aa r X i v : . [ m a t h . AG ] A ug REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA
HAOYU HU
Abstract.
In this article, we prove a conductor formula in a geometric situation that generalizesthe Grothendieck-Ogg-Shafarevich formula. Our approach uses the ramification theory of Abbesand Saito, and relies on Tsushima’s refined characteristic class.
Contents
1. Introduction 12. Notation 33. Preliminaries on étale cohomology 44. Cohomological correspondences 85. Ramification of ℓ -adic sheaves 136. Clean ℓ -adic sheaves and characteristic cycles 167. Tsushima’s refined characteristic class 218. The conductor formula 26References 331. Introduction ℓ -adic sheaves in a geometricsituation (1.3.1) which generalizes the classical Grothendieck-Ogg-Shafarevich formula ([11] X 7.1)as well as the index formula of Saito ([18] 3.8). It uses the ramification theory developed by Abbesand Saito and it relies on a previous work of Tsushima, who proved a special case ([23] 5.9).1.2. Let k be a perfect field of characteristic p ą , f : X Ñ Y a proper flat morphism of smoothconnected k -schemes and d the dimension of X . We assume that dim Y “ and let y be a closedpoint of Y , y a geometric point localized at y , Y p y q the strict localization of Y at y and η a geometricgeneric point of Y p y q . Put W “ Y ´ t y u , V “ f ´ p W q and that Q “ f ´ p y q . We assume thatthe canonical projection f V : V Ñ W is smooth and Q is a divisor with normal crossing on X .Let D be a divisor with simple normal crossing on X containing S “ Q red such that D X V is adivisor with simple normal crossing relatively to W . We put U “ X ´ D and let j : U Ñ X be thecanonical injection. We consider the diagram U ν / / f U (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ V l / / f V (cid:15) (cid:15) X l f (cid:15) (cid:15) Q o o (cid:15) (cid:15) W / / Y y o o where ν is the canonical injection and f U “ f V ˝ ν . We fix a prime number ℓ invertible in k , and anArtinian local Z ℓ -algebra Λ . Let F be a locally constant and constructible sheaf of free Λ -moduleson U such that(i) F is tamely ramified along the divisor D X V relatively to V ;(ii) the conductor R of F is effective with support contained in S ([5] 8.10) and F is isoclinicand clean along D ([5] 8.22 and 8.23).Condition (i) implies that f V is universally locally acyclic relatively to ν ! p F q ([6] Appendiceto Th. Finitude, [19] 3.14). Since f V is proper, all cohomology groups of R f U ! p F q are locallyconstant and constructible on W . We put ([6] Rapport 4.4) rk Λ p RΓ c p U η , F | U η qq “ Tr p id; RΓ c p U η , F | U η qq , sw y p RΓ c p U η , F | U η qq “ ÿ q P Z p´ q q sw y p R q Γ c p U η , F | U η qq , dimtot y p RΓ c p U η , F | U η qq “ rk Λ p RΓ c p U η , F | U η qq ` sw y p RΓ c p U η , F | U η qq , where sw y p R q Γ c p U η , F | U η qq denotes the Swan conductor of R q Γ c p U η , F | U η q at y .We denote by T ˚ X p log D q “ V p Ω X { k p log D q _ q the logarithmic cotangent bundle over X and by σ : X Ñ T ˚ X p log D q zero section. Under theconditions (i) and (ii), Abbes and Saito defined the characteristic cycles of F , denoted by CC p F q ,as a d -cycle on T ˚ X p log D q ([5] 1.12; [18] 3.6; cf. 6.15). The vertical part CC ˚ p F q of CC p F q isa d -cycle on T ˚ X p log D q ˆ X S such that CC p F q “ p´ q d p rk Λ p F qr σ p X qs ` CC ˚ p F qq . Theorem 1.3.
We keep the notation and assumptions of and assume moreover that S “ D (i.e., U “ V ) or that rk Λ p F q “ . Then, for any section s : X Ñ T ˚ X p log D q , we have thefollowing equality in Λ (1.3.1) dimtot y p RΓ c p U η , F | U η qq ´ rk Λ p F q ¨ dimtot y p RΓ c p U η , Λ qq “ p´ q d ` deg p CC ˚ p F q X r s p X qsq . The case where rk Λ p F q “ is due to Tsushima ([23] 5.9). Although we follow the same linesfor sheaves of higher ranks, the situation is technically more involved. Our approach requires theassumption that S “ D .1.4. To prove 1.3, we follow the strategy of Saito for the proof of an index formula for ℓ -adicsheaves on proper smooth varieties [18]. The latter can be schematically divided into two steps.The first step uses the theory of cohomological correspondences due to Grothendieck and Verdierto associate a cohomology class to the ℓ -adic sheaf, called the characteristic class , that computesits Euler-Poincaré characteristic by the Lefschetz-Verdier formula ([11] III). The second step ismore geometric. It consists of computing the characteristic class as an intersection product usingthe ramification theory developed by Abbes and Saito [2].1.5. The analogous approach for the proof of the conductor formula (1.3.1) was started byTsushima in [23]. He refined the characteristic class of an ℓ -adic sheaf into a cohomology classwith support in the wild locus, called in this article the refined characteristic class . He proveda Lefschetz-Verdier formula for this class ([23] 5.4) which amounts to say that it commutes withproper push-forward. On a smooth curve, the refined characteristic class gives the Swan conductor([23] 4.1). The main goal of this article is to prove an intersection formula that computes therefined characteristic class. EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 3 C S p j ! p F qq of j ! p F q is defined as an element in H S p X, K X q . The Lefschetz-Verdier formula impliesthe following relation sw y p RΓ c p U η , F | U η qq ´ rk Λ p F q ¨ sw y p RΓ c p U η , Λ qq “ ´ f ˚ p C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qqq in H t y u p Y, K Y q „ ÝÑ Λ , where f ˚ in the left hand side is the proper push-forward H S p X K X q Ñ H t y u p Y, K Y q (cf. 7.12). Assume that D “ S or that rk Λ p F q “ . Then, our main result is thefollowing formula (8.2) C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qq“ p´ q d rk Λ p F q ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log D q ¯ XS X r X s P H S p X, K X q , where c d p´q XS is a bivariant class built of localized Chern classes (cf. 2.4). The right hand side isthe image of a zero cycle class in CH p S q , whose degree is p´ q d deg p CC ˚ p F q X r s p X qsq (cf. 8.24),which implies theorem 1.3.1.7. Beyond Tsushima’s work already mentioned, there have been several works on the conductorformula. Abbes gave a conductor formula for an ℓ -adic sheaf on an arithmetic surface, under thecondition that the sheaf has no fierce ramification [1]. Vidal proved that the alternating sum of theSwan conductor of the cohomology groups with compact support of an ℓ -adic sheaf on a normalscheme over a local field only depends on its rank and its wild ramification [24]. For an ℓ -adic sheafon a smooth scheme over a local field of mixed characteristic, Kato and Saito defined its Swanclass, which is a -cycle class supported on the wild locus, that computes the Swan conductorof the cohomology groups with compact support [15]. In a recent work [20], Saito defined thecharacteristic cycle of an ℓ -adic sheaf on a smooth surface as a cycle on the cotangent bundlewithout the cleanliness condition. When the surface is fibered over a smooth curve, he proved aconductor formula conjectured by Deligne ([20] 3.16).1.8. This article is organized as follows. After preliminaries on étale cohomology, we brieflyintroduce the cohomological correspondences and the characteristic class of an ℓ -adic sheaf in § .We recall Abbes and Saito’s ramification theory in § and review the definition of clean sheavesand the characteristic cycle in § . We give the definition of Tsushima’s refined characteristic classand introduce the corresponding Lefschetz-Verdier formula in § . The last section is devoted tothe proof of the conductor formula. Acknowledgement.
This article is a part of the author’s thesis at Université Paris-Sud andNankai University. The author would like to express his deepest gratitude to his supervisorsAhmed Abbes and Lei Fu for leading him to this area and for patiently guiding him in solvingthis problem. The author would also like to thank professor Takeshi Saito for his stimulatingsuggestions toward to this article. This work is developed during a long visit to IHES supportedby Fonds Chern and Fondation Mathématiques Jacques Hadamard. The author is grateful to theseinstitutions for their support. 2.
Notation k denotes a perfect field of characteristic p ą . We fix a prime number ℓ invertible in k , an Artinian local Z ℓ -algebra Λ and a non-trivial additive character ψ : F p Ñ Λ ˆ .All k -schemes are assumed to be separated and of finite type over Spec p k q . HAOYU HU k -scheme X , we denote by D p X, Λ q the derived category of complexes of étale sheaves of Λ -modules on X and by D b ctf p X, Λ q (resp. D ´ p X, Λ q , resp. D ` p X, Λ q and resp. D bc p X, Λ q ) its fullsubcategory consisting of objects bounded of finite tor-dimension with constructible cohomologies(resp. of objects bounded above, resp. of objects bounded below and resp. of objects boundedwith constructible cohomologies). We denote by K X the complex R f ! Λ , where f : X Ñ Spec p k q is the structure map and by D X the functor R H om p´ , K X q on D b ctf p X, Λ q . For two k -schemes X and Y , and an étale sheaf of Λ -modules F (resp. G ) on X (resp. Y ), F b G denotes the sheaf pr ˚ F b pr ˚ G on X ˆ k Y .2.3. Let X be a scheme and E a sheaf of O X -modules of finite type. Following ([9] 1.7.8), wedenote by V p E q the vector bundle Spec p Sym O X p E qq over X .2.4. Let X be a k -scheme of equidimension e , Z a closed subscheme of X , E and E locally free O X -modules of rank e , f : E Ñ E an O X -linear map which is an isomorphism on X ´ Z , and E “ r E f ÝÑ E s the complex such that E is in degree . For i ą , we put ([14] 3.24) c i p E ´ E q XZ “ min p e,i ´ q ÿ j “ c j p E q X c i ´ jXZ p E q as a bivariant class, where c p E q denotes the Chern class of E ([8] 3.2) and c XZ p E q the localizedChern class of E ([8] 18.1).3. Preliminaries on étale cohomology X f / / g (cid:15) (cid:15) Y g (cid:15) (cid:15) X f / / Y be a commutative diagram of k -schemes. We have the base change maps ([10] XVII 4.1.5, XVIII3.1.13.2) g ˚ R f ˚ Ñ R f g , (3.1.2) R f ! R g ! Ñ R g ! R f ! . (3.1.3)Assume that diagram (3.1.1) is Cartesian. There exists a canonical base change isomorphism(the proper base change theorem) ([10] XVII 5.2.6)(3.1.4) g ˚ R f ! „ ÝÑ R f ! g . There exists a canonical isomorphism of functors ([10] XVIII 3.1.12.3)(3.1.5) R f R g ! „ ÝÑ R g ! R f ˚ . There exists a canonical morphism of functors ([10] XVIII 3.1.14.2)(3.1.6) g R f ! Ñ R f ! g ˚ . It is defined as the adjoint of the composed morphisms R f ! g R f ! „ ÝÑ g ˚ R f ! R f ! Ñ g ˚ , where the first arrow is induced by the inverse of the proper base change theorem and the secondarrow is induced by the adjunction map R f ! R f ! Ñ id . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 5 f : X Ñ Y be a morphism of k -schemes and F (resp. G ) an object of D b ctf p X, Λ q (resp. D b ctf p Y, Λ q ). There exists a canonical isomorphism (the projection formula) ([10] XVII 5.2.9)(3.2.1) R f ! p f ˚ G b L F q „ ÝÑ G b L R f ! F . f : X Ñ Y of k -schemes and two objects F and G of D b ctf p Y, Λ q , we have acanonical map(3.3.1) f ˚ F b L R f ! G Ñ R f ! p F b L G q , defined as follows. By the projection formula (3.2.1), we have a canonical isomorphism R f ! p f ˚ F b L R f ! G q „ ÝÑ F b L R f ! p R f ! G q . Composing with the adjunction map R f ! p R f ! G q Ñ G , we obtain a map R f ! p f ˚ F b L R f ! G q Ñ F b L G , which gives (3.3.1) by adjunction.If f is a closed immersion, (3.3.1) induces a cup product(3.3.2) H i p X, f ˚ F q ˆ H jX p Y, G q Y ÝÑ H i ` jX p Y, F b L G q . g : W Ñ X and f : X Ñ Y be closed immersions of k -schemes, and F and G objects of D ctf p Y, Λ q . Then, the following diagram is commutative(3.4.1) f ˚ F b L g ˚ R g ! p R f ! G q (cid:15) (cid:15) „ / / g ˚ pp f g q ˚ F b L R p f g q ! G q (3.3.1) / / g ˚ R g ! p R f ! p F b L G qq (cid:15) (cid:15) f ˚ F b L R f ! G (3.3.1) / / R f ! p F b L G q where the vertical maps are induced by the adjunction map g ˚ R g ! Ñ id , and the isomorphic map isinduced by the projection formula (3.2.1). Indeed, it is enough to show that the following diagramis commutative f ˚ p f ˚ F b L g ˚ R g ! p R f ! G qq (cid:15) (cid:15) „ / / p f g q ˚ pp f g q ˚ F b L R p f g q ! G q „ / / F b L p f g q ˚ R p f g q ! G (cid:15) (cid:15) f ˚ p f ˚ F b L R f ! G q „ / / F b L f ˚ p R f ! G q / / F b L G where the isomorphic maps are the projection formulae and the other maps are induced by ad-junction. Since the composition of the upper horizontal maps is the projection formula f ˚ p f ˚ F b L g ˚ R g ! p R f ! G qq „ ÝÑ F b L p f g q ˚ R p f g q ! G , we are reduced to show the following diagram is commutative f ˚ p f ˚ F b L p g ˚ R g ! p R f ! G qqq adj (cid:15) (cid:15) „ / / F b L f ˚ p g ˚ R g ! p R f ! G qq adj (cid:15) (cid:15) f ˚ p f ˚ F b L R f ! G q „ / / F b L f ˚ p R f ! G q which is obvious. HAOYU HU
Diagram (3.4.1) induces a commutative diagram(3.4.2) H i p X, f ˚ F q ˆ H jW p Y, G q Y W / / (cid:15) (cid:15) H i ` jW p Y, F b L G q (cid:15) (cid:15) H i p X, f ˚ F q ˆ H jX p Y, G q Y / / H i ` jX p Y, F b L G q where Y W is defined by the upper horizontal arrows of (3.4.1).3.5. Let f : X Ñ Y be a morphism of k -schemes, F an object of D ´ p X, Λ q and G an object of D ` p Y, Λ q . We have a canonical isomorphism ([10] XVIII 3.1.10, [7] 8.4)(3.5.1) R f ˚ R H om p F , Rf ! G q „ ÝÑ R H om p Rf ! F , G q . Taking G “ K Y , we obtain an isomorphism (2.2)(3.5.2) R f ˚ p D X p F qq „ ÝÑ D Y p R f ! F q . f : X Ñ Y of k -schemes and two objects F and G of D b ctf p Y, Λ q , we recallthe definition of the canonical isomorphism ([10] XVIII 3.1.12.2, [7] 8.4.7)(3.6.1) R H om p f ˚ F , R f ! G q „ ÝÑ R f ! R H om p F , G q . By the inverse of the projection formula (3.2.1), we have a canonical isomorphism F b L R f ! R H om p f ˚ F , Rf ! G q „ ÝÑ R f ! p f ˚ F b L R H om p f ˚ F , R f ! G qq . Composing with the canonical morphisms R f ! p f ˚ F b L R H om p f ˚ F , R f ! G qq Ñ R f ! R f ! G Ñ G , we obtain a morphism F b L R f ! R H om p f ˚ F , R f ! G q Ñ G . It induces the map (3.6.1) by adjunction. Taking G “ K Y , we obtain a canonical isomorphism(3.6.2) D X p f ˚ F q „ ÝÑ R f ! p D Y p F qq . k -schemes X and Y and objects F and G of D b ctf p X, Λ q and D b ctf p Y, Λ q , a canonicalisomorphism(3.7.1) R H om p pr ˚ G , Rpr !1 F q „ ÝÑ F b L D Y p G q is defined in ([11] III 3.1.1).Let g : X Ñ X and h : Y Ñ Y be open immersions. By (3.7.1), (3.5.2) and the Künnethformula, we have a canonical isomorphism on X ˆ k Y R H om p pr ˚ h ! G , Rpr !1 g ! F q „ ÝÑ p g ! F q b L D Y p h ! G q „ ÝÑ p g ! F q b L p R h ˚ p D Y p G qqq (3.7.2) „ ÝÑ p g ˆ q ! p R p ˆ h q ˚ p F b L D Y p G qqq „ ÝÑ p g ˆ q ! p R p ˆ h q ˚ R H om p pr ˚ G , Rpr !1 F qq . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 7 f : X Ñ Y be a flat morphism of k -schemes with fibers of equidimension d and F anobject of D b ctf p Y, Λ q . We have a canonical trace map ([10] XVIII 2.9) Tr f : R f ! f ˚ F p d qr d s Ñ F . Its adjoint(3.8.1) t f : f ˚ F p d qr d s Ñ R f ! F is called the class map ([10] XVIII 3.2.3). If f is smooth, t f (3.8.1) is an isomorphism (Poincaréduality) ([10] XVIII 3.2.5, [7] 8.5.2).3.9. Let f : X Ñ Y be a morphism of smooth k -schemes with the same equidimension d . For anyobject F of D b ctf p Y, Λ q , we recall the definition of the canonical map ([4] (1.9))(3.9.1) f ˚ F Ñ R f ! F . The map f is the composition of the graph Γ f : X Ñ X ˆ k Y of f and the projection pr : X ˆ k Y Ñ Y . Since Γ f is a section of the projection pr : X ˆ k Y Ñ X , there exists a canonicalisomorphism Λ Ñ R f ! Λ defined as the composition Λ „ ÝÑ RΓ ! f Rpr !1 Λ „ ÝÑ RΓ ! f Λ p d qr d s „ ÝÑ RΓ ! f Rpr !2 Λ „ ÝÑ R f ! Λ , where the second and the third arrows are induced by Poincaré duality. Then, the canonical map(3.3.1) induces (3.9.1).3.10. Let V be a k -scheme, Z an integral closed subscheme of V of equidimension d . The canonicalclass map Λ p d qr d s Ñ K Z induces a morphism(3.10.1) H p Z, Λ q Ñ H ´ dZ p V, K V p´ d qq . The cycle class r Z s P H ´ dZ p V, K V p´ d qq is defined as the image of P H p Z, Λ q by the map (3.10.1).We obtain a homomorphism Z d p V q Ñ H ´ d p V, K V p´ d qq , where Z d p V q denotes the free abeliangroup generated by integral closed subschemes of equidimension d of V . This map factors throughthe Chow group CH d p V q , and induces the cycle map(3.10.2) cl : CH d p V q Ñ H ´ d p V, K V p´ d qq . If V is a closed subscheme of a smooth k -scheme X of dimension d ` c , by Poincaré duality, wehave H ´ d p V, K V p´ d qq „ ÝÑ H ´ dV p X, K X p´ d qq „ ÝÑ H cV p X, Λ p c qq . Let Y be another smooth k -scheme of dimension e ` c , f : X Ñ Y a k -morphism and W aclosed subscheme of Y such that f ´ p W q is a closed subscheme of V . By ([14] 2.1.2), we have acommutative diagram(3.10.3) CH e p W q cl / / f ! (cid:15) (cid:15) H cW p Y, Λ p c qq f ˚ (cid:15) (cid:15) CH d p V q cl / / H cV p X, Λ p c qq where the map f ! denotes the refined Gysin homomorphism ([8] 6.6). HAOYU HU X be a k -scheme, Z a closed subscheme of X , V “ X ´ Z the complementary opensubscheme of Z in X , U an open subscheme of X , i : Z Ñ X , j : V Ñ X , i U : Z X U Ñ U and j U : U X V Ñ U the canonical injections, and F an object of D b ctf p X, Λ q . Assume that for anyinteger q , H q p F q| U is locally constant and constructible. Then, we have a canonical isomorphism([7] 6.5.5) p F | U q b L R j U ˚ Λ „ ÝÑ R j U ˚ j ˚ U p F | U q . Since R i ! U R j U ˚ “ (3.1.5), we have R i ! U pp F | U q b L R j U ˚ Λ q “ . In particular, for any integer q ,the canonical map H qZ ´ U p X, F b L R j ˚ p Λ V qq Ñ H qZ p X, F b L R j ˚ p Λ V qq is an isomorphism ([23] Lemma 3.1).3.12. Let X be a k -scheme, Z a closed subscheme of X , V “ X ´ Z the complementary opensubscheme of Z in X , i : Z Ñ X and j : V Ñ X the canonical injections, and F an object of D ´ p X, Λ q . Applying the functor R i ! p F b L ´q to the distinguished triangle i ˚ R i ! Λ X Ñ Λ X Ñ R j ˚ Λ V Ñ , we obtain a distinguished triangle ([23] Lemma 3.5)(3.12.1) i ˚ F b L R i ! p Λ X q a ÝÑ R i ! F b ÝÑ R i ! p F b L R j ˚ p Λ V qq Ñ , where a is the map (3.3.1). We denote the functor R i ! p´ b L R j ˚ p Λ V qq : D ´ p X, Λ q Ñ D ´ p Z, Λ q by ∆ i p´q .3.13. Let X be a k -scheme, δ : X Ñ X ˆ k X the diagonal map and S a closed subscheme of X such that its complement is dense in X . Consider the composed map(3.13.1) K X Ñ R δ ! δ ˚ K X Ñ ∆ δ p δ ˚ p K X qq , where the first arrow is the adjunction map and the second arrow is b in (3.12.1). If X is anequidimensional smooth k -scheme, the following map induced by (3.13.1) is an isomorphism ([4]5.2)(3.13.2) H S p X, K X q „ ÝÑ H S p X, ∆ δ p δ ˚ p K X qqq . Cohomological correspondences
Definition 4.1 ([11] III 3.2, [4],1.2.1) . Let X and Y be two k -schemes. A correspondence between X and Y is a k -scheme C equipped with k -morphisms c : C Ñ X and c : C Ñ Y . Let F and G be objects of D b ctf p X, Λ q and D b ctf p Y, Λ q , respectively. A cohomological correspondence is amorphism u : c ˚ G Ñ R c !1 F from G to F on C .We switch the factors compared to ([11] III 3.2).4.2. Let X and Y be two k -schemes and p C, c : C Ñ X, c : C Ñ Y q a correspondence between X and Y . We denote by c the map p c , c q : C Ñ X ˆ k Y and by pr : X ˆ k Y Ñ X and pr : X ˆ k Y Ñ Y the canonical projections. Let F and G be objects of D b ctf p X, Λ q and D b ctf p Y, Λ q ,respectively. We have a canonical isomorphism (3.6.1) R H om p c ˚ G , R c !1 F q „ ÝÑ R c ! R H om p pr ˚ G , Rpr !1 F q . Taking global sections on C , we get a canonical isomorphism(4.2.1) Hom p c ˚ G , R c !1 F q „ ÝÑ H p C, R c ! R H om p pr ˚ G , R pr !1 F qq , which shows that cohomological correspondences u : c ˚ G Ñ R c !1 F are in one to one corre-spond with morphisms Λ C Ñ R c ! R H om p pr ˚ G , Rpr !1 F q , and hence with morphisms R c ! p Λ C q Ñ R H om p pr ˚ G , Rpr !1 F q by adjunction. EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 9 If c : C Ñ X ˆ k Y is a closed immersion, and X and Y are smooth k -schemes of dimension d ,we have H p C, R c ! R H om p pr ˚ G , Rpr !1 F qq „ ÝÑ H dC p X ˆ k Y, R H om p pr ˚ G , pr ˚ F qp d qq . If we further assume that F and G are sheaves of free Λ -modules and that G is locally constantand constructible. Then the canonical map c ˚ R H om p pr ˚ G , pr ˚ F q Ñ R H om p c ˚ G , c ˚ F q is anisomorphism, and we have Hom p c ˚ G , c ˚ F q „ ÝÑ H p C, c ˚ R H om p pr ˚ G , pr ˚ F qq . Then, the cycle class map CH d p C q Ñ H dC p X ˆ k Y, Λ p d qq induces a pairing CH d p C q b Hom p c ˚ G , c ˚ F q Ñ H dC p X ˆ k Y, Λ p d qq b H p C, c ˚ R H om p pr ˚ G , pr ˚ F qq Y ÝÑ H C p X ˆ k Y, R H om p pr ˚ G , Rpr !1 F qq “ Hom p c ˚ G , R c !1 F q . In this case, for a cycle class Γ P CH d p C q and a homomorphism γ : c ˚ G Ñ c ˚ F , the pair p Γ , γ q induces a cohomological correspondence u p Γ , γ q from F to G on C .4.3. We consider a commutative diagram of k -schemes(4.3.1) X f (cid:15) (cid:15) C c / / c o o h (cid:15) (cid:15) Y g (cid:15) (cid:15) X C c / / c o o Y and let F and G be objects of D b ctf p X, Λ q and D b ctf p Y, Λ q , respectively. By (3.5.2), (3.7.1) and theKünneth formula, we have a canonical isomorphism(4.3.2) R p f ˆ g q ˚ R H om p pr ˚ G , Rpr !1 F q „ ÝÑ R H om p pr ˚ R g ! G , Rpr !1 R f ˚ F q . Diagram (4.3.1) gives a commutative diagram C h (cid:15) (cid:15) c / / X ˆ k Y f ˆ g (cid:15) (cid:15) C c / / X ˆ k Y We assume that f , g and h are proper. A cohomology correspondence u : c ˚ G Ñ R c !1 F is identifiedwith a map u : Λ C Ñ R c ! R H om p pr ˚ G , Rpr !1 F q (4.2). It induces a map(4.3.3) Λ C Ñ R h ˚ R c ! R H om p pr ˚ G , Rpr !1 F q . The base change map (3.1.3) gives(4.3.4) R h ˚ R c ! “ R h ! R c ! Ñ R c ! R p f ˆ g q ! “ R c ! R p f ˆ g q ˚ . Composing (4.3.2), (4.3.3) and (4.3.4), we obtain a map Λ C Ñ R c ! R p f ˆ g q ˚ R H om p pr ˚ G , R pr !1 F q „ ÝÑ R c ! R H om p pr ˚ R g ! G , Rpr !1 R f ˚ F q . By (4.2.1), we obtain a map c R g ! G “ c R g ˚ G Ñ R c !1 R f ˚ F , which is a correspondence form R g ˚ G to R f ˚ F on C , that we denote by h ˚ p u q and call thepush-forward of u by h . The map h ˚ p u q is equal to the composition of the maps c R g ˚ G Ñ R h ˚ c ˚ G R h ˚ p u q ÝÝÝÝÝÑ R h ˚ R c !1 F Ñ R c !1 R f ˚ F , where the left and right arrows are the base change maps.4.4. We consider a commutative diagram of k -schemes(4.4.1) U j U (cid:15) (cid:15) C c / / j C (cid:15) (cid:15) c o o V j V (cid:15) (cid:15) X C c / / c o o Y where all the vertical arrows are open immersions. Let F and G be objects of D b ctf p X, Λ q and D b ctf p Y, Λ q , respectively, and u : c ˚ G Ñ R c !1 F a cohomological correspondence on C . Denoteby F U and G V the restrictions of F and G to U and V , respectively. We have R j ! C “ j ˚ C and R j ! U “ j ˚ U . Hence, the restriction u of u to C defines a cohomological correspondence(4.4.2) u : c p G V q “ j ˚ C c ˚ G j ˚ C p u q ÝÝÝÑ j ˚ C R c !1 F “ R c !1 p F U q . We denote by j the map j U ˆ j V : U ˆ k V Ñ X ˆ k Y , by c the map p c , c q : C Ñ X ˆ k Y andby c the map p c , c q : C Ñ U ˆ k V . We have a commutative diagram(4.4.3) C c / / j C (cid:15) (cid:15) U ˆ k V j (cid:15) (cid:15) C c / / X ˆ k Y The base change map (3.1.3) gives a canonical morphism(4.4.4) R c ! p Λ C q “ R c ! R j ! C p Λ C q Ñ R j ! R c ! p Λ C q “ j ˚ R c ! p Λ C q . Put H “ R H om p pr ˚ p G V q , Rpr !1 p F U qq on U ˆ k V, (4.4.5) H “ R H om p pr ˚ G , Rpr !1 F q on X ˆ k Y. (4.4.6)By (4.2), we identify a cohomological correspondence u : c ˚ G Ñ R c !1 F with a map u : Λ C Ñ R c ! H and also with the associated map u : R c ! p Λ C q Ñ H . We identify the restriction u : c p G V q Ñ R c !1 p F U q of u with a map u : Λ C Ñ R c ! p H q also with the associated map u : R c ! p Λ C q Ñ H .Since R j ! U “ j ˚ U and R j ! “ j ˚ , by (3.6.1), we have a canonical isomorphism(4.4.7) j ˚ H „ ÝÑ H . Lemma 4.5 ([4] Lemma 1.2.2) . We take the notation and assumptions of (4.4) . Then, The map u : Λ C Ñ R c ! p H q coincides with the restriction of u : Λ C Ñ R c ! H to C bythe composed isomorphism j ˚ C R c ! H “ R j ! C R c ! H Ñ R c ! R j ! H “ R c ! j ˚ H “ R c ! p H q . The following diagram is commutative (4.5.1) j ˚ R c ! p Λ C q j ˚ u / / j ˚ H (4.4.7) (cid:15) (cid:15) R c ! p Λ C q (4.4.4) O O u / / H EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 11
Lemma 4.6 ([4] Lemma 1.2.3) . Consider diagram (4.4.1) again and assume moreover that its rightsquare is Cartesian. Let F and G be objects of D b ctf p U, Λ q and D b ctf p V, Λ q , and u : c p G q Ñ R c !1 p F q a cohomological correspondence on C . Then, there exist a unique cohomological corre-spondence u : c ˚ j V ! p G q Ñ R c !1 j U ! p F q on C such that its restriction to C is u . We call u in (4.6) the extension by zero of u and denote it by j C ! u .4.7. Let X , X , Y , Y be smooth equidimensional k -schemes such that dim X “ dim X and dim Y “ dim Y , f : X Ñ X and g : Y Ñ Y morphisms of k -schemes, p C, c : C Ñ X, c : C Ñ Y q a correspondence between X and Y , and F and G objects of D b ctf p X, Λ q and D b ctf p X, Λ q ,respectively. We denote by c the map p c , c q : C Ñ X ˆ k Y . By (3.6.1) and (3.9.1), we have amap p f ˆ g q ˚ R H om p pr ˚ G , Rpr !1 F q Ñ R p f ˆ g q ! R H om p pr ˚ G , Rpr !1 F q (4.7.1) Ñ R H om p pr ˚ g ˚ G , Rpr !1 R f ! F q . Let u : c ˚ G Ñ R c !1 F be a cohomological correspondence on C , that we identify with a map u : R c ! Λ C Ñ R H om p pr ˚ G , Rpr !1 F q . We define a correspondence c “ p c , c q : C Ñ X ˆ k Y bythe Cartesian diagram C l c / / h (cid:15) (cid:15) X ˆ k Y f ˆ g (cid:15) (cid:15) C c / / X ˆ k Y By the proper base change theorem, the base change map p f ˆ g q ˚ R c ! Λ C Ñ R c ! Λ C is an isomor-phism. The composed map R c ! p Λ C q „ ÝÑ p f ˆ g q ˚ R c ! p Λ C q Ñ p f ˆ g q ˚ R H om p pr ˚ G , Rpr !1 F qÑ R H om p pr ˚ g ˚ G , Rpr !1 R f ! F q , where the first arrow is the inverse of the base change isomorphism, corresponds to a cohomologicalcorrespondence p f ˆ g q ˚ p u q : c g ˚ G Ñ R c !1 R f ! F , called the pull-back of u by f ˆ g .4.8. Let X be a k -scheme, F an object of D b ctf p X, Λ q . We denote by δ : X Ñ X ˆ k X the diagonalmap and put H “ R H om p pr ˚ F , Rpr !1 F q . The canonical isomorphism H „ ÝÑ F b L D X p F q (3.7.1) induces an isomorphism δ ˚ H „ ÝÑ F b L D X p F q . Composed with the evaluation map F b L D X p F q Ñ K X , we get a map(4.8.1) ev : δ ˚ H Ñ K X , that we also call the evaluation map.Let C be a closed subscheme of X ˆ k X and u a cohomological correspondence of F on C . Wedenote by c : C Ñ X ˆ k X the canonical injection. By (4.2), u corresponds to a section u P H p C, R c ! H q “ H C p X ˆ k X, H q . We call the image of u by the following composed maps H C p X ˆ k X, H q δ ˚ ÝÝÑ H C X X p X, δ ˚ H q ev ÝÑ H C X X p X, K X q the characteristic class of the cohomological correspondence u , and denote it by C p F , C, u q P H C X X p X, K X q ([11] III, [4] 2.1.8). If C “ δ p X q , and u : F Ñ F is an endomorphism (resp. theidentity of F ), we abbreviate the notation of the characteristic class of u by C p F , u q P H p X, K X q (resp. C p F q P H p X, K X q , and call it the characteristic class of F ).4.9. Let X be a k -scheme, U an open subscheme of X , and F an object of D b ctf p U, Λ q . We denoteby j : U Ñ X the canonical open immersion, and by δ : X Ñ X ˆ k X and δ U : U Ñ U ˆ k U thediagonal maps. Put H “ R H om p pr ˚ F , Rpr !1 F q on U ˆ k U, H “ R H om p pr ˚ j ! F , Rpr !1 j ! F q on X ˆ k X. By (3.7.1) and the projection formula for j ! (3.2.1), we have δ ˚ p H q – p j ! F q b L D p j ! F q – j ! p F b L D p F qq – j ! p δ ˚ U H q . Then, the evaluation map ev : δ ˚ U H Ñ K U (4.8.1) induces a map ev : δ ˚ p H q Ñ j ! p K U q . Let C be a closed subscheme of U ˆ k U and u a cohomological correspondence of F on C . Wedenote by C the closure of C in X ˆ k X . We have a commutative diagram C c / / j C (cid:15) (cid:15) U ˆ k U j ˆ j (cid:15) (cid:15) C c / / X ˆ k X where j , c and c are the canonical injections. Assume C “ p X ˆ k U q X C . The extension by zero j C ! p u q of u (4.6) corresponds, by (4.2), to a section j C ! p u q P H p C, R c ! p H qq “ H C p X ˆ k X, H q . We denote by C ! p j ! F , C, j C ! p u qq the image of j C ! p u q by the composed map H C p X ˆ k X, H q δ ˚ ÝÝÑ H C X X p X, δ ˚ p H qq ev ÝÝÑ H C X X p X, j ! p K U qq . By ([4] 2.1.7), the characteristic class C p j U ! F , C, j C ! p u qq P H C X X p X, K X q is the canonical imageof C ! p j ! F , C, j C ! p u qq .4.10. Let X be an equidimensional smooth k -scheme, S a closed subscheme of X , U “ X ´ S the complementary open subscheme of S in X that we assume to be dense in X , j : U Ñ X thecanonical injection, δ : X Ñ X ˆ k X the diagonal map, and F an object of D b ctf p X, Λ q such thatfor any integer q , H q p F q| U is locally constant. Put H “ R H om p pr ˚ F , Rpr !1 F q on X ˆ k X , wehave R δ ! H „ ÝÑ R H om p F , F q (3.6.1). Hence id F P End p F q corresponds to a map Λ X Ñ R δ ! H ,and by adjunction to a map δ ˚ Λ X Ñ H . Let ev : H Ñ δ ˚ K X be the adjoint of the evaluationmaps (4.8.1). Applying the functor ∆ δ (3.13) to the composition of the two morphisms above, weobtain a map(4.10.1) ∆ δ p δ ˚ p Λ X qq Ñ ∆ δ p H q ∆ δ p ev q ÝÝÝÝÝÑ ∆ δ p δ ˚ p K X qq . Since, for each integer q , H q p H q| U ˆ k U is locally constant and constructible, by (3.11), we have(4.10.2) H S p X, ∆ δ p H qq „ ÝÑ H p X, ∆ δ p H qq . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 13
Hence, the canonical map Λ X “ R δ ! δ ˚ Λ X Ñ ∆ δ p δ ˚ Λ X q , (4.10.1), (4.10.2) and the inverse of(3.13.2) define a map H p X, Λ X q Ñ H p X, ∆ δ p H qq „ ÝÑ H S p X, ∆ δ p H qq Ñ H S p X, ∆ δ p δ ˚ K X qq „ ÝÑ H S p X, K X q . We denote the image of P H p X, Λ X q in H S p X, K X q by C S p F q and call it the localized character-istic class of F ([4] 5.2). In [23], the author gave another definition of the localized characteristicclass and proved that the two definitions are equivalent.5. Ramification of ℓ -adic sheaves K be a complete discrete valuation field, O K the integer ring, F the residue field of O K , K a separable closure of K , and G K the Galois group of K over K . Abbes and Saito definedtwo decreasing filtrations G rK and G rK, log ( r P Q ą ) of G K by closed normal subgroups called theramification filtration and the logarithmic ramification filtration, respectively ([2], 3.1, 3.2). Wedenote by G K, log the inertia subgroup of G K . For any r P Q ě , we put G r ` K, log “ ď s P Q ą r G sK, log and Gr r log G K “ G rK, log L G r ` K, log . By ([2] 3.15), P “ G ` K, log is the wild inertia subgroup of G K , i.e. the p -Sylow subgroup of G K, log .For every rational number r ą , the group Gr r log G K is abelian and is contained in the center of P { G r ` K, log ([3] Theorem 1).5.2. Let L be a finite separable extension of K . For a rational number r ě , we say that thelogarithmic ramification of L { K is bounded by r (resp. by r ` ) if G rK, log (resp. G r ` K, log ) acts triviallyon Hom K p L, K q via its action on K . The logarithmic conductor c of L { K is defined as the infimumof rational numbers r ą such that the logarithmic ramification of L { K is bounded by r . Then c is a rational number and the logarithmic ramification of L { K is bounded by c ` ([2] 9.5). If c ą ,the logarithmic ramification of L { K is not bounded by c . Lemma 5.3 ([16] 1.1) . Let M be a Λ -module on which P “ G ` K, log acts Λ -linearly through a finitediscrete quotient, say by ρ : P Ñ Aut Λ p M q . Then, (i) The module M has a unique direct sum decomposition (5.3.1) M “ à r P Q ě M p r q into P -stable submodules M p r q , such that M p q “ M P and for every r ą , p M p r q q G rK, log “ p M p r q q G r ` K, log “ M p r q . (ii) If r ą , then M p r q “ for all but the finitely many values of r for which ρ p G r ` K, log q ‰ ρ p G rK, log q . (iii) For any r ě , the functor M ÞÑ M p r q is exact. (iv) For M , N as above, we have Hom P ´ Mod p M p r q , N p r q q “ if r ‰ r . slope decomposition of M . The values r ě for which M p r q ‰ are called the slopes of M . We say that M is isoclinic if it has only one slope. If M isisoclinic of slope r ą , we have a canonical central character decomposition M “ ‘ χ M χ , where the sum runs over finite characters χ : Gr r log G K Ñ Λ ˆ χ such that Λ χ is a finite étale Λ -algebra([5] 6.7). K has characteristic p and that F is of finite type over k . Let Ω F p log q bethe F -vector space Ω F p log q “ p Ω F { k ‘ p F b Z K ˆ qq{p d¯ a ´ ¯ a b a ; a P O ˆ K q , where a denotes the residue class of an element a P F . We denote by O K the integral closureof O K in K , F the residue field of O K and by v the valuation of K normalized by v p K ˆ q “ Z .For a rational number r , we put m rK (resp. m r ` K ) the set of elements of K such that v p x q ě r (resp. v p x q ą r ). For any rational number r ą , Gr r log G K is a F p -vector space, and there exists acanonical injective homomorphism, called the refined Swan conductor ([18] 1.24),(5.5.1) rsw : Hom F p p Gr r log G K , F p q Ñ Ω F p log q b F m ´ rK { m ´ r ` K . X be a smooth k -scheme, D a divisor with simple normal crossing on X , t D i u i P I theirreducible components of D . A rational divisor on X with support in D is an element R “ ř i P I r i D i of the Q -vector space generated by t D i u i P I . We say that R is effective if r i ě for all i . We call generic points of R the generic points of the D i ’s such that r i ‰ . We denote by t nR u the divisor ř i P I t nr i u D i on X , where t nr i u is the integral part of nr i . For two rational divisors R and R with support in D , we say that R is bigger than R and use the notation R ě R if R ´ R is effective.Let u : P Ñ X a smooth separated morphism of finite type, s : X Ñ P a section of u and R an effective rational divisor with support on D . Put U “ X ´ D and denote by j : U Ñ X and j P : P U “ u ´ p U q Ñ P the canonical injections and by I X the ideal sheaf of O P associated to s . We call dilatation of P along s of thickening R and denote by P p R q the affine scheme over P defined by the quasi-coherent sub- O P -algebra of j P ˚ p O P U q (5.6.1) ÿ n ě u ˚ p O X p t nR u qq ¨ I nX . The image of the algebra (5.6.1) by the surjective homomorphism j P ˚ p O P U q Ñ s ˚ j ˚ p O U q is canon-ically isomorphic to s ˚ p O X q . Hence we have a canonical section s p R q : X Ñ P p R q lifting s ([5] 5.26).5.7. In the following of this section, let X be a smooth k -scheme, D a divisor with simple normalcrossing on X , t D i u i P I the irreducible components of D , and j : U “ X ´ D Ñ X the canonicalinjection. We denote by p X ˆ k X q i the blow-up of X ˆ k X along D i ˆ i D i , by p X ¸ k X q i thecomplement of the proper transform of D ˆ k X in p X ˆ k X q i and by p X ˚ k X q i the complementof the proper transform of D i ˆ k X and X ˆ k D i in p X ˆ k X q i . We denote by p X ˆ k X q the fiberproduct of tp X ˆ k X q i u i P I over X ˆ k X , which is also the blow-up of X ˆ k X along t D i ˆ k D i u i P I ([18] §2.3). We denote by X ¸ k X the fiber product of tp X ¸ k X q i u i P I that we call the left-framedself-product of X along D . We denote by X ˚ k X the fiber product of tp X ˚ k X q i u i P I over X ˆ k X ,which is the open subscheme of p X ˆ k X q obtained by removing the strict transforms of D ˆ k X and X ˆ k D in p X ˆ k X q , that we call the framed self-product of X along D ([5] 5.22).By the universality of the blow-up, the diagonal map δ : X Ñ X ˆ k X induces closed immersionsthat we denote by δ : X Ñ p X ˆ k X q and r δ : X Ñ X ˚ k X. We consider X ˚ k X as an X -scheme by the second projection. This projection is smooth ([18]§2.3). EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 15
We denote by D i the pull-back of δ p D i q by the canonical projection p X ˆ k X q i Ñ X ˆ k X andby D the pull-back of δ p D q by the canonical projection p X ˆ k X q Ñ X ˆ k X . By definition, D i Ñ D i is a P -bundle. For a subset J of I , we put D J “ Ş i P J D i and denote by n J thecardinality of J . Since p X ˆ k X q is the fiber product of tp X ˆ k X q i u i P I over X ˆ k X , D is theunion of p P q n J -bundles over D J ([23] 3.12).We denote by r D i the pull-back of δ p D i q by the canonical projection p X ˚ k X q i Ñ X ˆ k X andby r D the pull-back of δ p D q by the canonical projection X ˚ k X Ñ X ˆ k X . By definition, r D i Ñ D i is a G m -bundle. Since X ˚ k X is the fiber product of tp X ˚ k X q i u i P I over X ˆ k X , r D is the unionof p G m q n J -bundles over D J ([22] 2.1).5.8. For any effective rational divisor R on X with support on D , we denote by p X ˚ k X q p R q the dilatation of X ˚ k X along r δ of thickening R (5.6 and 5.7). If we consider X ˚ k X as an X -scheme by the first projection, then the dilatation of X ˚ k X along r δ of thickening R is equalto p X ˚ k X q p R q ([5] 5.31). There is a canonical morphism δ p R q : X Ñ p X ˚ k X q p R q lifting r δ , and a canonical open immersion j p R q : U ˆ k U Ñ p X ˚ k X q p R q . Moreover, the following diagram(5.8.1) U l δ U / / j (cid:15) (cid:15) U ˆ k U j p R q (cid:15) (cid:15) X δ p R q / / p X ˚ k X q p R q is Cartesian.If R has integral coefficients, then the canonical projection p X ˆ k X q p R q Ñ X is smooth ([5]4.6) and we have a canonical R -isomorphism ([5] 4.6.1)(5.8.2) p X ˚ k X q p R q ˆ X R „ ÝÑ V p Ω X { k p log D q b O X O X p R qq ˆ X R. F be a locally constant constructible sheaf of Λ -modules on U , R an effective rationaldivisor on X with support on D , and x a geometric point of X . Put H “ H om p pr ˚ F , pr ˚ F q on U ˆ k U . Then the base change map(5.9.1) α : δ p R q˚ j p R q˚ p H q Ñ j ˚ δ ˚ U p H q “ j ˚ p E nd p F qq relatively to the Cartesian diagram (5.8.1) is injective ([5] 8.2). We say that the ramification of F at x is bounded by R ` ([5] 8.3) if F satisfies the following equivalent conditions ([5] 8.2):(i) The stalk α x of the morphism α (5.9.1) at x is an isomorphism.(ii) The image of id F in j ˚ p E nd p F qq x is contained in the image of α x .We say that the ramification of F along D is bounded by R ` ([5] 8.3) if the ramification of F at x is bounded by R ` for every geometric point x P X . F be a locally constant constructible sheaf of Λ -modules on U , R an effective rationaldivisor on X with support in D , ξ a generic point of D , ξ a geometric point of X above ξ , X p ξ q the corresponding strictly local scheme, η its generic point and r the multiplicity of R at ξ . Thenthe following conditions are equivalent ([5] 8.8):(i) The ramification of F at ξ is bounded by R ` .(ii) The sheaf F | η is trivialized by a finite étale connected covering η of η such that thelogarithmic ramification of η { η is bounded by r ` (5.2).The conductor of F at ξ is defined to be the minimum of the set of rational numbers r ě such that F | η is trivialized by a finite étale connected covering η of η and that the logarithmicramification of η { η is bounded by r ` (5.2). The conductor of F relatively to X is defined to bethe effective rational divisor on X with support in D whose multiplicity at any generic point ξ of D is the conductor of F at ξ ([5] 8.10). Definition 5.11 ([12] 2.6) . Let Y be a k -scheme, Z a closed subscheme of Y , V “ Y ´ Z thecomplementary open subscheme of Z in Y that is connected and smooth over Spec p k q and F a locally constant and constructible sheaf of Λ -modules on V . For any geometric point y of Y , Y p y q denotes the strict localization of Y at y . Then F is tamely ramified along Z if the followingequivalent conditions are satisfied:(i) For each geometric point y of Y and each geometric point x P Y p y q ˆ X V , the p -sylowsub-groups of the étale fundamental group π p Y p y q ˆ Y V, x q act trivially on F x .(ii) For each geometric point y of Y , there exists an étale neighborhood W of y and a Galoisétale covering T of W ˆ Y V of order prime to p , such that the pull-back of F on T is aconstant sheaf.Moreover, if Y is smooth over Spec p k q and Z is a divisor with simple normal crossing on Y , F istamely ramified along Z if and only if(iii) For any geometric point ξ of Z localized at a generic point of Z , the pull-back of F on thegeneric point of the trait X p ξ q is tamely ramified in the usual sense. Lemma 5.12 ([18] 2.21) . Let F be a locally constant and constructible sheaf of Λ -modules on U .Then the following conditions are equivalent: (i) F is tamely ramified along D . (ii) The conductor of F vanishes. (iii) The ramification of F along D is bounded by ` . F be a locally constant and constructible sheaf of Λ -modules on U . Let ξ be a genericpoint of D , X p ξ q the henselization of X at ξ , η ξ the generic point of X p ξ q , η ξ a geometric genericpoint of X p ξ q and G ξ the Galois group of η ξ over η ξ . We say that F is isoclinic at ξ if therepresentation F η ξ of G ξ is isoclinic (5.4). We say that F is isoclinic along D if it is isoclinic atall generic points of D ([5] 8.22).6. Clean ℓ -adic sheaves and characteristic cycles Definition 6.1 ([5] 3.1) . Let X be a k -scheme, π : E Ñ X a vector bundle, and F a constructiblesheaf of Λ -modules on E . We say that F is additive if for every geometric point x of X and forevery e P E p x q , denoting by τ e the translation by e on E x “ E ˆ Y x , τ ˚ e p F | E x q is isomorphic to F | E x . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 17 L ψ be the Artin-Schreier sheaf of Λ -modules of rank over the additive group scheme A F p over F p , associated to the character ψ (2.2) ([17] 1.1.3). If µ : A F p ˆ F p A F p Ñ A F p denotes theaddition, we have an isomorphism µ ˚ L ψ „ ÝÑ pr ˚ L ψ b pr ˚ L ψ . Hence, L ψ is additive (6.1). If f : X Ñ A F p is a morphism of schemes, we put L ψ p f q “ f ˚ L ψ .6.3. Let X be a k -scheme, π : E Ñ X a vector bundle of constant rank d and ˇ π : ˇ E Ñ X its dualbundle. We denote by x , y : E ˆ X ˇ E Ñ A F p the canonical pairing, by pr : E ˆ X ˇ E Ñ E and pr : E ˆ X ˇ E Ñ ˇ E the canonical projections and by F ψ : D bc p E, Λ q Ñ D bc p ˇ E, Λ q the Fourier-Deligne transform defined by ([17] 1.2.1.1) F ψ p K q “ Rpr p pr ˚ K b L ψ px , yqq . Let π : E Ñ X be the bidual vector bundle of π : E Ñ X , a : E Ñ E the anti-canonicalisomorphism defined by a p x q “ ´x x, y , and F _ ψ the Fourier-Deligne transform for ˇ π : ˇ E Ñ X . Forevery object K of D bc p E, Λ q , we have a canonical isomorphism ([17] 1.2.2.1)(6.3.1) F _ ψ ˝ F ψ p K q „ ÝÑ a ˚ p K qp´ d qr´ d s . Let π : E Ñ X be a vector bundle of constant rank d , F ψ its Fourier-Deligne transform, f : E Ñ E a morphism of vector bundles, and ˇ f : ˇ E Ñ ˇ E its dual. For every object K of D bc p E , Λ q , we have canonical isomorphisms ([5] 3.4.6, 3.4.7) R ˇ f ! ˝ F ψ p K qp d qr d s „ ÝÑ F ψ ˝ f ˚ p K qp d qr d s , (6.3.2) R ˇ f ˚ ˝ F ψ p K q „ ÝÑ F ψ ˝ R f ! p K q . (6.3.3)6.4. Let X be a k -scheme and K an object of D bc p X, Λ q . The support of K is the subset of pointsof X where the stalks of the cohomology sheaves of K are not all zero. It is constructible in X . Proposition 6.5 ([5] 3.6) . Let X be a k -scheme, π : E Ñ X a vector bundle of constant rank, ˇ π : ˇ E Ñ X its dual bundle, F a constructible sheaf of Λ -modules on E and S Ă ˇ E the support of F ψ p F q . Then, F is additive if and only if for every x P X , the set S X ˇ E x is finite. Definition 6.6 ([5] 3.8) . Let X be a k -scheme, π : E Ñ X a vector bundle of constant rank, ˇ π : ˇ E Ñ X its dual bundle, and F an additive constructible sheaf of Λ -modules on E . We call the Fourier dual support of F the support of F ψ p F q in ˇ E . We say that F is non-degenerated if theclosure of its Fourier dual support does not meet the zero section of ˇ E .If we replace ψ by aψ for an element a P F ˆ p , then the Fourier dual support of F will bereplaced by its inverse image by the multiplication by a on ˇ E . In particular, the notion of beingnon-degenerated dose not depend on ψ . Lemma 6.7 ([18] 2.6) . Let X be a k -scheme, π : E Ñ X a vector bundle of constant rank, s : X Ñ E the zero section of π , µ : E ˆ X E Ñ E the addition and F and G constructible sheavesof Λ -modules on E , where F is additive. Let e P Γ p X, s ˚ F q be a section and u : F b G Ñ µ ˚ G amap such that the composed map u | s p X qˆ E ˝ p e ˆ id G q : G Ñ s ˚ F b G Ñ G is the identity. Then G is additive and the Fourier dual support of G is a subset of that of F . Lemma 6.8 ([5] 3.10) . Let X be a k -scheme, π : E Ñ X a vector bundle of constant rank, and F an additive constructible sheaf of Λ -modules on E . If F is non-degenerate, R π ˚ F “ R π ! F “ . It follows form (6.3.1), (6.3.2) and (6.3.3) by applying f to the zero section of the dual bundle ˇ E of E and K “ F ψ p F q .6.9. Let X be a connected smooth k -scheme of dimension d , D a divisor with simple normalcrossing on X , t D i u i P I the irreducible components of D , R an effective Cartier divisor of X withsupport in D , U “ X ´ D and V “ X ´ R the complementary open subschemes of D and R in X respectively, j : U Ñ X , j V : V Ñ X and ν : U Ñ V the canonical injections. We denote by X ˚ k X (resp. V ˚ k V ) the framed self-product of X along D (resp. of V along D X V ) (5.7), by r δ : X Ñ X ˚ k X the canonical lifting of the diagonal δ : X Ñ X ˆ k X (5.7) and by p X ˚ k X q p R q the dilatation of X ˚ k X along r δ of thickening R , and we take the notation of (5.8). Moreover, wedenote by r ν : U ˆ k U Ñ V ˚ k V and j p R q V : V ˚ k V Ñ p X ˚ k X q p R q the canonical injections, by E p R q the vector bundle p X ˚ k X q p R q ˆ X R over R (5.8.2), and by ˇ E p R q its dual bundle.Let F be a locally constant and constructible sheaf of free Λ -modules on U . We put H “ H om p pr ˚ F , pr ˚ F q on U ˆ k U . Proposition 6.10 ([5] 8.15, 8.17) . We keep the assumptions and notation of , moreover, weassume that the ramification of F along D is bounded by R ` (5.9) . Then, (i) j p R q˚ H | E p R q is additive. Let S R p F q Ă ˇ E p R q be its Fourier dual support. (ii) S R p F q is the underlying space of a closed subscheme of ˇ E p R q which is finite over R . Proposition 6.11.
We keep the assumptions and notation of , moreover, we assume that theramification of F along D is bounded by R ` . Then, for any integer q ě , R q j p R q V ˚ p r ν ˚ p H qq| E p R q is additive. Let S qR p F q Ă ˇ E p R q be the Fourier dual support of R q j p R q V ˚ p r ν ˚ p H qq| E p R q , we have S qR p F q Ď S R p F q .Proof. We focus on the situation q ě since the case q “ is due to 6.10. For a scheme Y over X ˆ k X , we denote by f , f : Y Ñ X the maps induced by the projections pr , pr : X ˆ k X Ñ X ,respectively. We denote the fiber product Y ˆ f ,X,f Y simply by Y ˆ X Y .By ([5] 5.34, [18] 2.24), there exists a morphism λ : p X ˚ k X q ˆ X p X ˚ k X q Ñ X ˚ k X thatlifts the composed map p X ˆ k X q ˆ X p X ˆ k X q „ ÝÑ X ˆ k X ˆ k X pr ÝÝÝÑ X ˆ k X , and a smoothmorphism µ : p X ˚ k X q p R q ˆ X p X ˚ k X q p R q Ñ p X ˚ k X q p R q that makes the diagram p X ˚ k X q p R q ˆ X p X ˚ k X q p R q µ / / (cid:15) (cid:15) p X ˚ k X q p R q (cid:15) (cid:15) p X ˚ k X q ˆ X p X ˚ k X q λ / / X ˚ k X commutative, where the vertical arrows are the canonical projections. Moreover, the pull-back of µ by the canonical injection E p R q Ñ p X ˚ k X q p R q µ p R q : E p R q ˆ R E p R q Ñ E p R q EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 19 is the addition of the bundle E p R q ([5] 5.35). Hence, we have a canonical commutative diagramwith Cartesian squares p U ˆ k U q ˆ X p U ˆ k U q “ U ˆ k U ˆ k U l ν (cid:15) (cid:15) pr / / U ˆ k U j p R q (cid:127) (cid:127) r ν (cid:15) (cid:15) p V ˚ k V q ˆ X p V ˚ k V q l µ V / / j p R q V (cid:15) (cid:15) V ˚ k V j p R q V (cid:15) (cid:15) p X ˚ k X q p R q ˆ X p X ˚ k X q p R q µ / / p X ˚ k X q p R q where ν and j p R q V are canonical injections. By adjunction, we have canonical maps r ν ˚ p H q b L r ν ˚ p H q Ñ ν ˚ p H b H q , (6.11.1) R j p R q V ˚ p r ν ˚ p H qq b L R j p R q V ˚ p r ν ˚ p H qq Ñ R j p R q V ˚ p r ν ˚ p H q b L r ν ˚ p H qq . (6.11.2)On p U ˆ k U q ˆ X p U ˆ k U q “ U ˆ k U ˆ k U , we have H b H “ H om p pr ˚ F , pr ˚ F q b H om p pr ˚ F , pr ˚ F q , that gives a map(6.11.3) H b H Ñ H om p pr ˚ F , pr ˚ F q “ pr ˚ H . Since µ is smooth, by the smooth base change theorem, we have an isomorphism(6.11.4) µ ˚ p R j p R q V ˚ p r ν ˚ p H qqq „ ÝÑ R j p R q V ˚ p ν ˚ p pr ˚ p H qqq . The maps (6.11.1), (6.11.2), (6.11.3) and the inverse of (6.11.4) induce a map(6.11.5) R j p R q V ˚ p r ν ˚ p H qq b L R j p R q V ˚ p r ν ˚ p H qq Ñ µ ˚ p R j p R q V ˚ p r ν ˚ p H qqq . Consider the following commutative diagram with Cartesian squares U ˆ U p U ˆ k U q δ U ˆ id / / r ν (cid:15) (cid:15) l p U ˆ k U q ˆ X p U ˆ k U q ν (cid:15) (cid:15) V ˆ V p V ˚ k V q r δ V ˆ id / / j p R q V (cid:15) (cid:15) l p V ˚ k V q ˆ X p V ˚ k V q j p R q V (cid:15) (cid:15) X ˆ X p X ˚ k X q p R q δ p R q ˆ id / / p X ˚ k X q p R q ˆ X p X ˚ k X q p R q Notice that µ ˝p δ p R q ˆ id q “ id ([5] 5.35). Pulling back (6.11.5) by δ p R q ˆ id , we obtain a commutativediagram p δ p R q ˆ id q ˚ p j p R q˚ p H q b R q j p R q V ˚ p r ν ˚ p H qqq / / θ (cid:15) (cid:15) p δ p R q ˆ id q ˚ µ ˚ p R q j p R q V ˚ p r ν ˚ p H qqq j ˚ δ ˚ U p H q b R q j p R q V p r ν ˚ p H qq ϑ / / R q j p R q V p r ν ˚ p H qq where θ is an isomorphism induced by the base change isomorphism (5.9.1) δ p R q˚ j p R q˚ p H q „ ÝÑ j ˚ δ ˚ U p H q . On U ˆ U p U ˆ k U q “ U ˆ k U , we have δ ˚ U p H q b H “ H om p pr ˚ F , pr ˚ F q b H om p pr ˚ F , pr ˚ F q , which induces a map(6.11.6) δ ˚ U p H q b H Ñ H om p pr ˚ F , pr ˚ F q “ H . The morphism ϑ is the following composed map j ˚ δ ˚ U p H q b R q j p R q V p r ν ˚ p H qq Ñ R q j p R q V ˚ r ν ˚ p δ ˚ U p H q b H q Ñ R q j p R q V p r ν ˚ p H qq , where the second arrow is induced by (6.11.6). The map(6.11.7) ǫ : Λ Ñ j ˚ δ ˚ U p H q associated to the element id F P Γ p X, j ˚ δ ˚ U p H qq “ End p F q induces the identity H „ ÝÑ Λ b H ǫ | U ˆ id ÝÝÝÝÑ δ ˚ U p H q b H (6.11.6) ÝÝÝÝÝÑ H . Hence ǫ and ϑ induce the identity of R q j p R q V p r ν ˚ p H qq . Restrict (6.11.5) to E p R q ˆ R E p R q , we obtaina map(6.11.8) p j p R q˚ p H q| E p R q q b p R q j p R q V ˚ p r ν ˚ p H qq| E p R q q Ñ µ p R q˚ p R q j p R q V ˚ p r ν ˚ p H qq| E p R q q . Notice that the zero section s p R q : R Ñ E p R q is just the pull-back of δ p R q : X Ñ p X ˚ k X q p R q by E p R q Ñ p X ˚ k X q p R q . After restricting (6.11.8) to s p R q p R q ˆ R E p R q , the map ǫ | R (6.11.7)induces the identity of R q j p R q V ˚ p r ν ˚ p H qq| E p R q . Hence, the proposition follows from (6.7) (appliedwith F “ j p R q˚ p H q| E p R q and G “ R q j p R q V ˚ p r ν ˚ p H qq| E p R q ). (cid:3) Definition 6.12 ([5] 8.23) . We keep the assumptions and notation of 6.9, moreover, we assumethat the conductor of F relatively to X is the effective divisor R (5.10) and that F is isoclinicalong D (5.13). We say that F is clean along D if the following conditions are satisfied:(i) the ramification of F along D is bounded by R ` ;(ii) the additive sheaf j p R q˚ H | E p R q on E p R q is non-degenerated (6.6). Proposition 6.13.
We keep the assumptions and notation of , moreover, we assume that theconductor of F relatively to X is the effective divisor R and that F is isoclinic and clean along D (6.12) . Then, we have RΓ E p R q pp X ˚ k X q p R q , j p R q˚ p H qp d qq “ . Proof.
We denote by i : E p R q Ñ p X ˚ k X q p R q the canonical injection and π : E p R q Ñ R thecanonical projection. Notice that ([5] 5.26) V ˚ k V “ p X ˚ k X q p R q ˆ X V “ p X ˚ k X q p R q ´ E p R q , then R q i ! p j p R q˚ H q “ " q ď i ˚ R q ´ j p R q V ˚ p r ν ˚ p H qq when q ě . Since F is clean along D , for any integer q , the sheaf i ˚ R q ´ j p R q V ˚ p ν ˚ p H qq on E p R q is additive andnon-degenerated (6.11). Hence, for any integer q , R π ˚ R q i ! p j p R q˚ p H qq “ (6.8). Hence, RΓ E p R q pp X ˚ k X q p R q , j p R q˚ p H qp d qq “ RΓ p R, R π ˚ R i ! p j p R q˚ p H qqp d qq “ . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 21 (cid:3)
Remark 6.14.
Proposition 6.13 is used in the proof of ([18] 3.4). However, the proof of loc. cit.relies on ([18] 2.25) which is not enough. We reinforce it in 6.11.6.15. We keep the assumptions and notation of 6.9 and we denote by T ˚ X p log D q “ V p Ω X { k p log D q _ q , the logarithmic cotangent bundle of X , by σ : X Ñ T ˚ X p log D q the zero section, for i P I , by ξ i the generic point of D i , by F i the residue field of O X,ξ i , by S i “ Spec p O K i q the henselization of X at ξ i , by η i “ Spec p K i q the generic point of S i , by K i a separable closure of K i and by G i theGalois group Gal p K i { K i q .We assume moreover that the conductor of F is R , and that F is isoclinic and clean along D .We denote by M i the Λ r G i s -module corresponding to F | η i . Since F is isoclinic along D , M i hasjust one slope r i . We put I w “ t i P I ; r i ą u and S “ ř i P I w D i . For i P I w , let M i “ ‘ χ M i,χ be the central character decomposition of M i (5.4). Note that M i,χ is a free Λ -module of finite typefor all χ . By enlarging Λ , we may assume that for all central characters χ of M i , we have Λ χ “ Λ .Since Gr r i log G i is abelian and killed by p ([18] 1.24), each χ factors uniquely as Gr r i log G i Ñ F p ψ ÝÑ Λ ˆ ,where ψ is the non-trivial additive character fixed in 2.1. We denote also by χ the induced characterand by rsw p χ q : m r i K i { m r i ` K i Ñ Ω F i p log q b F i F i its refined Swan conductor (5.5.1) (where the notation are defined as in 5.5). Let F χ be the field ofdefinition of rsw p χ q , which is a finite extension of F i contained in F i . The refined Swan conductor rsw p χ q defines a line L χ in T ˚ X p log D q b X Spec p F χ q . Let L χ be the closure of the image of L χ in T ˚ X p log D q . For i P I w , we put(6.15.1) CC i p F q “ ÿ χ r i ¨ rk Λ p M i,χ qr F χ : F i s r L χ s , which is a d -cycle on T ˚ X p log D q ˆ X D i . We define a d -cycle CC ˚ p F q on T ˚ X p log D q ˆ X S by(6.15.2) CC ˚ p F q “ ÿ i P I w CC i p F q . We define the characteristic cycle of F and denote by CC p F q , the d -cycle on T ˚ X p log D q definedby ([18] 3.6) CC p F q “ p´ q d p rk Λ p F qr σ p X qs ` CC ˚ p F qq . Tsushima’s refined characteristic class X denotes a connected smooth k -scheme of dimension d , D a divisor withsimple normal crossing on X and t D i u i P I the irreducible components of D . We assume that I “ I t š I w , and we put S “ Ť i P I w D i , T “ Ť i P I t D i , U “ X ´ D and V “ X ´ S . We denote by j : U Ñ X , j V : V Ñ X and ν : U Ñ V the canonical injections.We denote by p X ˆ k X q the blow-up of X ˆ k X along t D i ˆ k D i u i P I , by p X ˆ k X q : the blow-upof X ˆ k X along t D i ˆ k D i u i P I t , by X ¸ k X the left-framed self-product of X along D and by X ˚ k X the framed self-product of X along D (5.7). For any open subschemes Y and Z of X , weput p Y ˆ k Z q “ p Y ˆ k Z q ˆ p X ˆ k X q p X ˆ k X q , p Y ˆ k Z q : “ p Y ˆ k Z q ˆ p X ˆ k X q p X ˆ k X q : ,Y ˚ k Z “ p Y ˆ k Z q ˆ p X ˆ k X q p X ˚ k X q . Notice that p V ˆ k V q “ p V ˆ k V q : and V ˚ k V “ ś i P I t pp X ˚ k X q i ˆ p X ˆ k X q p V ˆ k V qq . We havethe following commutative diagram with Cartesian squares(7.1.1) U ˆ k U l U ˆ k U l r ν / / ν (cid:15) (cid:15) V ˚ k V ϕ (cid:15) (cid:15) l V ˚ k V / / (cid:15) (cid:15) l X ˚ k X ϕ (cid:15) (cid:15) U ˆ k U l / / U ˆ k V l ν ¸ / / V ¸ k V ϕ (cid:15) (cid:15) U ˆ k U l / / U ˆ k V l ν : / / p V ˆ k V q : l h (cid:15) (cid:15) j : / / p V ˆ k X q : l j : / / g (cid:15) (cid:15) p X ˆ k X q : f (cid:15) (cid:15) U ˆ k U ν / / U ˆ k V ν / / V ˆ k V j / / V ˆ k X j / / X ˆ k X where all horizontal arrows are open immersions. We denote by r j : U ˆ k U Ñ X ˚ k X the canonicalinjection.We denote by δ : X Ñ X ˆ k X the diagonal map. By the universality of the blow-up, δ inducesclosed immersions(7.1.2) δ : : X Ñ p X ˆ k X q : and r δ : X Ñ X ˚ k X, and hence, by pull-back, the following closed immersions(7.1.3) δ : V : V Ñ p V ˆ k V q : and r δ V : V Ñ V ˚ k V. F denotes a locally constant and constructible sheaf of free Λ -modules on U , tamely ramified along T X V relatively to V . We put H “ H om p pr ˚ F , pr ˚ F q on U ˆ k U, H “ R H om p pr ˚ F , Rpr !1 F q on U ˆ k U, H “ R H om p pr ˚ j ! F , Rpr !1 j ! F q on X ˆ k X, Ă H “ r j ˚ H p d qr d s on X ˚ k X. We have a canonical isomorphism H „ ÝÑ H p d qr d s .7.3. We denote by H V the restriction of H to V ˆ k V and by Ă H V the restrictions of Ă H to V ˚ k V . Notice that H V „ ÝÑ R H om p pr ˚ ν ! F , Rpr !2 ν ! F q . We put H : V “ ϕ p R ϕ ˚ p Ă H V qq on p V ˆ k V q : . Since ν is an open immersion, the base change maps give by composition an isomorphism (7.1.1) ν :˚ p H : V q “ ν :˚ ϕ p R ϕ ˚ p Ă H V qq „ ÝÑ ν ¸˚ p R ϕ ˚ p Ă H V qq „ ÝÑ R ν ˚ H . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 23
By (3.7.2), we have h ˚ p H V q „ ÝÑ h ˚ ν R ν ˚ H „ ÝÑ ν : p R ν ˚ H q „ ÝÑ ν : p ν :˚ p H : V qq . It induces a canonical map(7.3.1) h ˚ p H V q Ñ H : V , that extends the identity of H on U ˆ k U . Since F is tamely ramified along the divisor T X V relatively to V , the adjoint map H V Ñ R h ˚ p H : V q is an isomorphism by ([4] 2.2.4).7.4. We put H : “ j : p R j : ˚ p H : V qq on p X ˆ k X q : , and we consider the following composition of maps(7.4.1) f ˚ H “ f ˚ j R j ˚ p H V q „ ÝÑ j : g ˚ R j ˚ p H V q Ñ j : R j : ˚ h ˚ p H V q (7.3.1) ÝÝÝÝÑ j : p R j : ˚ p H : V qq “ H : , where the second and the third arrows are induced by the base change maps. Lemma 7.5 ([23] Lemma 3.13) . The adjoint map of (7.4.1) H Ñ R f ˚ H : is an isomorphism. X : “ f ´ p δ p X qq and S : “ f ´ p δ p S qq . We denote by γ : : p X ˆ k X q : z X : Ñ p X ˆ k X q : the canonical injection, which is an open immersion and put(7.6.1) L : “ R γ :˚ p Λ q . The map (7.4.1) induces by pull-back a map(7.6.2) H X p X ˆ k X, H q Ñ H X : pp X ˆ k X q : , H : q . The adjunction Λ Ñ L : induces a map(7.6.3) H X : pp X ˆ k X q : , H : q Ñ H X : pp X ˆ k X q : , H : b L L : q . By (3.6.1), we have a canonical isomorphism
End p j ! F q „ ÝÑ H X p X ˆ k X, H q . We denote also id j ! F the image of id j ! F P End p j ! F q in H X p X ˆ k X, H q . Its image in H X : pp X ˆ k X q : , H : b L L : q bythe composition of the maps (7.6.2) and (7.6.3) will be denoted by α p j ! F q . Proposition 7.7 ([23], 3.14 and 3.15) . The canonical map (7.7.1) H S : pp X ˆ k X q : , H : b L L : q Ñ H X : pp X ˆ k X q : , H : b L L : q is injective and there exists a unique element (7.7.2) α p j ! F q P H S : pp X ˆ k X q : , H : b L L : q whose image by (7.7.1) is α p j ! F q . X l δ (cid:15) (cid:15) V l j V o o (cid:15) (cid:15) V l δ : V (cid:15) (cid:15) V l r δ V (cid:15) (cid:15) U ν o o δ U (cid:15) (cid:15) p X ˆ k X q : p V ˆ k X q : j : o o p V ˆ k V q : j : o o V ˚ k V ϕ ˝ ϕ o o U ˆ k U r ν o o are Cartesian and all the horizontal arrows are open immersions. By (5.9) and (5.12), since F istamely ramified along T X V relatively to V , we have an isomorphism(7.8.1) r δ ˚ V p r ν ˚ p H qq „ ÝÑ ν ˚ p δ ˚ U p H qq . The base change maps give by composition an isomorphism(7.8.2) δ :˚ p H : q „ ÝÑ j V ! δ :˚ V p H : V q „ ÝÑ j V ! r δ ˚ V p Ă H V q „ ÝÑ j V ! ν ˚ p E nd p F qqp d qr d s , where the third arrow is (7.8.1). There exists a unique map(7.8.3) Tr V : ν ˚ p E nd p F qq Ñ Λ V which extends the trace map Tr : E nd p F q Ñ Λ U ([4] (2.9)). The maps (7.8.2) and (7.8.3) give anevaluation map(7.8.4) ev : : δ :˚ p H : q Ñ j V ! p K V q . Composing with the canonical map j V ! p K V q Ñ K X , we obtain a morphism(7.8.5) H S p X, δ :˚ p H : q b L δ :˚ L : q Ñ H S p X, K X b L δ :˚ L : q . The pull-back by δ : gives a morphism(7.8.6) H S : pp X ˆ k X q : , H : b L L : q Ñ H S p X, δ :˚ p H : q b L δ :˚ L : q . Composing (7.8.5) and (7.8.6), we get a map(7.8.7) H S : pp X ˆ k X q : , H : b L L : q Ñ H S p X, K X b L δ :˚ L : q . Lemma 7.9 ([22] Lemma 2.3) . The canonical map (7.9.1) H S p X, K X q Ñ H S p X, K X b L δ :˚ L : q induced by the canonical map Λ Ñ δ :˚ L : , is an isomorphism. κ : H S : pp X ˆ k X q : , H : b L L : q Ñ H S p X, K X q . We call κ p α p j ! F qq P H S p X, K X q the refined characteristic cycle of j ! F , and we denote it by C S p j ! F q . Remark 7.11. If T “ H , we have p X ˆ k X q : “ X ˆ k X , p V ˆ k V q : “ V ˚ k V “ U ˆ k U , X : “ X , S : “ S and, by (3.7.2), H : “ H . It is easy to see that (4.10) C S p j ! p F qq “ C S p j ! p F qq P H S p X, K X q . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 25 Y denotes a connected smooth k -scheme, Z a closed sub-scheme of Y and W “ Y ´ Z the complementary open subscheme of Z in Y . We assume that thereexists a proper flat morphism π : X Ñ Y such that V “ π ´ p W q , Q “ π ´ p Z q is a divisor withnormal crossing, that S “ Q red , that the canonical projection π V : V Ñ W is smooth and that T X V is a divisor with simple normal crossing relatively to W . We have a commutative diagramwith Cartesian squares U ν / / π U ❅❅❅❅❅❅❅❅ V l j V / / π V (cid:15) (cid:15) X l π (cid:15) (cid:15) Q i Q o o π Q (cid:15) (cid:15) W j W / / Y Z i Z o o We make the following remarks:(i) For any locally constant and constructible sheaf of Λ -modules G tamely ramified alongthe divisor T X V relatively to V , π V is universally locally acyclic relatively to ν ! p G q ([6]Appendice to Th. Finitude, [19] 3.14). Since π V is proper, all cohomology groups of R π U ! p G q are locally constant and constructible on W .(ii) Since π is proper, we have a push-forward(7.12.1) H S p X, K X q „ ÝÑ H Q p X, K X q Ñ H Z p Y, K Y q defined by applying the functor H p Z, ´q to the following composed map R π Q ˚ p K Q q „ ÝÑ R π Q ˚ R π ! Q p K Z q „ ÝÑ R π Q ! R π ! Q p K Z q Ñ K Z , where the third arrow is induced by the adjunction. Theorem 7.13 (Localized Lefschetz-Verdier trace formula, [23] 5.4) . We have (4.10, 7.10) π ˚ p C S p j ! p F qqq “ C Z p j W ! p R π U ! p F qqq in H Z p Y, K Y q . Y is of dimension and that Z is a closed point y of Y , and we denote by y a geometric point of Y localized at y , by Y p y q the strict localization of Y at y , by η a geometricgeneric point of Y p y q . For any object G of D b ctf p W, Λ q with locally constant cohomology groups.We put ([6] Rapport 4.4) rk Λ p G η q “ Tr p id; G η q , sw y p G η q “ ÿ q P Z p´ q q sw y pp H q p G qq η q , dimtot y p G η q “ rk Λ p G η q ` sw y p G η q , where sw y pp H q p G qq η q denotes the Swan conductor of p H q p G qq η at y ([21] 19.3). By ([23] 4.1),we have(7.14.1) C t y u p j W ! p G qq ´ rk Λ p G | η q ¨ C t y u p j W ! p Λ W qq “ ´ sw y p G η q in H t y u p Y, K Y q „ ÝÑ Λ . In fact, the proof of (7.14.1) is simpler than the general case treated in ([23]4.1), since Y is of dimension , we can use the usual Swan conductor rather than the generalizedone ([14] 4.2.2). Corollary 7.15 ([23] 5.5) . Keep the notation and assumptions of . We have (7.15.1) sw y p RΓ c p U η , F | U η qq ´ rk Λ p F q ¨ sw y p RΓ c p U η , Λ qq “ ´ π ˚ p C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qqq in H t y u p Y, K Y q „ ÝÑ Λ .Proof. Since F is tamely ramified along T X V relatively to V , we have ([12] 2.7) rk Λ p R π U ! p F q| η q “ rk Λ p F q ¨ rk Λ p R π U ! p Λ U q| η q . Then, by (7.13) and (7.14.1), π ˚ p C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qqq“ C t y u p j W ! R π U ! p F qq ´ rk Λ p F q ¨ C t y u p j W ! R π U ! p Λ U qq“ C t y u p j W ! R π U ! p F qq ´ rk Λ p R π U ! p F q| η q ¨ C t y u p j W ! p Λ W qq´ rk Λ p F q ¨ ´ C t y u p j W ! R π U ! p Λ U qq ´ rk Λ p R π U ! p Λ q| η q ¨ C t y u p j W ! p Λ W qq ¯ “ ´ sw y p R π U ! p F q| η q ` rk Λ p F q ¨ sw y p R π U ! p Λ q| η q . By the proper base change theorem (3.1.4), we have R π U ! p F q| η „ ÝÑ RΓ c p U η , F | U η q and R π U ! p Λ q| η „ ÝÑ RΓ c p U η , Λ q . Then (7.15.1) follows. (cid:3) The conductor formula § . Let R be the conductor of F (5.9) that we assume having integral coefficients ofsupport in S . We assume also that F is isoclinic and clean along D (5.13 and 6.12). Notice thatif R “ , a sheaf F is tamely ramified along D and is automatically isoclinic and clean. Theorem 8.2.
Let F be a sheaf on U as in . Assume that T X S “ H or that rk Λ p F q “ .Then, we have (2.4) C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qq“ p´ q d rk Λ p F q ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log D q ¯ XS X r X s P H S p X, K X q , where the right hand side is considered as an element of H S p X, K X q by the cycle map (3.10.2) . The theorem will be proved in 8.20 after some preliminaries. We will deduce from it the theorem1.3 in 8.26. The case where rk Λ p F q “ is due to Tsushima ([23] 5.9). Remark 8.3.
Although we follow the same lines as [22] for sheaves of higher ranks, the situationis technically more involved. The assumption S X T “ H is required for the injectivity of a map λ defined in (8.6.4), which is a crucial step in my proof (cf. 8.17). We don’t know if it holds withoutthis assumption.8.4. We consider X ˚ k X as an X -scheme by the second projection, and we denote by p X ˚ k X q p R q the dilatation of X ˚ k X along r δ of thickening R (5.6). We have a Cartesian diagram (5.8.1) U l δ U / / j (cid:15) (cid:15) U ˆ k U j p R q (cid:15) (cid:15) X δ p R q / / p X ˚ k X q p R q We denote by f p R q : p X ˚ k X q p R q Ñ X ˆ k X and ϕ p R q : p X ˚ k X q p R q Ñ p X ˆ k X q : the canonicalprojections. We put X p R q “ f p R q´ p δ p X qq and S p R q “ f p R q´ p δ p S qq . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 27
We put H p R q “ j p R q˚ p H qp d qr d s on p X ˚ k X q p R q . Notice that H p R q | V ˚ k V “ Ă H V (7.3). Proposition 8.5 ([18] Corollary 3.3) . There exists a unique homomorphism (7.2)(8.5.1) f p R q˚ p H q Ñ H p R q extending the identity of H on U ˆ k U . H X p X ˆ k X, H q Ñ H X p R q pp X ˚ k X q p R q , H p R q q . We put (7.6.1)(8.6.2) L p R q “ ϕ p R q˚ p L : q . The canonical map Λ Ñ L p R q induces a map(8.6.3) H X p R q pp X ˚ k X q p R q , H p R q q Ñ H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q . The canonical injection S p R q Ñ X p R q induces a map(8.6.4) λ : H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q Ñ H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q . r V “ X p R q ´ S p R q the complementary open subscheme of S p R q in X p R q , by r ι V : r V Ñ V ˚ k V , r γ V : p V ˚ k V qz r V Ñ V ˚ k V and γ U : p U ˆ k U qz δ U p U q Ñ U ˆ k U the canonicalinjections. We put D p R q “ f p R q´ p δ p D qq and T p R q “ f p R q´ p δ p T qq . Notice that T p R q Y S p R q “ D p R q and that δ U p U q is the complementary open subscheme of D p R q in X p R q . We put Ă L V “ R r γ V ˚ p Λ q . Notice that Ă L V „ ÝÑ L : | V ˚ k V “ L p R q | V ˚ k V (7.6.1) and (8.6.2). Proposition 8.8 ([22] 2.2) . If the sheaf F on U has rank , the map λ (8.6.4) is an isomorphism.Proof. It is sufficient to show that, for any integer q , H q r V p V ˚ k V, Ă H V b L Ă L V q “ . Since F is of rank and is tamely ramified along T X V relatively to V , r ν ˚ H is a locally constant andconstructible sheaf on V ˚ k V ([4] 4.2.2.1). By ([7] 6.5.5), we have Ă H V b L Ă L V „ ÝÑ R r γ V ˚ p r γ ˚ V p Ă H V qq . Since R r ι ! R r γ ˚ “ (3.1.5), for any integer q , H q r V p V ˚ k V, Ă H V b L Ă L V q “ H q p r V , R r ι ! R r γ V ˚ p r γ ˚ V p Ă H V qqq “ . (cid:3) Proposition 8.9. If T X S “ H , the map λ (8.6.4) is injective.Proof. Since H is locally constant, by ([7] 6.5.5), for any integer q , H qU p U ˆ k U, H b L R γ U ˚ p Λ qq „ ÝÑ H qU p U ˆ k U, R γ U ˚ γ ˚ U p H qq „ ÝÑ H q p U, R δ ! U R γ U ˚ γ ˚ U p H qq “ . Hence, we have a canonical isomorphism H D p R q pp X ˚ k X q p R q , H p R q b L L p R q q „ ÝÑ H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q . Since T X S “ H , we have T p R q X S p R q “ H . Hence, for any object G of D b ctf pp X ˚ k X q p R q , Λ q , H D p R q pp X ˚ k X q p R q , G q “ H S p R q pp X ˚ k X q p R q , G q ‘ H T p R q pp X ˚ k X q p R q , G q . In particular, the canonical map H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q Ñ H D p R q pp X ˚ k X q p R q , H p R q b L L p R q q is injective. Hence λ is injective. (cid:3) δ p R q gives a map H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q Ñ H S p X, δ p R q˚ p H p R q q b L δ :˚ p L : qq . Since the conductor of F is R and F is isoclinic and clean along D (8.1), the ramification of F along D is bounded by R ` (6.12). Hence, we have an isomorphism (5.9) δ p R q˚ p j p R q˚ p H qq „ ÝÑ j ˚ p δ ˚ U p H qq . We have an evaluation map δ p R q˚ p H p R q q „ ÝÑ j ˚ p δ ˚ U p H qqp d qr d s “ j ˚ p E nd p F qqp d qr d s Ñ p j ˚ p Λ U qqp d qr d s “ Λ X p d qr d s “ K X , where the third arrow is the push-forward of the trace map Tr : E nd p F q Ñ Λ U . It induces a map ev p R q : H S p X, δ p R q˚ p H p R q q b L δ :˚ p L : qq Ñ H S p X, K X b L δ :˚ p L : qq „ ÝÑ H S p X, K X q , where second arrow is the inverse of the isomorphism (7.9.1).8.11. By (3.4.2), we have a map H p X, δ p R q˚ j p R q˚ p H qq ˆ H S p X, δ :˚ p L : qp d qr d sq Y S ÝÝÑ H S p X, δ p R q˚ p H p R q q b L δ :˚ p L : qq . In the following of this section, we denote by e P H p X, δ p R q˚ j p R q˚ p H qq the unique pre-image of id F P End p F q “ H p X, j ˚ δ ˚ p H qq . The following diagram is commutative(8.11.1) H S p X, δ p R q˚ p H p R q q b L δ :˚ p L : qq ev p R q / / H S p X, K X q (7.9.1) (cid:15) (cid:15) H S p X, δ :˚ p L : qp d qr d sq e Y S ´ O O ¨ rk Λ p F q / / H S p X, K X b L δ :˚ p L : qq since the composition of the following morphisms H p X, δ p R q˚ j p R q˚ p H qq Ñ H p X, j ˚ δ ˚ U p H qq ev ÝÑ H p X, Λ q maps e P H p X, δ p R q˚ j p R q˚ p H qq to rk Λ p F q P Λ “ H p X, Λ q .8.12. We have a commutative diagram with Cartesian squares (7.1)(8.12.1) V ˚ k V l j p R q V (cid:15) (cid:15) V ˚ k V l (cid:15) (cid:15) ϕ / / V ¸ k V ϕ / / p V ˆ k V q : j : / / p V ˆ k X q : j : (cid:15) (cid:15) p X ˚ k X q p R q ϕ p R q / / X ˚ k X ϕ / / p X ˆ k X q : The base change maps give by composition the following isomorphism ϕ p R q˚ p H : q “ ϕ p R q˚ j : R j : ˚ p H : V q „ ÝÑ j p R q V ! p j : ˝ ϕ ˝ ϕ q ˚ R j : ˚ p H : V q „ ÝÑ j p R q V ! Ă H V , which induces a map(8.12.2) ϕ p R q˚ p H : q „ ÝÑ j p R q V ! p Ă H V q Ñ j p R q V ˚ p r ν ˚ p H qqp d qr d s “ H p R q . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 29
By (8.5), the composed map f p R q˚ p H q “ ϕ p R q˚ p f ˚ p H qq (7.4.1) ÝÝÝÝÑ ϕ p R q˚ p H : q (8.12.2) ÝÝÝÝÝÑ H p R q is equal to (8.5.1). We deduce by pull-back a commutative diagram(8.12.3) H X p X ˆ k X, H q (7.6.2) / / (8.6.1) * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ H X : pp X ˆ k X q : , H : q (cid:15) (cid:15) H X p R q pp X ˚ k X q p R q , H p R q q H X : pp X ˆ k X q : , H : q (cid:15) (cid:15) (7.6.3) / / H X : pp X ˆ k X q : , H : b L L : q (cid:15) (cid:15) H X p R q pp X ˚ k X q p R q , H p R q q (8.6.3) / / H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q H X pp X ˚ k X q p R q , Λ p d qr d sq e Y´ O O θ / / H X pp X ˚ k X q p R q , L p R q p d qr d sq e Y´ O O (8.13.2) H X : pp X ˆ k X q : , H : b L L : q (cid:15) (cid:15) H S : pp X ˆ k X q : , H : b L L : q ? _ (7.7.1) o o (cid:15) (cid:15) H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q p q H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q λ o o H X pp X ˚ k X q p R q , L p R q p d qr d sq e Y´ O O H S pp X ˚ k X q p R q , L p R q p d qr d sq e Y S ´ O O λ o o (8.13.3) H S : pp X ˆ k X q : , H : b L L : q (cid:15) (cid:15) (7.8.6) / / H S p X, δ :˚ p H : q b L δ :˚ p L : qq (7.8.5) / / (cid:15) (cid:15) H S p X, K X q H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q / / H S p X, δ p R q˚ p H p R q q b L δ :˚ p L : qq ev p R q / / (8.11.1) H S p X, K X q H S pp X ˚ k X q p R q , L p R q p d qr d sq e Y S ´ O O / / H S p X, δ :˚ p L : qp d qr d sq ¨ rk Λ p F q / / e Y S ´ O O H S p X, K X q wherei. The arrows from the upper row to the middle row are the pull-backs by ϕ p R q ;ii. The arrows e Y ´ are the cup products (3.3.2) and the arrows e Y S ´ are given in (3.4.2);iii. The arrows θ are induced by the canonical map Λ Ñ L p R q ;iv. The arrows λ is the canonical map induced by the injection S Ñ X ;v. The square p q is commutative by (3.4.1); vi. The arrows under (7.8.6) are the pull-back by δ p R q ;vii. We use the canonical isomorphism H S p X, K X q „ ÝÑ H S p X, K X b L δ :˚ p L : qq (cf. 7.9). Lemma 8.14.
For any integer q , the canonical maps H qS pp X ˚ k X q p R q , L p R q p d qq Ñ H qX pp X ˚ k X q p R q , L p R q p d qq , are isomorphisms. In particular, the map λ in (8.13.2) is an isomorphism.Proof. It is sufficient to show that, for any integer q , H qV p V ˚ k V, R r γ V ˚ Λ p d qq “ , which follows from the fact that R r δ ! V R r γ V ˚ Λ p d q “ (3.1.5). (cid:3) q , any object G of D b ctf pp X ˚ k X q p R q , Λ q and any closed subscheme Z Pp X ˚ k X q p R q , we denote by H qZ pp X ˚ k X q p R q , G q Ñ H qZ pp X ˚ k X q p R q , G b L L p R q q , x ÞÑ x a , the morphism induced by the canonical map Λ Ñ L p R q . For any closed immersion Z Ñ Y ofclosed subschemes of p X ˚ k X q p R q , we denote abusively by H qZ pp X ˚ k X q p R q , G q Ñ H qY pp X ˚ k X q p R q , G q , x ÞÑ x, the canonical map. Proposition 8.16 ([18] 3.3, 3.4) . We denote by r X s P H X pp X ˚ k X q p R q , Λ p d qr d sq the cycle classof δ p R q p X q . Then we have (7.6) , (8.6.1) f p R q˚ p id j ! p F q q “ e Y r X s P H X p R q pp X ˚ k X q p R q , H p R q q . The proof in ([18] 3.4) should be modified as in 6.13.
Proposition 8.17. If T X S “ H or if rk Λ p F q “ , we have (7.10)(8.17.1) C S p j ! p F qq “ rk Λ p F q ¨ δ p R q˚ p λ ´ pr X s a qq P H S p X, K X q . Proof.
By (8.12.3), (8.13.1) and 8.16, we have (7.6) ϕ p R q˚ p α p id j ! p F q qq “ e Y pr X s a q P H X p R q pp X ˚ k X q p R q , H p R q b L L p R q q . Then, by 8.8, 8.9, 8.14 and (8.13.2), we have (7.7.2) ϕ p R q˚ p α p j ! p F qqq “ e Y S p λ ´ pr X s a qq P H S p R q pp X ˚ k X q p R q , H p R q b L L p R q q . Equation (8.17.1) follows form (8.13.3). (cid:3)
Lemma 8.18.
We put r X p R q “ ϕ p R q´ p δ : p X qq and r S p R q “ ϕ p R q´ p δ : p S qq (8.12.1) . Then (i) There exists a unique element τ P CH d p r S p R q q which maps to r X s ´ ϕ p R q ! r X s P CH d p r X p R q q ,and we have (8.18.1) δ p R q ! p τ q “ p´ q d ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log T q ¯ XS X r X s P CH p S q . (ii) We consider τ as an element in H r S p R q pp X ˚ k X q p R q , Λ p d qr d sq by the cycle map (3.10) . Wehave (8.18.2) τ a “ λ ´ pr X s a q P H r S p R q pp X ˚ k X q p R q , L p R q p d qr d sq . The proof of this lemma is similar to that of ([22] 3.7), in which the author consider the casewhere supp p R q “ S . It is an immediate application of ([14] 3.4.9). EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 31
Corollary 8.19.
We have (8.19.1) δ p R q˚ p λ ´ pr X s a qq “ p´ q d ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log T q ¯ XS X r X s P H S p X, K X q , where the right hand side is considered as an element of H S p X, K X q by the cycle map.Proof. Applying (3.10.3) to the map δ p R q : X Ñ p X ˚ k X q p R q , we have(8.19.2) δ p R q ! p τ q “ δ p R q˚ p τ q P H S p X, K X q , where we consider δ p R q ! p τ q as an element of H S p X, K X q by the cycle map. Since the followingdiagram H r S p R q pp X ˚ k X q p R q , Λ p d qr d sq (cid:15) (cid:15) / / H S p X, K X q (7.9.1) (cid:15) (cid:15) H r S p R q pp X ˚ k X q p R q , L p R q p d qr d sq / / H S p X, K X b L δ :˚ L : q is commutative, where the horizontal arrows are the pull-backs by δ p R q , we have(8.19.3) δ p R q˚ p τ q “ δ p R q˚ p τ a q P H S p X, K X q . Hence, (8.19.1) follows form (8.18.1), (8.18.2), (8.19.2) and (8.19.3). (cid:3)
Proof of Theorem . By 8.17 and 8.19, we have C S p j ! p F qq “ p´ q d rk Λ p F q ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log T q ¯ XS X r X s ,C S p j ! p Λ U qq “ p´ q d ¨ c d ´ Ω X { k p log D q ´ Ω X { k p log T q ¯ XS X r X s in H S p X, K X q . Hence, C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qq“ p´ q d rk Λ p F q ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log D q ¯ XS X r X s P H S p X, K X q . l Remark 8.21 ([4] 4.2.1, [22] remark after 3.9) . Observe that we have p´ q d ¨ c d ´ Ω X { k p log D q b O X O X p R q ´ Ω X { k p log D q ¯ XS X r X s (8.21.1) “ ´t c p Ω X { k p log D q _ q X p ` c p O X p R qqq ´ X r R su dim 0 “ p´ q d ¨ t c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X r R su dim 0 P CH p S q . T ˚ X p log D q “ V p Ω X { k p log D q _ q , the logarithmic cotangent bundle of X . Since the R is supported in S , R “ ř i P I w r i D i , where r i P Z ě . By ([18] 3.16), For i P I w , we have (6.15.1) CC i p F q “ r i ¨ rk Λ p F q ¨ t c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X r T ˚ X p log D q ˆ X D i su dim d in CH d p T ˚ X p log D q ˆ X D i q . Hence, we have (6.15.2)(8.22.1) CC ˚ p F q “ rk Λ p F q ¨ t c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X r T ˚ X p log D q ˆ X R su dim d in CH d p T ˚ X p log D q ˆ X S q . Y is ofdimension and that Z is a closed point y of Y , and we denote by y a geometric point localizedat y , by Y p y q the strict localization of Y at y and by η a geometric generic point of Y p y q . Theorem 8.24.
We assume that S “ D (i.e., T “ H ) or that rk Λ p F q “ . Then, for any section s : X Ñ T ˚ X p log D q , we have (8.24.1) sw y p RΓ c p U η , F | U η qq ´ rk Λ p F q ¨ sw y p RΓ c p U η , Λ qq “ p´ q d ` deg p CC ˚ p F q X r s p X qsq in H t y u p Y, K Y q „ ÝÑ Λ .Proof. We denote by ̟ : T ˚ X p log D q Ñ X the canonical projection. Since ̟ ˝ s “ id X , we have CC ˚ p F q X r s p X qs“ rk Λ p F q ¨ t c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X ̟ ˚ r R s X r s p X qsu dim “ “ rk Λ p F q ¨ ̟ ˚ pt c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X r R su dim “ q X r s p X qs“ rk Λ p F q ¨ t c p Ω X { k p log D qq X p ´ c p O X p R qqq ´ X r R su dim “ P CH p S q . By 8.2 and (8.21.1), we get (3.10) p´ q d p CC ˚ p F q X r s p X qsq “ C S p j ! p F qq ´ rk Λ p F q ¨ C S p j ! p Λ U qq P H S p X, K X q . Hence, by (7.15), we have sw y p RΓ c p U η , F | U η qq ´ rk Λ p F q ¨ sw y p RΓ c p U η , Λ qq “ p´ q d ` π ˚ p CC ˚ p F q X r s p X qsq in H t y u p Y, K Y q „ ÝÑ Λ . It is easy to see that the composed map CH p S q cl ÝÑ H S p X, K X q Ñ H t y u p Y, K Y q „ ÝÑ Λ , where the second arrow is the push-forward (7.12.1), is just the degree map of zero cycles. Weobtain (8.24.1). (cid:3) Remark 8.25.
Since π : X Ñ Y is proper and T X V is a divisor with simple normal crossingrelatively to W (7.12), the condition S X T “ H in 8.2 implies T “ H .8.26. Proof of Theorem . Since F is tamely ramified along T X V relatively to V , F | U η istamely ramified along p T X V q η relatively to V η . By ([12] 2.7, [18] 3.2), we have rk Λ p RΓ c p U η , F | U η qq “ p´ q d ´ rk Λ p F q ¨ c d ´ p Ω V η { η p log p T X V q η qq X r V η s“ rk Λ p F q ¨ rk Λ p RΓ c p U η , Λ qq in H p V η , K V η q „ ÝÑ Λ . Hence, we obtain (1.3.1) by 8.24. l Remark 8.27.
We denote by K the function field of Y p y q , by K a separable closure of K and by P the wild inertia subgroup of Gal p K { K q . We assume that T “ H and Q is reduced. Notice that X ˆ Y Y p y q is semi-stable over the strict trait Y p y q . Then the cohomology group H ˚ p U η , Λ q “ H ˚ c p U η , Λ q is tame, i.e., the action of P is trivial ([13] 3.3). Hence, sw y p RΓ c p U η , Λ qq “ . EFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 33
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