Ind-constructible Weil sheaves

Chapter VIII: Ind-constructible Weil sheaves

8.1 Ind-constructible Weil sheaves

LetF^𝑞 be a finite field of characteristic 𝑝 > 0, and fix an algebraic closureF. Let 𝑋1, . . . , 𝑋_𝑛be schemes of finite type overF^𝑞. LetΛbe a condensed ring associated with the one of the following topological rings: a discrete coherent torsion ring (for example, a discrete finite ring), an algebraic field extension𝐸 ⊃ Q^ℓ, or its ring of integersO𝐸. We write𝑋 := 𝑋

1×_F𝑞. . .×_F𝑞 𝑋_𝑛, and denote by𝑋_𝑖,

F := 𝑋_𝑖×_F𝑞SpecF and 𝑋

F := 𝑋×_F𝑞 SpecFthe base change. Recall that under these assumptions, by Chapter 5 Item viii, we have a fully faithful embedding

Ind D_cons(𝑋

F,Λ)

−→D_indcons(𝑋

F,Λ) ⊂ D(𝑋

F,Λ), (8.1) and likewise for (ind-)lisse sheaves.

Definition 8.1.1. An object 𝑀 ∈ D(𝑋Weil

1 × . . . × 𝑋Weil

𝑛 ,Λ) is called ind-lisse, resp. ind-constructible if the underlying sheaf𝑀

F ∈D(𝑋

F,Λ)is ind-lisse, resp. ind- constructible in the sense of Definition 5.0.1.

We denote by

D_indlis 𝑋^Weil

1 ×. . .×𝑋^Weil

𝑛 ,Λ

⊂ D_indcons 𝑋^Weil

1 ×. . .× 𝑋^Weil

𝑛 ,Λ the resulting full subcategories of D(𝑋^Weil

1 ×. . .×𝑋^Weil

𝑛 ,Λ) consisting of ind-lisse, resp. ind-constructible objects. Both categories are naturally commutative algebra objects in Pr^St_Λ∗(see the notation from Chapter 4), that is, presentable stableΛ_∗-linear symmetric monoidal∞-categories whereΛ_∗ := Γ(∗,Λ)is the ring underlyingΛ. It is immediate from Definition 8.1.1 that the equivalence (6.6) restricts to an equivalence

D^• 𝑋^Weil

1 ×. . .× 𝑋^Weil

𝑛 ,Λ Fix

D^•(𝑋

F,Λ), 𝜙^∗

𝑋1

, . . . , 𝜙^∗

𝑋_𝑛

for• ∈ {indlis,indcons}.

Remark 8.1.2. Note that that we have a fully faithful embedding of 𝐷

cons(𝑋^Weil) intoD_indcons(𝑋^Weil) whose image consists of compact objects. However, the latter category is not generated by this image. Indeed, even in the case of a point, the ind-cons category consists ofΛ-modules with an action of an endomorphism, whereas the image of the embedding consists of Λ-modules with an action of an automorphism. This automorphism does not have to fix any finitely generated submodule, which would be the case for any objects generated by the image of the constructible Weil complexes.

Our goal in this chapter is to obtain a categorical Künneth formula for the categories of ind-lisse, resp. ind-constructible Weil sheaves. In order to state the result, we need the following terminology. Under our assumptions onΛ, each cohomology sheaf H^𝑗(𝑀), 𝑗 ∈Zfor 𝑀 ∈D_lis(𝑋

F,Λ

is naturally a continuous representation of the proétale fundamental groupoid𝜋^pro´et

1 (𝑋

F)on a finitely presentedΛ-module, see 6.5.4. Further, the projections𝑋

F → 𝑋_𝑖,

Finduce a full surjective map of topological groupoids

𝜋^pro´et

1 (𝑋

F) → 𝜋^pro´et

1 (𝑋

1,F) ×. . .×𝜋^pro´et

1 (𝑋_𝑛,

F). (8.2)

Definition 8.1.3. Let𝑀 ∈D(𝑋

F,Λ).

1. The sheaf 𝑀 is called split lisse if it is lisse and the action of 𝜋^pro´et

1 (𝑋

F) on H^𝑗(𝑀)factors through(8.2)for all 𝑗 ∈Z.

2. The sheaf𝑀is called split constructible if it is constructible and there exists a finite subdivision into locally closed subschemes 𝑋_𝑖,𝛼 ⊆ 𝑋_𝑖 such that for each 𝑋_𝛼 =Î

𝑖𝑋_𝑖,𝛼 ⊆ 𝑋, each restriction𝑀|𝑋𝛼 is split lisse.

Definition 8.1.4. An object𝑀 ∈D(𝑋^Weil

1 ×. . .×𝑋^Weil

𝑛 ,Λ)is called ind-(split lisse), resp. ind-(split constructible) if the underlying object𝑀

F ∈D(𝑋

F,Λ)is a colimit of split lisse, resp. split constructible objects.

As the category D^•(𝑋

F,Λ), • ∈ {indlis,indcons} is cocomplete, every ind-(split lisse) object is ind-lisse, and likewise, every ind-(split constructible) object is ind- constructible.

Theorem 8.1.5. Assume thatΛis either a finite discrete ring of prime-to-𝑝torsion, an algebraic field extension𝐸 ⊃ Q^ℓ forℓ ≠ 𝑝, or its ring of integersO𝐸. Then the functor induced by the external tensor product

D^•(𝑋^Weil

1 ,Λ) ⊗_ModΛ∗. . .⊗_ModΛ∗ D^•(𝑋^Weil

𝑛 ,Λ) → D^•(𝑋^Weil

1 ×. . .×𝑋^Weil

𝑛 ,Λ) (8.3) is fully faithful for • ∈ {indlis, indcons}. For • = indlis, resp. • = indcons the essential image contains the ind-(split lisse), resp. ind-(split constructible) objects.

Proof. For full faithfulness, it is enough to consider the case• = indcons. Using Lemma 4.2.5, it remains to show that the functor

Ì

𝑖

D_indcons 𝑋_𝑖,

F,Λ) Ind Ì

𝑖

D_cons(𝑋_𝑖,

F,Λ)

!

→D^•(𝑋

F,Λ) (8.4) is fully faithful. In view of Equation (8.1), this is immediate from the Künneth formula for constructibleΛ-sheaves as explained in Section 7.3.

To identify objects in the essential image, we note that the fully faithful functors Equation (8.3) and Equation (8.4) induce a Cartesian diagram (see Lemma 4.2.5):

Ë

𝑖D^• 𝑋Weil

𝑖 ,Λ) ^//

D^•(𝑋Weil

1 ×. . .×𝑋Weil 𝑛 ,Λ)

Ë

𝑖D^•(𝑋_𝑖,

F,Λ) ^//D^•(𝑋

F,Λ),

(8.5)

for• ∈ {indlis, indcons}. Thus, it is enough to show that the object 𝑀

Funderlying an ind-split object 𝑀 lies in the image of the lower horizontal arrow. Since this essential image is closed under colimits, it remains to show it contains the split lisse objects for•=indlis, resp. the split constructible objects for•=indcons.

By the full faithfulness of Equation (8.4), the split constructible case reduces to the split lisse case, see also the proof of 7.1.2 in Section 7.6. So assume•=indlis and let 𝑀

F ∈ D(𝑋

F,Λ) be split lisse. As each cohomology sheaf H^𝑗(𝑀

F), 𝑗 ∈ Zis at least ind-lisse, an induction on the cohomological length of 𝑀

Freduces us to show that H^𝑗(𝑀

F) lies in the essential image. By definition, being split lisse implies that the action of𝜋^pro´et

1 (𝑋

F)on H^𝑗(𝑀

F)factors through𝜋^pro´et

1 (𝑋

1,F) ×. . .×𝜋^pro´et

1 (𝑋_𝑛,

F). Then the arguments of Section 7.6 show that H^𝑗(𝑀

F)lies is in the essential image of the lower horizontal arrow in Equation (8.5). We leave the details to the reader. □ Remark 8.1.6. The functor Equation(8.3) is not essentially surjective in general.

To see this, note that the functor D_indcons(𝑋^Weil,Λ) → D_indcons(𝑋

F,Λ) admits a left adjoint 𝐹 that adds a free partial Frobenius action. Explicitly, for an object 𝑀 ∈ D_indcons(𝑋

F,Λ) the object 𝐹(𝑀) has underlying sheaf 𝐹(𝑀)_F given by a countable direct sum of copies of 𝑀. If 𝑀 was not originally in the image of the external tensor product, then 𝐹(𝑀) will not be either. This is, however, the only obstacle for essential surjectivity: as noted in the proof of Theorem 8.1.5, the diagram Equation(8.5)is Cartesian.