GEOMETRIC LOCAL ε -FACTORS IN HIGHER DIMENSIONS

We prove a product formula for the determinant of the cohomology of an ´etale sheaf with (cid:2) -adic coeﬃcients over an arbitrary proper scheme over a perfect ﬁeld of positive characteristic p distinct from (cid:2) . The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local ε -factors. We provide several applications of our higher dimensional product formula, such as twist formulas for global ε -factors

1. Introduction 1.1.Let k be a perfect field of positive characteristic p and let k be an algebraic closure of k.Let Λ be either F ℓ or Q ℓ for a fixed prime number ℓ distinct from p.The ε-factor associated to a pair (X, F ), where X is a proper k-scheme and F is a bounded constructible complex of étale sheaves of Λ-modules on X, is the Galois line given by ε k (X, F ) = det(RΓ(X k , F )) −1 .
In this paper, we simply consider this object as a homomorphism from G k = Gal(k/k) to Λ × .We will also consider the Euler characteristic χ(X, F ) = rk(RΓ(X k , F )).
In the case where X is a smooth curve over a finite field, it was conjectured by Langlands that the global ε-factor ε k (X, F ) should split as a product of local contributions.This conjecture was motivated by the corresponding theory of automorphic local ε-factors.Partial results were obtained by Dwork [Dw56] and Deligne [De73], and the product formula for ℓ-adic sheaves on a smooth curve over a finite field was proved in its full generality by Laumon ([La87], Th. 3.2.1.1).The case of a smooth curve over an arbitrary perfect field of positive characteristic was handled in [Gu19].An alternative approach by "spreading out" in order to reduce to the finite field case was also given by Yasuda in [Ya1], [Ya2], [Ya3].
In the geometric setting of [Gu19], the local ε-factors are realized as determinants of certain Galois modules arising in the cohomology of Gabber-Katz extensions, cf.([Gu19], 9.2).These Galois modules are thus linearizations of local ε-factors: these are given by additive functors, whose determinants yield the local factors.In this paper we show how such linearizations, henceforth labelled refined Artin conductors, allow to split higher dimensional global ε-factors into a finite product of local contributions.The resulting higher local ε-factors arise as iteration of vanishing cycle functors and of these refined Artin conductors.Our main result is the following: Theorem 1.2.Let k be a perfect field of positive characteristic, and let X be a proper k-scheme.Let Λ be either F ℓ or Q ℓ , where ℓ is a prime number invertible in k.Then there exists a collection (E(X (x) , −)) x∈|X| , of triangulated functors E(X (x) , −) : D b c (X (x) , Λ) → D b c (x, Λ), indexed by the set of closed points of X, where we denoted by X (x) the henselization of X at a point x, such that: (1) for any object F of D b c (X, Λ), we have E(X (x) , F |X (x) ) ≃ 0 for all but finitely many closed points x of X; (2) for any closed point x of X, any object F of D b c (X (x) , Λ), and any object G of D b c (x, Λ), we have an isomorphism where sp : X (x) → x is the canonical specialization morphism, and this isomorphism is functorial in F and G; (3) for any object F of D b c (X, Λ), we have where G k = Gal(k/k) is the Galois group of an algebraic closure of k and G x is the Galois group of k/k(x).
This theorem will be proved in Section 5.It will be clear from the proof that the resulting functors E(X (x) , −) are not uniquely determined by the conditions in the conclusion of Theorem 1.2.The factorisation of ε k (X, F ) in Theorem 1.2 can be written as x/k det(E(X (x) , F |X (x) ) • ver x/k , where δ x/k : G k → {±1} is the signature homomorphism for the left action of G k on G k /G k(x) , and ver x/k : G ab x → G ab k is the transfer homomorphism associated to the finite index subgroup G x of G k .The factors det E(X (x) , F |X (x) ) , thus play the role of local ε-factors in higher dimension.

Theorem 1.2 admits the following immediate consequence:
Theorem 1.4.Let X be a proper k-scheme.Let F 1 and F 2 be objects of D b c (X, Λ) such that F 1|X (x) is isomorphic to F 2|X (x) for any closed point x of X.Then we have The conclusion of 1.4 regarding Euler characteristics is originally due to Deligne, whose proof was written by Illusie in [Ill81].The part concerning ε-factors appears to be new, though the case of a finite base field alternatively follows from the Grothendieck-Lefschetz trace formula, as was pointed out to the author by Takeshi Saito.
Theorem 1.5.Let S be a henselian trait of equicharacteristic p, with closed point s such that k(s) is perfect.Let f : X → S be a proper morphism and let F 1 and F 2 be objects of D b c (X, Λ) such that F 1|X (x) is isomorphic to F 2|X (x) for any closed point x of the special fiber X s .Then we have for any non zero meromorphic 1-form ω on S.Here we denoted by a(S, −) the Artin conductor, cf.([Gu19], 7.2), and by ε s (S, −, ω) the geometric local ε-factor, cf.
This a consequence of Theorem 6.1, whose proof relies on Theorem 1.2.
Theorem 1.6 (Twist Formula).Let X be a proper k-scheme and let (E(X (x) , −) x∈|X| be as in Theorem 1.2.Let F be an object of D b c (X, Λ), and let G be a Λ-local system of constant rank r on X.We then have This is an immediate consequence of Theorem 1.2.A similar twist formula was obtained in [UYZ] in the case of a projective smooth scheme over a finite field.Let us set as a 0-cycle on X. Theorem 1.2 yields deg(cc X (F )) = χ(X, F ), and the conclusion of Theorem 1.6 can be written as where the pairing −, − is defined by for any Λ-local system L of rank 1 on X and any 0-cycle c = i n i [x i ].When k is finite and when X is projective smooth, then the main result of [UYZ] and the class field theory of Kato-Saito [KS] yield that cc X (F ) coincides in the Chow group CH 0 (X) with the characteristic class defined in ([Sa17], Def.5.7).When k is arbitrary and X is projective smooth, one can simply assert that the difference between cc X (F ) and Saito's characteristic class belong to the kernel of the pairing 1.6.1.As in ([UYZ], 5.26), this implies that the formation of Saito's characteristic class on projective smooth k-schemes commutes with proper pushforward, up to a 0-cycle in the kernel of the pairing 1.6.1.Finally, we notice that Theorem 1.2 actually yields a slightly stronger twist formula: Theorem 1.7.Let X be a proper k-scheme and let (E(X (x) , −) x∈|X| be as in Theorem 1.2.Let r be an integer.Let F 1 and F 2 be objects of D b c (X, Λ) such that for any closed point x of X, we have F 2|X (x) ≃ F 1|X (x) ⊗ G x for some free Λ-module G x of rank r endowed with an admissible action of Gal(k/k(x)).We then have 1.8.Theorem 1.2, as well as its consequences 1.4, 1.5, 1.6, 1.7, can be generalized to (complexes of) étale ℓ-adic sheaves twisted by a 2-cocycle on G k , cf. ([Gu19], Sect.3).We choose to restrict the exposition to non twisted sheaves for the sake of simplicity and concision.Let us however notice that in the twist formula 1.6, we can allow G to be a twisted local system, and this ultimately yields a more precise conclusion in the discussion following Theorem 1.6, regarding the commutation of the formation of Saito's characteristic class with proper pushforward.
1.9.Let us now describe the organization of this paper.In Section 2, we give a few sorites on Galois equivariant étale sheaves on schemes.We also study transformations of such sheaves afforded by Λ-linear contractions, cf.2.11.
In Section 3, we give a brief review of nearby and vanishing cycles functors, as presented in ([SGA7], XIII), and we state a few results regarding the composition of these functors with a Λ-linear contraction.
In Section 4, we review the theory of Gabber-Katz extensions, and we use it to provide three families of Λ-linear contractions: Katz's cohomological construction of the Swan module from ([Ka86], 1.6), cf.4.5, a functor refining geometric class field theory, cf.4.6, and a refined Artin conductor, cf.4.8, which linearizes Artin conductors and geometric local ε-factors.Only the second and the third of these constructions will be used in the proof of our main results, though Katz's construction was inspirational to us.
Finally, we prove the main Theorem 1.2 in Section 5. We first provide a stronger result in the case of a projective line in 5.1, by using the product formula from [Gu19].We then prove in 5.2 and 5.3 using Chow's lemma that it is sufficient to prove Theorem 1.2 in the case of a projective space.The latter case is handled in 5.4 by choosing an arbitrary pencil and by applying to the latter the product formula from 5.1, as well as the functoriality properties from Section 3. 1.10.Acknowledgements.This work was prepared at the Institut des Hautes Études Scientifiques and the École Normale Supérieure while the author benefited from their hospitality and support.This text is a continuation of the part [Gu19] of the author's doctoral thesis, realized under the supervision of Ahmed Abbes.The author would like to thank Ahmed Abbes, Lei Fu, Ofer Gabber, Dennis Gaitsgory, Adriano Marmora, Deepam Patel, Takeshi Saito, Daichi Takeuchi, Enlin Yang for useful discussions.1.11.Conventions and notation.Throughout this paper, we fix distinct prime numbers p, ℓ.We fix a non trivial homomorphism ψ : F p → Z ℓ × , and every ℓ-adic coefficient ring Λ will be assumed to contain the image of ψ.We denote by L ψ the corresponding Artin-Schreier sheaf on A 1 Fp .For any F p -scheme X and any global function f in Γ(X, O X ), we denote by L ψ {f } the pullback of L ψ by the morphism f : X → A 1 Fp .
We write "qcqs" for "quasi-compact quasi-separated", and for any scheme S we denote by Ét S the category of qcqs étale S-schemes, and by S ét the étale topos of S, namely the topos of sheaves of sets on the site Ét S .

Preliminaries
In this section, we record various sorites regarding equivariant sheaves and transformations thereof.Throughout this section, we let s be the spectrum of a field and we let s be a separable closure of s.We denote by G s the Galois group of k(s) over k(s).In particular, the group G s acts on the right on s.
2.1.Let θ : Q → G s be a continuous homomorphism of profinite groups and let Y be an s-scheme.Definition 2.2 ([SGA7] XIII 1.1).Let Λ be a ring.Let F be an étale sheaf of sets (resp. of Λ-modules) on Y s .
(2) An action of Q compatible with θ on F is continuous if for any qcqs étale Y -scheme U , the resulting left action of Q on F (U s ) is continuous, when the latter is endowed with the discrete topology.
no confusion can arise from this abuse of notation, the category of étale sheaves (resp.étale sheaves of Λ-modules) on Y s endowed with a continuous action of Q compatible with θ.The morphisms in this category are the morphisms of sheaves on Y s which commute with the action of Q.
If one takes Y to be the base scheme s, then the category Sh(Y, Q, θ) (resp.Sh(Y, Q, θ, Λ)) is simply the category of sets (resp. of Λ-modules) endowed with a continuous left action of Q, which we also denoted by Remark 2.3.One can alternatively describe Sh(Y, Q, θ) (resp.Sh(Y, Q, θ, Λ)) as the product topos BQ × s ét Y ét (resp.as the category of Λ-modules in the topos BQ × s ét Y ét ), where BQ is the topos of sets endowed with a continuous left action of Q.In particular, the abelian category Sh(Y, Q, θ, Λ) has enough injectives.

This is ([SGA7] XIII 1.1.3(ii))
. A quasi-inverse to this functor can be described as follows: to any object F of Sh(Y, G s , id) one associates the étale sheaf on Y whose sections on a quasi-compact étale Y -scheme U are given by the G s -invariants F (U s ) Gs .2.5.Let Λ be a ring.Let Y be an s-scheme and let s → s ′ → s be a factorization of s → s, where s ′ is a finite extension of s.Let us consider a commutative diagram of continuous homomorphisms of profinite groups, with injective vertical arrows.We then have a restriction functor Then for any j > 0, the Čech cohomology group H j (U, F ) vanishes.
Since I is finite, and since any object of Ét Y s is qcqs, there exists a factorization s → s ′ → s as in 2.5 and a cover For any open subgroup where the last nonzero term is placed in degree 0, cf.Example 2.6 for the notation.If Č• (U, F ) is the Čech complex of F with respect to the cover U, then the natural homomorphism ) and by exactness of the induction functor Ind Q Q ′ .By applying the exact functor Hom(−, F ), this yields that the natural homomorphism from

The conclusion then follows by taking the colimit over all open subgroups
2.8.Let θ : Q → G s be a continuous homomorphism of profinite groups and let Λ be a ring.Let f : Y ′ → Y be a quasi-compact morphism of s-schemes.For any object F of Sh(Y, Q, θ, Λ), with corresponding action (σ(q)) q∈Q , the data (f * σ(q)) q∈Q is a continous action of Q on f * F compatible with θ.We thus obtain a functor We similarly have a functor Remark 2.9.With the description from Remark 2.3, this pair of adjoint functors is induced by the morphism of toposes id 2.10.Let θ : Q → G be a surjective continuous homomorphism of profinite groups and let I be the its kernel.Let Λ be a ring.For any element q of Q and any object M of I − Mod Λ , we denote by q * M the object of I − Mod Λ with underlying Λ-module M , on which an element t of I acts as m → qtq −1 m.This yields a Λ-linear functor q * : I − Mod Λ → I − Mod Λ , and we have (q 1 q 2 ) * = q * 2 q * 1 .For any element q of I, we have a Λ-linear natural transformation λ q : id → q * , given on an object M of I − Mod Λ by the homomorphism m → qm.
Definition 2.11.A Λ-linear (Q, I)-contraction is a Λ-linear functor endowed with a collection τ = (τ (q)) q∈Q of Λ-linear natural transformations τ (q) : T q * → T, satisfying the following conditions: (1) the functor T is exact and commutes with small filtered colimits; (2) if an object M of I − Mod Λ is finitely generated as a Λ-module, then T (M ) is a finitely generated Λ-module, and there exists an open subgroup Q ′ of Q such that for any element q of Q ′ , we have q * M = M and τ (q) M = id T (M) ; (3) for any elements q 1 , q 2 of Q, then the natural transformation τ (q 1 ) • τ (q 2 )q * 1 from T q * 2 q * 1 to T coincides with τ (q 1 q 2 ); (4) for any element q of I, we have τ (q) • T (λ q ) = id T .
We will give non trivial examples of Λ-linear (Q, I)-contractions in the next section, cf.4.5, 4.6 and 4.8.2.12.Let θ : Q → G be a surjective continuous homomorphism of profinite groups with kernel I and let Λ be a ring.Let (T, τ ) be a Λ-linear (Q, I)-contraction, cf.2.11.Let G ′ be a closed subgroup of G and let Q ′ be its inverse image by θ.Then one can construct a functor as follows.For any element q of Q ′ , we have a Λ-linear natural transformation λ q : id → q * , of endofunctors of Q ′ − Mod Λ , cf. 2.10.We then obtain a Λ-linear natural transformation where φ is the forgetful functor from Q ′ − Mod Λ to I − Mod Λ .Lemma 2.13 below then ensures that for any object M of Q ′ − Mod Λ , one obtains a continuous left action of Q ′ on T (M ) by letting an element q of Q ′ act as (τ (q) • T (λ q )) M .By the axiom 2.11 (4), this action factors through θ and therefore yields an object of G ′ − Mod Λ .Lemma 2.13.For any elements q 1 , q 2 of Q ′ , we have Moreover, for any object M of Q ′ − Mod Λ and element m of T (M ), there exists an open subgroup Q ′′ of Q ′ such that τ (q)T (λ q )m = m for any element q of Q ′′ .
The first assertion follows from the identities The first equality stems from the fact that τ (q 2 ) is a natural transformation, the second follows from 2.11 (3), and the third identity is tautological.For the last assertion of Lemma 2.13, we can assume by 2.11 (1) that M is finitely generated as a Λ-module, and the conclusion then follows from 2.11 (2).
2.14.Since T and T G are exact functors, we still denote by T and T G the functors induced by them on the corresponding derived categories.
Proposition 2.15.We have a bifunctorial isomorphism We only prove the first assertion, the second one being similar.Let C be the Λ-linear abelian category of bounded above chain complexes with terms in the full subcategory of Mod Λ consisting of Λ-modules of the form i<κ Λ for some cardinal κ.Since T is Λ-linear and commutes with small direct sums by 2.11 (1), we have a functorial isomorphism and N in C.This isomorphism factors through the quotient D − (Mod Λ ) of C since for any homomorphism λ : N 1 → N 2 in C, which is homotopic to 0, the corresponding homomorphisms in D − (Mod Λ ).In particular, the complex T (M ) L ⊗ Λ N is acyclic in positive degrees, hence the flatness of M .2.17.Let θ : Q → G s be a surjective continuous homomorphism of profinite groups with kernel I and let Λ be a ring.Let us consider a Λ-linear (Q, I)-contraction (T, τ ), cf.2.11.Let Y be an s-scheme.Definition 2.18.For any object F of Sh(Y, Q, Λ), we define T (F ) to be the étale sheaf of Λ-modules on Y whose sections on a qcqs étale Y -scheme U are given by the G s -invariants T Gs (F (U s )) Gs (cf.2.12).
The mentioned presheaf is indeed a sheaf by left exactness of the functor M → T Gs (M ) Gs from Q − Mod Λ to the category of Λ-modules.
Proposition 2.19.We have a bifunctorial isomorphism Ler us consider a factorization s ϕ0 −→ s ′ → s where s ′ is a finite extension of s, and an object U ′ of Ét Y s ′ , such that U is isomorphic to U ′ × s ′ ,ϕ0 s.We henceforth assume that U is equal to U ′ × s ′ ,ϕ0 s.We have a decomposition and in particular Taking into consideration the action of G s on the left hand side, cf.2.12, we obtain an isomorphism , where s ′′ is a finite extension of s ′ , then replacing s ′ and U ′ by s ′′ and U ′ × s ′ s ′′ above yields By taking the colimit over s ′′ , we obtain Corollary 2.20.For any geometric point ȳ of Y , we have a natural isomorphism for F in Sh(Y, Q, Λ).In particular, the functor is exact and commutes with small filtered colimits.
Indeed, we can consider ȳ as a geometric point of Y s and the conclusion follows by taking the (filtered) colimit of the isomorphisms with notation as in 2.8, is an isomorphism.This is an immediate consequence of 2.20.Proposition 2.22.Let F be an injective object in Sh(Y, Q, Λ).Then for any U in Ét Ys and any j > 0, we have H j (U, T (F )) = 0.
For any étale cover U = (U i → U ) i∈I in Ét Ys with I finite, the Čech complex Č• (U, F ) is acyclic in nonzero degree by Proposition 2.7.By Proposition 2.19 and by exactness of T , we obtain that is acyclic as well in nonzero degree.The conclusion then follows from ([Stacks] 03F9).
2.23.Let θ, Λ and (T, τ ) be as in 2.17.Let f : Y ′ → Y be a qcqs morphism of s-schemes.For any object U of Ét Y , the fiber product U × Y Y ′ is an object of Ét Y ′ .Moreover, for any object F of Sh(Y, Q, Λ), we have, with the notation from 2.8, Proposition 2.24.We have a natural isomorphism Indeed, for any injective object F of Sh(Y ′ , Q, Λ), the object f * F of Sh(Y, Q, Λ) is T -acyclic by exactness of T , cf. 2.20.Moreover, for any integer j > 0 the restriction of R j f * T (F ) to Y s is the sheaf associated to the presheaf which vanishes by 2.22.Thus T (F ) is f * -acyclic.This implies that the canonical transformations are isomorphisms, and the conclusion then follows from the equality

Nearby and vanishing cycles
Let S be a scheme and let s be a point of S such that the local ring O S,s is a discrete valuation ring.We denote by s a separable closure of s, by η the generic point of the henselization S (s) of S at s, and by η a separable closure of the generic point η s of the strict henselization S (s) .
3.1.Let us denote by G η and G s the Galois group of the extensions η → η and s → s respectively.We have a natural specialization homomorphism θ : G η → G s .This homomorphism is surjective and we denote by I η its kernel, so that we have an exact sequence of profinite groups.In this section, we apply the definitions and results of Section 2 to Q = G η and θ as above.
3.2.Let f : X → S be an S-scheme.Let us consider the commutative diagram of S-schemes, where all squares are cartesian.

Definition 3.3 ([SGA7]
, XIII 1.3.2).Let Λ be a ring and let F be an object of Sh(X, Λ).We denote by Ψ s f (F ) the object of Sh(X s , G η , Λ), cf.2.2, obtained the pullback by i of the sheaf of Λ-modules on X × S S (s) whose sections on an object U of Ét X×S S (s) are given by Γ(U × S (s) η, F ).

This defines a left exact functor
thereby inducing a right derived functor , namely the nearby cycles functor, cf.([SGA7], XIII 2.1.1).3.4.Let i : X s → X be the natural closed immersion.The natural cospecialization homomorphism We denote its cone by ), the vanishing cycles functor.We thus have a functorial distinguished triangle for F in D + (X, Λ).
3.5.Let Λ be a noetherian ring annihilated by an integer invertible on s.Let (T, τ ) be a Λ-linear (G η , I η )-contraction, cf.2.11.The composition of the functor T from 2.14 with the vanishing cycles functor from 3.4 yields a triangulated functor We now reference a few of its properties.
Proposition 3.6.Let F be an object of D b c (X, Λ).If f is locally of finite type, then T RΦ s f belongs to D b c (X s , Λ).Under these assumptions, it follows from ([SGA 4 1 2 ], 7.3.2) that RΨ s f (F ) belongs to D b c (X s , Λ), hence RΦ s f (F ) belongs as well to D b c (X s , Λ) and so does T RΦ s f (F ) by 2.26 (2).Proposition 3.7.Let h : X ′ → X be a smooth morphism of S-schemes and let F be an object of D + (X, Λ).Then the natural homomorphism ), is an isomorphism, where h s : X ′ s → X s is the morphism induced by h.In particular, if f is smooth then T RΦ s f (Λ) vanishes.This follows from the corresponding property of RΨ s f , cf. ([SGA7], XIII 2.1.7.1), and from Corollary 2.21.Proposition 3.8.Let h : X ′ → X be a proper morphism of S-schemes and let F be an object of D + (X ′ , Λ).Then we have a natural homomorphism This follows from the corresponding property of RΨ s f , cf. ([SGA7], XIII 2.1.7.2), and from Proposition 2.24 since h is qcqs.
(1) If Λ is the ring of integers in a finite extension of Q ℓ , with maximal ideal m, we set (2) If Λ is a finite extension of Q ℓ , with ring of integers Λ 0 , and if M 0 is an I ηs -stable finitely generated sub-Λ 0 -module such that ΛM 0 = M , we set Let f : X → S be a morphism locally of finite type, with ℓ-invertible on S. Proposition 3.6 allows to extend the functors from 3.10 to triangulated functors , for any admissible ℓ-adic coefficient ring, in such a way that its formation commutes with base change along homomorphisms Λ → Λ ′ of admissible ℓ-adic coefficient rings, and so that it coincides with the functor T RΦ s f from 3.5 when Λ is a finite Z ℓ -algebra.Propositions 3.7 and 3.8 still hold in this context.

Gabber-Katz extensions and examples of contractions
In this section, we provide examples of linear contractions, cf.2.11, whose construction are based on the theory of Gabber-Katz extensions from [Ka86].We review the latter in 4.1 by placing these extensions in what we believe to be their rightful context, namely as pullbacks along a morphism of toposes, the canonical retraction, cf 4.4.Throughout this section, we fix a perfect field k of characteristic p > 0.
4.1.Let S be a smooth k-scheme purely of dimension 1, and let s be a closed point of S. Thus the henselization S (s) of S at s is a henselian trait whose residue field k(s) is a finite extension of k.Let π be a uniformizer of S (s) , thereby inducing a morphism s , the Gabber-Katz extension functor, cf.([Gu19], 4.2): for any étale S (s) -scheme U , the étale A 1 sscheme π ♦ U is the unique (up to isomorphism) such A 1 s -scheme whose pullback by π is isomorphic to U and whose restriction along the open immersion G m,s → A 1 s is a special finite étale G m,s -scheme, namely it is tamely ramified above ∞ and its geometric monodromy group has a unique p-Sylow, cf.([Gu19], 4.3).We will also label special an étale A 1 s -scheme whose restriction along the open immersion G m,s → A 1 s is special, or equivalently an object of the essential image of π ♦ .Proposition 4.2.The functor π ♦ has the following properties: (1) it sends a final object of Ét S (s) to a final object of Ét A 1 s ; (2) for any covering (U i → U ) i∈I of the site Ét S (s) , the family (π s is special and its pullback by π is S (s) , hence π ♦ S (s) isomorphic to A 1 s .This proves (1).In order to prove (2), we can assume that U is connected, since the functor π ♦ commutes with finite disjoint unions.Moreover, we have that be the morphism under consideration.Then π −1 α is an isomorphism, and we infer from the characterization of π ♦ , cf. 4.1, that (3) holds if and only if s -scheme.Since π ♦ (U ) and π ♦ (W ) are tamely ramified above infinity, so is X.Moreover, the geometric monodromy group of j −1 X → G m,s is a subgroup of the product of those of j −1 π ♦ (U ) and j −1 π ♦ (U ), and the class of finite groups which admit a unique p-Sylow is stable by finite products and subobjects, hence the geometric monodromy group of j −1 X → G m,s belongs to that class.Thus X is a special A 1 s -scheme, and (3) holds.Corollary 4.3.The functor π ♦ : Ét S (s) → Ét A 1 s is a morphism of sites.This follows from Proposition 4.2 and from the fact that Ét S (s) is an essentially small category.Definition 4.4.Let (S, s, π) be as in 4.1.The Gabber-Katz retraction associated to this data is the morphism of toposes r s,π : A 1 s,ét → S (s),ét associated to the morphism of sites π ♦ , cf. 4.3.The composition of r s,π with the morphism π : S (s),ét → A 1 s,ét of toposes induced by π, is (equivalent to) the identity morphism of S (s),ét , hence the terminology "retraction".The pair r s,π = (r −1 s,π , r s,π * ) of adjoint functors can be described as follows.For an object F of A 1 s,ét , the object r s,π * F is the sheaf on Ét S (s) given by U → F (π ♦ U ).Moreover, the functor r −1 s,π : S (s),ét → A 1 s,ét is the Gabber-Katz extension functor, sending a representable sheaf U to the representable sheaf π ♦ U .4.5.Let Λ be an admissible ℓ-adic coefficient ring, cf.([Gu19], 2.2), and let (S, s, π) be as in 4.1.Let η s be the generic point of S (s) , let s be a separable closure of s, and let η s be a separable closure of η s .Let G ηs and G s be the Galois group of the extensions s → s and η s → η s .Let I ηs be the kernel of the sujective specialization homomorphism G ηs → G s .For an object M of G ηs − Mod Λ , we denote by M ! the extension by zero to S (s) of the corresponding sheaf on η s .In ( [Ka86], 1.6),Katz considered the functor defined by Sw(M ) = H 1 (A 1 s , r −1 s,π M ! ), for M in I ηs − Mod Λ , where r s,π is the canonical retraction from 4.4.This is naturally a Λ-linear (G ηs , I ηs )-retraction if Λ is finite, and yields a compatible system as in 3.9 in general.When Λ is a field, this functor has the remarkable property that rk(Sw(M )) = sw(M ), where sw(M ) is the Swan conductor of M , hence Sw can be considered as a refined Swan conductor.This particular Λ-linear contraction will not be appear in the rest of this text.4.6.We retain the notation from 4.5.Let us set π = (1 − π −1 ) −1 .The uniformizer π is uniquely determined by π, since we have π = (1 − π−1 ) −1 .Let 1 : s → A 1 s be the s-point 1 of A 1 s .Let us set CFT π (M ) = r −1 s,π (M ! ) 1 , for M in I ηs − Mod Λ .This is naturally a Λ-linear (G ηs , I ηs )-retraction if Λ is finite, and yields a compatible system as in 3.9 in general.The functor CFT π can be considered as constituting a refined class field theory for η s , as indicated by the following result.4.8.We still retain the notation from 4.5.Let Λ be an admissible ℓ-adic coefficient ring, cf.([Gu19], 2.2).The profinite group I ηs admits a unique p-Sylow subgroup P , the wild inertia group associated to (S (s) , η s ).Definition 4.9.A continuous representation of G ηs on a Λ-module M is totally wildly ramified if the submodule M P of P -invariants elements vanishes.
Example 4.10.Let ψ : F p → Λ × be a non trivial homomorphism and let f be an element of k(η s ) whose valuation is negative and prime to p. Then the continuous representation of G η corresponding to L ψ {f } is totally wildly ramified.The latter assertion holds as well for F ⊗ L ψ {f } for any tamely ramified lisse étale sheaf of Λ-modules F on η, since P acts trivially on F η .Proposition 4.11.Let M be a Λ-module endowed with a continuous Λ-linear left action of G η , and let F be the corresponding lisse étale sheaf of Λ-modules on η.If M is totally wildly ramified (cf.4.9) then the canonical homomorphism This amounts to the acyclicity of the fiber of Rj * F at s, i.e. to the vanishing of H j (I ηs , M ) for each integer j.We have the Hochschild-Serre spectral sequence H i (I ηs /P, H j (P, M )) ⇒ H i+j (I ηs , M ).
The groups H j (P, M ) vanish for j = 0 since P is a p-group and M is of ℓ-torsion, while H 0 (P, M ) = M P vanishes since M is assumed to be totally wildly ramified.Thus H j (P, M ) vanishes for any j, and consequently so does H j (I ηs , M )., ι * G). 5.4.Let us prove Theorem 1.2.By 5.2 and 5.3, it is sufficient to consider the case where X = P(V ) is the projective space parametrizing hyperplanes in a k-vector space V .We denote by d + 1 the dimension of V , and we argue by induction on d, the case d = 0 being immediate.We henceforth assume that d is greater than 0.

Let
Let e 1 , e 2 be a basis of a 2-dimensional sub-k-vector space W of V and let X ′ be the flag variety parametrizing pairs ([t 1 : t 2 ], H), where [t 1 : t 2 ] is a point of P 1 k and H is an hyperplane in V containing t 1 e 1 + t 2 e 2 .Let f : X ′ → P 1 k and r : X ′ → P(V ) be the first and second projections respectively.Let F be an object of D b c (P(V ), Λ).The morphism r is an isomorphism outside the closed subscheme ι : P(V /W ) → P(V ), and is a P 1 -bundle above P(V /W ), hence where G = Rf * r −1 F .Moreover, by 3.8 we have a functorial isomorphism k , the fiber X ′ s is the projective space associated to the quotient of V ⊗ k k(s) by the k(s)-line k(s)(t 1 e 1 + t 2 e 2 ).In particular, the induction hypothesis applies to X ′ s , and we therefore have a collection (E(X ′ s,(x ′ ) , −)) x ′ ∈|X ′ s | of triangulated functors satisfying the conclusion of Theorem 1.2 for the s-scheme X ′ s .Thus One should note that r −1 F is universally locally acyclic with respect to f above a dense open subscheme of P 1 k by ([SGA 4 1 2 ], [Th.finitude] Th. 2.13), hence RΦ s f (r −1 F ) vanishes for all but finitely many closed points s of P 1 k .This and the induction hypothesis ensure that only finitely many terms contribute in the above sum.The conclusion then follows from 5.4.2 and from the induction hypothesis applied to P(V /W ); in particular, the twist formula 1.2(2) follows from 2.15 and the induction hypothesis.

Proof of Theorem 1.5
In this section, we prove Theorem 6.1 below, of which Theorem 1.5 is an immediate consequence.Theorem 6.1.Let Λ be either F ℓ or Q ℓ .Let S be a henselian trait of equicharacteristic p, with closed point s such that k(s) is perfect, and let ω be a nonzero meromorphic differential 1-form on S. Let f : X → S be a proper morphism.Then there exists a collection (E f,ω,x ) x∈|Xs| , of triangulated functors E f,ω,x : D b c (X (x) , Λ) → D b c (x, Λ), indexed by the set of closed points of the special fiber X s , such that: (1) for any object F of D b c (X, Λ), we have E f,ω,x (F |X (x) ) ≃ 0 for all but finitely many closed points x of X; (2) for any closed point x of X, any object F of D b c (X (x) , Λ), and any object G of D b c (x, Λ), we have an isomorphism where sp : X (x) → x is the canonical specialization morphism, and this isomorphism is functorial in F and G; (3) for any object F of D b c (X, Λ), we have Let π be a uniformizer of S, and let us write ω = αdπ for some nonzero element α of the function fields k(η) of S. We denote by v(α) the valuation of α in the discretely valued field k(η) In order to construct (E 1 f,x ) x∈|Xs| , we first notice that Corollary 4.15 and Proposition 3.8 yields that ε s (S, Rf * F , dπ) and a(S, Rf * F ) are respectively the determinant and rank of the complex Art π RΦ s id (Rf * F ) ∼ = RΓ(X s , Art π RΦ s f (F )).If (E(X s,(x) , −)) x∈|Xs| is as in Theorem 1.2 applied to the proper s-scheme X s , then we can set E 1 f,x (F ) = E(X s,(x) , Art π RΦ s f (F )).For (E 2 f,x ) x∈|Xs| , we can assume by additivity that α is a uniformizer on S, in which case Propositions 4.7 ensures that it is enough to set E 2 f,x (F ) = E(X s,(x) , CFT α RΨ s f (F )(−1)).Remark 6.2.Let X be a smooth projective curve of a field k of positive characteristic p and let F be an object of D b c (X, Λ).The product formula for curves expresses the determinant of RΓ(X k , F ) as a product of finitely many local ε-factors, which are invariants attached to Galois representations of 1-dimensional local fields.If k is itself a 1-dimensional local field, then the method described in this section decomposes the local ε-factor of RΓ(X k , F ) as a product of finitely many contributions, namely 2-dimensional local ε-factors, which are invariants attached to Galois representations of 2-dimensional local fields.By iterating this procedure, one can attach an n-dimensional local ε-factor to Galois representations of n-dimensional local fields, such that if k is an n-dimensional local field then the n-dimensional local ε-factor of RΓ(X k , F ) factors as the product of finitely many (n + 1)-dimensional local ε-factors.

Corollary 2 .
16. Let M be an object of I − Mod Λ which is flat as a Λ-module.Then T (M ) is a flat Λ-module.Indeed, for any Λ-module N , the complex M L ⊗ Λ N is quasi-isomorphic to M ⊗ Λ N [0] by flatness of M , hence Proposition 2.15 yields an isomorphism

from 2 .
19, where U runs over qcqs étale neighbourhoods of ȳ over Y s .Corollary 2.21.Let f : Y ′ → Y be a morphism of s-schemes.For any object F of Sh(Y, Q, Λ), the natural homomorphism
ι : X → P d k be a closed immersion into a projective space P d k , for some integer d.If Theorem 1.2 holds for P d k , so that we have a collection (E(P d k,(x) , −)) x∈|P d k | of triangulated functors, then Theorem 1.2 also holds for X by setting