Compatible systems and ramification

Abstract

We show that compatible systems of ℓ-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on ℓ-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

$$\ell $$-independence of the trace of local monodromy in a relative case (with an appendix by Qing Lu and Weizhe Zheng)

“…Thus we may assume G = {1}. We conclude the proof of Theorem A.1 by the last assertion of [5,Corollary 3.10] applied to R i h ! Q and R i h * Q .…”

“…In this "Appendix" we prove results on -independence and integrality of -adic cohomology over Henselian valuation fields with not necessarily discrete valuations, for the action of inertia and Weil subgroups of the Galois group. We deduce these results from relative results on compatible systems and integral sheaves over discrete valuation fields ( [7,25]) using the valuative criteria of [5]. Our result for inertia action (Theorem A.1) slightly generalizes Hiroki Kato's Theorem 6.1 and our method is different from his.…”

“…The rest of the proof is similar to the proof of [5,Theorem 1.4], except that the base here may have mixed characteristics. By standard limit arguments, there exists a finitely generated sub-algebra R ⊆ K 0 over Z such that X and the action of G are defined over B = Spec(R): there exists an algebraic space X of finite presentation over B, equipped with an action of G by B-automorphisms, such that we have a Gequivariant isomorphism X X × B Spec(K ) over K .…”

“…In equal characteristic case, there are some recent developments. In [4], Chiarellotto and Lazda discuss various forms of -independence including = p. Especially, in [4,Theorem 6.1], it is stated that, if X is a smooth proper variety over a local field of equal characteristic p > 0, then the trace of each element of the Weil group acting on each H i is independent of , including = p. Though their proof included a mistake as pointed out in the introduction of [5], it has been corrected in [6].…”

“…They proved that, for arbitrary S of characteristic p > 0 and for smooth proper X/L, the trace of each σ ∈ Gal(L/L) in Vidal's ramified part acting on the cohomology group of each degree is independent of = p [5, Theorem 1.4 and Remark 2.18 (3)]. Further, after the first version of this article was circulated, they informed us that Theorem 1.1.1 can be deduced from the results [5,Corollary 1.3] and [7,Proposition 4.15] on compatibility of -adic sheaves and that the assumptions on S and Spec L → S can be removed, i.e., Theorem 1.1.1 holds for an arbitrary scheme S and an arbitrary morphism Spec L → S with L being a field. We include their comments as an "Appendix".…”