Next we review global ramification theory. Let
X, C be as in the introductionand fix a Cartier divisor D with | D | ⊂ C . We recall the definition of π ab ( X, D ). Wewrite H ( U ) for the ´etale cohomology group H ( U, Q / Z ) which is identified withthe group of continuous characters π ab ( U ) → Q / Z . Definition 2.4.
We define fil D H ( U ) to be the subgroup of χ ∈ H ( U ) satisfyingthe condition: for all integral curves Z ⊂ X not contained in C , its restriction χ | Z : π ab ( Z ) → Q / Z satisfies the following inequality of Cartier divisors on Z N : X y ∈ Z ∞ art y ( χ | Z )[ y ] ≤ φ ∗ Z D, where art y ( χ | Z ) ∈ Z ≥ is the Artin conductor of χ | Z at y ∈ Z ∞ and φ ∗ Z D is thepullback of D by the natural map φ Z : Z N → X . Define(2.4) π ab ( X, D ) = Hom(fil D H ( U ) , Q / Z ) , endowed with the usual pro-finite topology of the dual. MORITZ KERZ AND SHUJI SAITO
For the rest of this section we assume that X is smooth and C is simple normalcrossing. Let I be the set of generic points of C and let C λ = { λ } for λ ∈ I . Write(2.5) D = X λ ∈ I m λ C λ . For λ ∈ I , let K λ be the henselization of K = k ( X ) at λ . Note that K λ is a henseliandiscrete valuation field with residue field k ( C λ ). Proposition 2.5.
We have fil D H ( U ) = Ker (cid:0) H ( U ) → M λ ∈ I H ( K λ ) / fil m λ H ( K λ ) (cid:1) . Proof.
This is a consequence of ramification theory developed in [Ka] and [Ma]. See[KeS, Cor.2.7] for a proof. (cid:3)
Proposition 2.6.
Fix λ ∈ I such that m λ > in (2.5) . The refined Artin conductor art K λ (cf. Theorem 2.3) induces a natural injective map art C λ : fil D H ( U ) / fil D − C λ H ( U ) ֒ → H ( C λ , Ω X ( D ) ⊗ O X O C λ ) which is compatible with pullback along maps f : X ′ → X of smooth varieties withthe property that f − ( C ) is a reduced simple normal crossing divisor.Proof. This follows from the integrality result [Ma, 4.2.2] of the refined Artin con-ductor. (cid:3)
Proposition 2.6 motivates us to introduce the following log-variant of fil D H ( U ). Definition 2.7.
We define fil log D H ( U ) asfil log D H ( U ) = Ker (cid:0) H ( U ) → M λ ∈ I H ( K λ ) / fil log m λ H ( K λ ) (cid:1) . Lemma 2.8. (1) fil C H ( U ) is the subgroup of tamely ramified characters. (2) fil D H ( U ) ⊂ fil log D H ( U ) ⊂ fil D + C H ( U ) . (3) fil D H ( U ) = fil log D − C H ( U ) if ( m λ , p ) = 1 for all λ ∈ I .Proof. This is a direct consequence of Lemma 2.2. (cid:3) Proof of the main theorem
Let X be a smooth projective variety over a perfect field of characteristic p > C ⊂ X a reduced simple normal crossing divisor on X . Let j : U = X \ C ⊂ X be the open immersion. We use the same notation as in the last part of the previoussection. Take an effective Cartier divisor D = X λ ∈ I m λ C λ with m λ ≥ . Let I ′ = { λ ∈ I | p | m λ } and put D ′ = X λ ∈ I ′ ( m λ + 1) C λ + X λ ∈ I \ I ′ m λ C λ . Let Y be a smooth hypersurface section such that Y × X C is a reduced simplenormal crossing divisor on Y . Definition 3.1.
EFSCHETZ THEOREM FOR ABELIAN FUNDAMENTAL GROUP WITH MODULUS 5 (1) Assuming dim( X ) ≥
3, we say that Y is sufficiently ample for ( X, D ) if thefollowing conditions hold:( A H i ( X, Ω dX ( − Ξ + Y )) = 0 for any effective Cartier divisor Ξ ≤ D and for i = d, d − , d − A
2) For any λ ∈ I ′ , we have H ( C λ , Ω X ( D ′ − Y ) ⊗ O C λ ) = H ( C λ , O C λ ( D ′ − Y )) = H ( C λ , O C λ ( D ′ − Y )) = 0 . (2) Assuming dim( X ) = 2, we say that Y is sufficiently ample for ( X, D ) if thefollowing condition holds:( B ) H i ( X, Ω dX ( − Ξ + Y )) = 0 for any effective Cartier divisor Ξ ≤ D and for i = 1 , N such that any smooth Y of degree ≥ N issufficiently ample for ( X, D ).Theorem 1.1 is a direct consequence of the following.
Theorem 3.2.
Let Y be sufficiently ample for ( X, D ) . Write E = Y × X D . (1) Assuming d := dim( X ) ≥ , we have isomorphisms fil D H ( U ) ∼ = −→ fil E H ( U ∩ Y ) and fil log D H ( U ) ∼ = −→ fil log E H ( U ∩ Y ) . (2) Assuming d = 2 , we have injections fil D H ( U ) ֒ → fil E H ( U ∩ Y ) and fil log D H ( U ) ֒ → fil log E H ( U ∩ Y ) . For an abelian group M , we let M { p ′ } denote the prime-to- p torsion part of M . Lemma 3.3. (1)
Assuming d := dim( X ) ≥ , we have an isomorphism fil D H ( U ) { p ′ } ∼ = −→ fil E H ( U ∩ Y ) { p ′ } and the same isomorphism for fil log D . (2) Assuming d = 2 , we have an injection fil D H ( U ) { p ′ } ֒ → fil E H ( U ∩ Y ) { p ′ } and the same injection for fil log D .Proof. Notingfil D H ( U ) { p ′ } = fil C H ( U ) { p ′ } = fil log C H ( U ) { p ′ } = fil log D H ( U ) { p ′ } , this follows from the tame case of Theorem 1.1 due to [SS]. (cid:3) By the above lemma, Theorem 3.2 is reduced to the following.
Theorem 3.4.
Let the assumption be as in Theorem 3.2. Take an integer n > . (1) Assuming d := dim( X ) ≥ , we have isomorphisms fil D H ( U )[ p n ] ∼ = −→ fil E H ( U ∩ Y )[ p n ] and the same isomorphism for fil log D . (2) Assuming d = 2 , we have an injection fil D H ( U )[ p n ] ֒ → fil E H ( U ∩ Y )[ p n ] and the same injection for fil log D . MORITZ KERZ AND SHUJI SAITO
In what follows we consider an effective Cartier divisor with Z [1 /p ]-coefficient: D = X λ ∈ I m λ C λ , m λ ∈ Z [1 /p ] ≥ . We put [ D ] = X λ ∈ I [ m λ ] C λ with [ m λ ] = max { i ∈ Z | i ≤ m λ } and F ( ± D ) = F ⊗ O X O X ( ± [ D ]) for an O X -module. For D as above, let fil log D W n O X be the subsheaf of j ∗ W n O U of local sections a ∈ W n O U such that a ∈ fil log m λ W n ( K λ ) for any λ ∈ I, where fil log m λ W n ( K λ ) := fil log[ m λ ] W n ( K λ ) is defined in § K λ of K = k ( X ) at λ . We note O X ( D ) = fil log D W n O X for n = 1 . The following facts are easily checked: • The Frobenius F induces F : fil log D/p W n O X → fil log D W n O X . • The Verschiebung V induces V : fil log D W n − O X → fil log D W n O X . • The restriction R induces R : fil log D W n O X → fil log D/p W n − O X . • The following sequence is exact:(3.1) 0 → O X ( D ) V n − −→ fil log D W n O X R −→ fil log D/p W n − O X → . We define an object ( Z /p n Z ) X | D of the derived category D b ( X ) of bounded com-plexes of ´etale sheaves on X :( Z /p n Z ) X | D = Cone (cid:0) fil log D/p W n O X − F −→ fil log D W n O X (cid:1) [ − . We have a distinguished triangle in D b ( X ):(3.2) ( Z /p n Z ) X | D → fil log D/p W n O X − F −→ fil log D W n O X + −→ . Lemma 3.5.
There is a distinguished triangle ( Z /p Z ) X | D → ( Z /p n Z ) X | D → ( Z /p n − Z ) X | D/p + −→ . Proof.
The lemma follows from the commutative diagram0 / / O X ( D/p ) V n − / / − F (cid:15) (cid:15) fil log D/p W n O X R / / − F (cid:15) (cid:15) fil log D/p W n − O X − F (cid:15) (cid:15) / / / / O X ( D ) V n − / / fil log D W n O X R / / fil log D/p W n − O X / / (cid:3) Lemma 3.6.
There is a canonical isomorphism fil log D H ( U )[ p n ] ≃ H ( X, ( Z /p n Z ) X | D ) . Proof.
Noting that the restriction of ( Z /p n Z ) X | D to U is Z /p n Z on U , we have thelocalization exact sequence(3.3) H ( X, ( Z /p n Z ) X | D ) → H ( U, Z /p n Z ) → H C ( X, ( Z /p n Z ) X | D ) . For the generic point λ of C λ , (3.2) gives us an exact sequence H λ ( X, fil log D/p W n O X ) − F −→ H λ ( X, fil log D W n O X ) → H λ ( X, ( Z /p n Z ) X | D ) → H λ ( X, fil log D/p W n O X ) . EFSCHETZ THEOREM FOR ABELIAN FUNDAMENTAL GROUP WITH MODULUS 7
By [Gr, Cor.3.10] and (3.1) we have H iλ ( X, fil log D/p W n O X ) = H iλ ( X, fil log D W n O X ) = 0 for i ≥ H λ ( X, fil log D/p W n O X ) ≃ W n ( K λ ) / fil log m λ /p W n ( K λ ) ,H λ ( X, fil log D W n O X ) ≃ W n ( K λ ) / fil log m λ W n ( K λ ) . Thus we get H λ ( X, ( Z /p n Z ) X | D ) ≃ H ( K λ )[ p n ] / fil log m λ H ( K λ )[ p n ] . Hence Lemma 3.6 follows from (3.3) and the injectivity of H C ( X, ( Z /p n Z ) X | D ) → M λ ∈ I H λ ( X, ( Z /p n Z ) X | D ) . This injectivity is a consequence of
Claim 3.7.
For x ∈ C with dim( O X,x ) ≥ we have H x ( X, ( Z /p n Z ) X | D ) = 0 . By Lemma 3.5 it suffices to show Claim 3.7 in case n = 1. Triangle (3.2) gives usan exact sequence H x ( X, O X ( D )) → H x ( X, ( Z /p Z ) X | D ) → H x ( X, O X ( D/p )) − F −→ H x ( X, O X ( D )) . If dim( O X,x ) > H x ( X, O X ( D )) = 0 and H x ( X, O X ( D/p )) = 0 by [Gr, Cor.3.10],which implies H x ( X, ( Z /p Z ) X | D ) = 0 as desired.We now assume dim( O X,x ) = 2. Let ( Z /p Z ) X denote the constant sheaf Z /p Z on X and put F X | D = Coker (cid:0) O X ( D/p ) − F −→ O X ( D ) (cid:1) . Note that F X | D = 0 for D = 0. By definition we have a distinguished triangle( Z /p Z ) X → ( Z /p Z ) X | D → F X | D + −→ . By [SGA1, X, Theorem 3.1], we have H x ( X, ( Z /p Z ) X ) = 0. Hence we are reducedto showing(3.4) H x ( X, F X | D ) = 0 . Without loss of generality we can assume that D has integral coefficients. Weprove (3.4) by induction on multiplicities of D reducing to the case D = 0. Fixan irreducible component C λ of C with the multiplicity m λ ≥ D and put D ′ = D − C λ . We have a commutative diagram with exact rows and columns( Z /p Z ) X (cid:15) (cid:15) ( Z /p Z ) X (cid:15) (cid:15) / / O X ( D ′ /p ) / / − F (cid:15) (cid:15) O X ( D/p ) / / − F (cid:15) (cid:15) L F (cid:15) (cid:15) / / / / O X ( D ′ ) / / O X ( D ) / / O C λ ( D ) / / . Here O C λ ( D ) = O X ( D ) ⊗ O C λ , and L = O C λ ( D/p ) if p | m λ , and L = 0 otherwise.Thus we get short exact sequences0 → F X | D ′ → F X | D → O C λ ( D ) → p m λ , → F X | D ′ → F X | D → O C λ ( D ) / O C λ ( D/p ) p → p | m λ . MORITZ KERZ AND SHUJI SAITO
We may assume H x ( X, F X | D ′ ) = 0 by the induction hypothesis. Hence (3.4) followsfrom(3.5) H x ( C λ , O C λ ( D )) = 0 , (3.6) H x ( C λ , O C λ ( D ) / O C λ ( E ) p ) = 0 , where we put E = [ D/p ]. We may assume x ∈ C λ so that dim( O C λ ,x ) = 1 by theassumption dim( O X,x ) = 2. (3.5) is a consequence of [Gr, Cor.3.10]. In view of anexact sequence0 → O C λ ( pE ) / O C λ ( E ) p → O C λ ( D ) / O C λ ( E ) p → O C λ ( D ) / O C λ ( pE ) → , (3.6) follows from H x ( C λ , O C λ ( pE ) / O C λ ( E ) p ) = 0 and H x ( C λ , O C λ ( D ) / O C λ ( pE )) = 0 . The first assertion follows from [Gr, Cor.3.10] noting that O C λ ( pE ) / O C λ ( E ) p isa locally free O pC λ -module. The second assertion holds since O C λ ( D ) / O C λ ( pE ) issupported in a proper closed subscheme T of C λ and x is a generic point of T if x ∈ T . This completes the proof of Lemma 3.6. (cid:3) (cid:3) In view of the above results, the assertions for fil log D of Theorem 3.4(1) and (2)follows from the following. Theorem 3.8.
Let the assumption be as in Theorem 3.2. The natural map H ( X, ( Z /p n Z ) X | D ) → H ( Y, ( Z /p n Z ) Y | D ) is an isomorphism for d := dim( X ) ≥ , and it is injective for d = 2 .Proof. By Lemma 3.5 we have a commutative diagram:0 (cid:15) (cid:15) (cid:15) (cid:15) H ( X, ( Z /p Z ) X | D ) / / (cid:15) (cid:15) H ( Y, ( Z /p Z ) Y | E ) (cid:15) (cid:15) H ( X, ( Z /p n Z ) X | D ) / / (cid:15) (cid:15) H ( Y, ( Z /p n Z ) Y | D ) (cid:15) (cid:15) H ( X, ( Z /p n − Z ) X | D/p ) / / (cid:15) (cid:15) H ( Y, ( Z /p n − Z ) Y | E/p ) (cid:15) (cid:15) H ( X, ( Z /p Z ) X | D ) / / H ( Y, ( Z /p Z ) Y | E )The theorem follows by the induction on n from the following. (cid:3) Lemma 3.9.
Let the assumption be as Theorem 3.2. (1)
Assuming d ≥ , the natural map H i ( X, ( Z /p Z ) X | D ) → H i ( Y, ( Z /p Z ) Y | E ) is an isomorphism for i = 1 and injective for i = 2 . (2) Assuming d = 2 , the natural map H ( X, ( Z /p Z ) X | D ) → H ( Y, ( Z /p Z ) Y | E ) is injective. EFSCHETZ THEOREM FOR ABELIAN FUNDAMENTAL GROUP WITH MODULUS 9
Proof.
We define an object K of D b ( X ): K = Cone (cid:0) O X ( D/p − Y ) − F −→ O X ( D − Y ) (cid:1) [ − . By the commutative diagram with exact horizontal sequences:0 / / O X ( D/p − Y ) / / − F (cid:15) (cid:15) O X ( D/p ) / / − F (cid:15) (cid:15) O Y ( E/p ) / / − F (cid:15) (cid:15) / / O X ( D − Y ) / / O X ( D ) / / O Y ( E ) / / D b ( X ): K → ( Z /p Z ) X | D → ( Z /p Z ) Y | E + −→ . Hence it suffices to show H i ( X, K ) = 0 for i = 1 , d ≥ H ( X, K ) = 0in case d = 2. We have an exact sequence H ( O X ( D − Y )) → H ( X, K ) → H ( O X ( D/p − Y )) → H ( O X ( D − Y )) → H ( X, K ) → H ( O X ( D/p − Y ))By Serre duality, for a divisor Ξ on X , we have H i ( X, O X (Ξ − Y )) = H d − i ( X, Ω dX ( − Ξ + Y )) ∨ . Thus the desired assertion follows from Definition 3.1( A
1) and ( B ). (cid:3) It remains to deduce the assertions for fil D of Theorem 3.4(1) and (2) from thatfor fil log D . Let D ′ be as in the beginning of this section and E ′ = D ′ × X Y . Notingthat the multiplicities of D ′ are prime to p , we have by Lemma 2.8(3)fil D ′ H ( U ) = fil log D ′ − C H ( U ) and fil E ′ H ( U ∩ Y ) = fil log E ′ − C ∩ Y H ( U ∩ Y ) . Thus the assertions for fil log D ′ − C of Theorem 3.4 implies that for fil D ′ . Since fil D ⊂ fil D ′ , it immediately implies the injectivity offil D H ( U ) → fil E H ( U ∩ Y ) . It remains to deduce its surjectivity from that offil D ′ H ( U ) → fil E ′ H ( U ∩ Y )assuming d ≥
3. For this it suffices to show the injectivity offil D ′ H ( U ) / fil D H ( U ) → fil E ′ H ( U ∩ Y ) / fil E H ( U ∩ Y ) . By Proposition 2.6 we have a commutative diagramfil D ′ H ( U ) / fil D H ( U ) (cid:31) (cid:127) / / (cid:15) (cid:15) L λ ∈ I ′ H ( C λ , Ω X ( D ′ ) ⊗ O X O C λ ) (cid:15) (cid:15) fil E ′ H ( U ∩ Y ) / fil E H ( U ∩ Y ) (cid:31) (cid:127) / / L λ ∈ I ′ H ( C λ ∩ Y, Ω Y ( D ′ ) ⊗ O Y O C λ ∩ Y )Thus we are reduced to showing the injectivity of the right vertical map. Putting L = Ker(Ω X → i ∗ Ω Y ) where i : Y ⊂ X , the assertion follows from H ( C λ , L ( D ′ ) ⊗ O X O C λ ) = 0 . Note that we used the fact that Y and C λ intersect transversally. We have an exactsequence 0 → Ω X ( − Y ) → L → O X ( − Y ) ⊗ O Y → . From this we get an exact sequence0 → Ω X ( D ′ − Y ) ⊗ O X O C λ → L ( D ′ ) ⊗ O X O C λ → O C λ ( D ′ − Y ) ⊗ O C λ ∩ Y → . We also have an exact sequence0 → O C λ ( D ′ − Y ) → O C λ ( D ′ − Y ) → O C λ ( D ′ − Y ) ⊗ O C λ ∩ Y → . Therefore the desired assertion follows from Definition 3.1( A (cid:3) References [EK] H. Esnault, M. Kerz,
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Moritz Kerz, NWF I-Mathematik, Universit¨at Regensburg, 93040 Regensburg, Ger-many
E-mail address : [email protected] Shuji Saito, Interactive Research Center of Science, Graduate School of Scienceand Engineering, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551,Japan
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