Partial-Frobenius stability

Chapter VI: Constructible Weil Sheaves

6.3 Partial-Frobenius stability

To give a reasonable definition of lisse and constructible Weil sheaves, we need to understand the relation between partial-Frobenius invariant constructible subsets in 𝑋 and constructible subsets in the single factors 𝑋_𝑖:

Definition 6.3.1. A subset 𝑍 ⊂ 𝑋 is called partial-Frobenius invariant if for all 1≤ 𝑖 ≤ 𝑛we haveFrob^𝑋𝑖(𝑍) =𝑍.

The composition Frob^𝑋₁ ◦ · · · ◦Frob^𝑋𝑛 is the absolute 𝑞-Frobenius on 𝑋 and thus induces the identity on the topological space underlying 𝑋. Therefore, in order to check that 𝑍 ⊂ 𝑋 is partial-Frobenius invariant, it suffices that, for any fixed𝑖, the subset 𝑍 is Frob^𝑋𝑗-invariant for all 𝑗 ≠ 𝑖. This remark, which also applies to 𝑋F = 𝑋

1×_F𝑞. . .×_F𝑞 𝑋_𝑛×_F𝑞 SpecF, will be used below without further comment.

We first investigate the case of two factors with one being a separably closed field.

This eventually rests on Drinfeld’s descent result (V. G. Drinfeld, 1987, Proposition 1.1) for coherent sheaves:

Lemma 6.3.2. Let𝑋 be a qcqsF^𝑞-scheme, and let𝑘/F^𝑞be a separably closed field.

Denote by𝑝: 𝑋_𝑘 → 𝑋the projection. Then 𝑍 ↦→ 𝑝⁻¹(𝑍)induces a bijection {constructible subsets in𝑋} ↔ {partial-Frobenius invariant, constructible subsets in𝑋_𝑘}

Proof. The injectivity is clear because 𝑝 is surjective. It remains to check the surjectivity. Without loss of generality we may assume that𝑘is algebraically closed, and replace Frob^𝑋 by Frob^𝑘 which is an automorphism. Given that 𝑍 ↦→ 𝑝⁻1(𝑍) is compatible with passing to complements, unions and localizations on 𝑋, we are reduced to proving the bijection for constructible closed subsets𝑍 and for𝑋 affine overF^𝑞. By Noetherian approximation (6.3.4), we reduce further to the case where 𝑋is of finite type overF^𝑞and still affine. Now we choose a locally closed embedding 𝑋 → P^𝑛_F𝑞 into projective space. A closed subset𝑍^′ ⊂ 𝑋_𝑘 is𝜙_𝑘-invariant if and only if its closure insideP^𝑛𝑘is so. Hence, it is enough to consider the case where𝑋 =P^𝑛_F𝑞 is the projective space. Let 𝑍^′ be a closed Frob^𝑘-invariant subset of 𝑋_𝑘. When viewed as a reduced subscheme, the isomorphism 𝜙_𝑘 restricts to an isomorphism of 𝑍^′. In particular,O𝑍^′ is a coherentO𝑋𝑘-module equipped with an isomorphism O𝑍^′ 𝜙^∗

𝑘O𝑍^′. Hence, Drinfeld’s descent result (V. G. Drinfeld, 1987, Proposition 1.1) (see also (Kedlaya, 2019, Section 4.2) for a recent exposition) yields 𝑍^′ = 𝑍_𝑘

for a unique closed subscheme𝑍 ⊂ 𝑋. □

The following proposition generalizes the results (Lau, 2004, Lemma 9.2.1) and (V. Lafforgue, 2018, Lemme 8.12) in the case of curves.

Proposition 6.3.3. Let 𝑋

1, . . . , 𝑋_𝑛 be qcqs F^𝑞-schemes, and denote 𝑋 = 𝑋

1 ×_F

𝑞

. . .×_F𝑞𝑋_𝑛. Then any partial-Frobenius invariant constructible closed subset𝑍 ⊂ 𝑋 is a finite set-theoretic union of subsets of the form𝑍

1×_F𝑞. . .×_F𝑞𝑍_𝑛, for appropriate constructible closed subschemes𝑍_𝑖 ⊂ 𝑋_𝑖.

In particular, any partial-Frobenius invariant constructible open subscheme𝑈 ⊂ 𝑋 is a finite union of constructible open subschemes of the form𝑈

1×_F𝑞. . .×_F𝑞𝑈_𝑛, for appropriate constructible open subschemes𝑈_𝑖 ⊂ 𝑋_𝑖.

Proof. By induction, we may assume 𝑛 = 2. By Noetherian approximation (Lemma 6.3.4), we reduce to the case where both 𝑋

1, 𝑋

2 are of finite type over F^𝑞. In the following, all products are formed overF^𝑞, and locally closed subschemes are equipped with their reduced subscheme structure. Let𝑍 ⊂ 𝑋

1×𝑋

2be a partial- Frobenius invariant closed subscheme. The complement𝑈 = 𝑋

1× 𝑋

2\ 𝑍 is also partial-Frobenius invariant.

In the proof, we can replace𝑋

1(and likewise𝑋

2) by a stratification in the following sense: Suppose 𝑋

1 = 𝐴^′⊔ 𝐴^′′ is a set-theoretic stratification into a closed subset 𝐴^′ with open complement 𝐴^′′. Once we know 𝑍 ∩ 𝐴^′× 𝑋

2 = Ð

𝑗 𝑍^′

1𝑗 × 𝑍^′

2𝑗 and 𝑍∩𝐴^′′×𝑋

2=Ð

𝑗𝑍^′′

1𝑗×𝑍^′′

2𝑗for appropriate closed subschemes𝑍^′

1𝑗 ⊂ 𝐴^′,𝑍^′′

1𝑗 ⊂ 𝐴^′′

and𝑍^′

2𝑗

, 𝑍^′′

2𝑗 ⊂ 𝑋

2, we have the set-theoretic equality 𝑍 =Ø

𝑗

𝑍^′

1𝑖×𝑍^′

2𝑗 ∪Ø

𝑗

𝑍^′′

1𝑗 ×𝑍^′′

2𝑗,

where 𝑍^′′

1𝑗 ⊂ 𝑋

1 denotes the scheme-theoretic closure. Here we note that taking scheme-theoretic closures commutes with products because the projections 𝑋

1 × 𝑋2 → 𝑋_𝑖 are flat, and that the topological space underlying the scheme-theoretic closure agrees with the topological closure because all schemes involved are of finite type.

The proof is now by Noetherian induction on 𝑋

2, the case 𝑋

2=∅being clear (or, if the reader prefers the case where 𝑋

2is zero dimensional reduces to Lemma 6.3.2).

In the induction step, we may assume, using the above stratification argument, that both𝑋_𝑖are irreducible with generic point𝜂_𝑖. We let𝜂

𝑖be a geometric generic point over𝜂_𝑖, and denote by 𝑝_𝑖: 𝑋

1×𝑋

2 → 𝑋_𝑖the two projections. Both𝑝_𝑖are faithfully

flat of finite type and in particular open, so that 𝑝_𝑖(𝑈) is open in 𝑋_𝑖. We have a set-theoretic equality

𝑍 = (𝑋

1\ 𝑝

1(𝑈)) ×𝑋

2

∪ 𝑋

1× (𝑋

2\𝑝

2(𝑈))

∪ 𝑍 ∩𝑝

1(𝑈) ×𝑝

2(𝑈) . Once we know𝑍∩𝑝

1(𝑈)×𝑝

2(𝑈) =Ð

𝑗𝑍

1𝑗×𝑍

2𝑗for appropriate closed𝑍_{𝑖 𝑗} ⊂ 𝑝_𝑖(𝑈), we are done. We can therefore replace𝑋_𝑖by𝑝_𝑖(𝑈)and assume that both𝑝_𝑖:𝑈 → 𝑋_𝑖 are surjective.

The base change𝑈×𝑋

2𝜂

2is a𝜙_𝜂

2-invariant subset of𝑋

1×𝜂

2. By Lemma 6.3.2, it is thus of the form𝑈

1×𝜂

2for some open subset𝑈

1 ⊂ 𝑋

1. There is an inclusion (of open subschemes of𝑋

1×𝜂

2):𝑈×𝑋

2𝜂

2 ⊂𝑈

1×𝜂

2. It becomes a set-theoretic equality, and therefore an isomorphism of schemes, after base change along𝜂

2 → 𝜂

2. By faithfully flat descent, this implies that the two mentioned subsets of𝑋

1×𝜂

2agree.

We claim𝑈

1 = 𝑋

1. Since the projection 𝑈 → 𝑋

2 is surjective, in particular its image contains 𝜂

2, so that𝑈

1 is a non-empty subset, and therefore open dense in the irreducible scheme 𝑋

1. Let𝑥

1 ∈ 𝑋

1be a point. Since the projection𝑈 → 𝑋

1

is surjective,𝑈∩ ({𝑥

1} ×𝑋

2)is a non-empty open subscheme of {𝑥

1} ×𝑋

2. So it contains a point lying over(𝑥

1, 𝜂

2). We conclude𝑋

1×𝜂

2 ⊂𝑈. We claim that there is a non-empty open subset 𝐴

2 ⊂ 𝑋

2such that 𝑋1× 𝐴

2 ⊂𝑈or, equivalently, 𝑋

1× (𝑋

2\ 𝐴

2) ⊃ 𝑋

1×𝑋

2\𝑈 . The underlying topological space of𝑉 = 𝑋

1× 𝑋

2\𝑈 is Noetherian and thus has finitely many irreducible components𝑉_𝑗. The closure of the projection𝑝

2(𝑉_𝑗) ⊂ 𝑋

2

does not contain𝜂

2, since𝑋

1×𝜂

2 ⊂ 𝑈. Thus, 𝐴

2 :=Ñ

𝑗 𝑋

1\ 𝑝

2(𝑉_𝑗) satisfies our requirements.

Now we continue by Noetherian induction applied to the stratification 𝑋

2 = 𝐴

2⊔ (𝑋

2\𝐴

2): We have𝑍∩𝑋

1×𝐴

2=∅, so that we may replace𝑋

2by the proper closed subscheme 𝑋

2\ 𝐴

2. Hence, the proposition follows by Noetherian induction. □ The following lemma on Noetherian approximation of partial Frobenius invariant subsets is needed for the reduction to finite type schemes:

Lemma 6.3.4. Let𝑋

1, . . . , 𝑋_𝑛be qcqsF^𝑞-schemes, and denote𝑋 = 𝑋

1×_F𝑞. . .×_F𝑞𝑋_𝑛. Let𝑋_𝑖 =lim^𝑗𝑋_{𝑖 𝑗} be a cofiltered limit of finite typeF^𝑞-schemes with affine transition maps, and write 𝑋 = lim^𝑗 𝑋_𝑗, 𝑋_𝑗 := 𝑋

1𝑗 ×_F𝑞 . . . ×_F𝑞 𝑋_{𝑛 𝑗} (see Let 𝑍 ⊂ 𝑋 be a constructible closed subset. Then the intersection

𝑍^′=

𝑛

Ù

𝑖=1

Ù

𝑚∈Z

Frob^𝑚𝑋𝑖

(𝑍)

is partial Frobenius invariant, constructible closed and there exists an index 𝑗 and a partial Frobenius invariant closed subset 𝑍^′

𝑗 ⊂ 𝑋_𝑗 such that 𝑍^′ = 𝑍^′

𝑗 ×𝑋𝑗 𝑋 as sets.

We note that each Frob^𝑋𝑖 induces a homeomorphism on the underlying topological space of 𝑋 so that 𝑍^′is well-defined. This lemma applies, in particular, to partial Frobenius invariant constructible closed subsets 𝑍 ⊂ 𝑋 in which case we have 𝑍 =𝑍^′.

Proof. As 𝑍 is constructible, there exists an index 𝑗 and a constructible closed subscheme 𝑍_𝑗 ⊂ 𝑋_𝑗 such that 𝑍 = 𝑍_𝑗 ×𝑋𝑗 𝑋 as sets. We put 𝑍^′

𝑗 = ∩^𝑛

𝑖=1 ∩_𝑚∈Z Frob^𝑚𝑋_{𝑖 𝑗}(𝑍_𝑗). As 𝑋_𝑗 is of finite type over F^𝑞, the subset 𝑍^′

𝑗 is still constructible closed. As partial Frobenii induce bijections on the underlying topological spaces, one checks that Frob^𝑚𝑋𝑖 𝑗(𝑍_𝑗) ×𝑋_𝑗 𝑋 = Frob^𝑚𝑋𝑖(𝑍) as sets for all 𝑚 ∈ Z. Thus, 𝑍^′=𝑍^′

𝑗 ×𝑋_𝑗 𝑋 which, also, is constructible closed because𝑋 → 𝑋_𝑗 is affine. □