CHAPTER 4

The Situation over a Finite Field

Let us now turn our attention to the case of a finite field k, and a groupscheme G/k which is a form of G_m. Concretely, it is either G_m/k itself, or it is the nonsplit form, defined in terms of the unique quadratic extension k₂/k inside the chosen k as follows: for any k-algebra A,

G(A) := {x ∈ A ⊗_k k₂| Norm_A⊗kk2/A(x) = 1}.

We denote by P_arith the full subcategory of the category Perv_arith of all perverse sheaves on G/k consisting of those perverse sheaves on G/k which, pulled back to G/k, lie in P. And we denote by Neg_arith the full subcategory of Perv_arith consisting of those objects which, pulled back to G/k, lie in Neg. Then once again we have

P_arith≅ Perv_arith/Neg_arith,

which endows P_arith with the structure of abelian category. Thus a sequence

of objects of P_arith is exact if and only if it is exact when pulled back to G/k; equivalently, if and only if α is injective, β is surjective, β ◦α = 0, and Ker(β)/Im(α) lies in Neg_arith.

An irreducible object in P_arith is an irreducible object in Perv_arith which is not negligible, i.e., whose pullback to G/k has nonzero Euler characteristic.

Middle convolution then endows P_arith with the structure of a neutral Tannakian category, indeed a subcategory of P such that the inclusion P_arith ⊂ P is exact and compatible with the tensor structure.

Recall [BBD, 5.3.8] that an object N in P_arith which is ι-pure of weight zero is geometrically semisimple, i.e., semisimple when pulled back to G/k. Recall also that the objects N in P_arith which are ι-pure of weight zero are stable by middle convolution and form a full Tannakian subcategory P_{arith,ι wt=0} of P_arith. [Indeed, if M and N in P_arith are both ι-pure of weight zero, then M £ N is ι-pure of weight zero on G_m × G_m, hence by Deligne’s main theorem [De-Weil II, 3.3.1] M_?!N is ι-mixed of weight ≤ 0, and hence M_?midN, as a perverse quotient of M_?!N, is also ι-mixed of weight ≤ 0, cf. [BBD, 5.3.1]. But the Verdier dual D(M_?midN) is the middle convolution DM_?midDN (because duality interchanges Rπ_! and Rπ_?), hence it too is ι-mixed of weight ≤ 0.]

This same stability under middle convolution holds for the objects N in P_arith which are geometrically semisimple, resp. which are arithmetically semisimple, giving full Tannakian subcategories P_arith,gss, resp. P_arith,ss. Thus we have full Tannakian subcategories

P_{arith,ι wt}=0 ⊂ P_arith,gss ⊂ P_arith

and

P_arith,ss ⊂ P_arith,gss ⊂ P_arith.

Also the objects which are both ι-pure of weight zero and arithmetically semisimple form a full Tannakian subcategory P_{arith,ι wt=0, ss}.

Pick one of the two possible isomorphisms of G/k with G_m/k, viewed as A¹/k 0. Denote by j₀ : G_m/k ⊂ A¹/k the inclusion, and denote by ! the fibre functor on P_arith defined by

N !(N) := H⁰(A¹/k, j_0!N).

The definition of this ! depends upon the choice of one of the two k-points at infinity on the complete nonsingular model, call it X, of G/k. If G/k is already G_m/k, then these two points at infinity on X are both k-rational. But if G/k is nonsplit, these two points at infinity are only k₂-rational, and they are interchanged by Frob_k. So it is always the case that Frob_k2 acts on !(N), but Frob_k may not act.

Theorem 4.1. Suppose that N in P_arith is ι-pure of weight zero and arithmetically semisimple. Then the following six conditions are equivalent.

(1) For j : G/k ⊂ X/k the inclusion of G/k into its complete non-singular model, the “forget supports” map is an isomorphism j_!N ≅ Rj_? N.

(2) The natural “forget supports” and restriction maps

H⁰_c(G/k, N) → !(N) → H⁰(G/k, N)

are both isomorphisms.

(3) The cohomology group !(N) is ι-pure of weight zero for the action of Frob_k2.

(3bis) The cohomology group H⁰_c(G/k, N) is ι-pure of weight zero for the action of Frob_k (or equivalently, for the action of Frob_k2).

(4) For every object M in <N>_arith, !(M) is ι-pure of weight zero for the action of Frob_k2.

(4bis) For every object M in <N>_arith, the cohomology group H⁰_c(G/k, M) is ι-pure of weight zero for the action of Frob_k (or equivalently, for the action of Frob_k2).

(5) For every object M in <N>_arith, the “forget supports” map is an isomorphism j_!M ≅ Rj_? M.

(6) For every object M in <N>_arith, the natural “forget supports” and restriction maps

H⁰_c(G/k, M) !(M) → H⁰(G/k, M)

are both isomorphisms.

When these equivalent conditions hold, the construction

M H⁰_c(G/k, M)

is a fibre functor on <N>_arith, on which Frob_k acts and is ι-pure of weight zero.

Proof. We first show that (1), (2), and (3) are equivalent. Since N is arithmetically semisimple, we reduce immediately to the case when N is arithmetically irreducible. If N is punctual, each of (1), (2), and (3) holds, so there is nothing to prove. So it suffices to treat the case when M is G[1] for an arithmetically irreducible middle extension sheaf G which is ι-pure of weight –1. As we saw in the proof of Lemma 3.3, we have four term exact sequences

0 → G^I(∞) → H⁰_c(G_m/k, G[1])→ H⁰(A¹/k, j_0!G[1]) → H¹(I(∞),G) → 0

and

0 → G^I(0) → H⁰(A¹/k, j_0!G[1]) → H⁰_c(G_m/k, G[1]) → H¹(I(∞);,G) → 0

In particular, we have an injection

G^I(0) ⊂ H⁰(A¹/k, j_0!G[1])

and a surjection

H⁰(A¹/k, j_0!G[1]) ³ H¹(I(∞), G).

Now (1) holds for G[1] if and only if we have either of the following two equivalent conditions:

(a) G^I(0) = 0 = H¹(I(0); G) and G^I(∞) = 0 = H¹(I(∞); G).

(b) G^I(0) = 0 = H¹(I(∞), G) = 0.

[That (a) and (b) are equivalent results from the fact that dim G^I(0) = dim H¹(I(0); G) and dim GI(∞) = dim H¹(I(∞); G).] So if (1) holds, then (a) holds, and the above four term exact sequences show that (2) holds. If (2) holds, then the “forget supports” map

H⁰_c(G/k, N) → H⁰(G/k, N)

is an isomorphism, which implies that each is ι-pure of weight zero (the source being mixed of weight ≤ 0 and the target being mixed of weight ≥ 0), and hence (3) holds. If (3) holds, we claim that (b) holds. Indeed, the four term exact sequences above give us an injection

G^I(0) ⊂ H⁰(A¹/k, j_0!G[1])

and a surjection

H⁰(A¹/k, j_0!G[1]) ³ H¹(I(∞), G).

But G^I(0) has ι-weight ≤ –1, so must vanish, and H¹(I(∞), G) has ι-weight ≥ 1, so also must vanish, exactly because !(M) = H⁰(A¹/k, j_0!G[1]) is ι-pure of weight zero.

We next show that (1) and (3bis) are equivalent. We have (1) implies (3bis) (the source is mixed of weight ≤ 0, and the target is mixed of weight ≥ 0). If (3bis) holds, we infer (1) as follows. The question is geometric, so we may reduce to the case where G is G_m.

Then on P¹ we have the short exact sequence

0 → j_!G → j_? G → δ₀ ⊗ V M δ_∞ ⊗ W → 0,

with V and W representations of Gal(k/k) which, by [De-Weil II, 1.81], are mixed of weight ≤ –1. Because G[1] has P, G has no constant subsheaf, and so the group H⁰(P¹, j_? G) vanishes. So the cohomology sequence gives an inclusion

V ⊕ W ⊂ H¹_c(G/k, G) := H⁰_c(G/k, N).

As the second group is pure of weight 0, we must have V = W = 0. Thus we have

j_!G ≅ j_?G,

and this in turn implies that

j_!G ≅ Rj_? G.

Each object M in <N>_arith is itself ι-pure of weight zero and arithmetically semisimple, so applying the argument above object by object we get the equivalence of (4), (4bis), (5), and (6). Trivially (5) implies (2). Conversely, if (2) holds, then (5) holds. Indeed, it suffices to check (5) for each arithmetically irreducible object M in <N>_arith, (i.e., for any irreducible representation of the reductive group G_arith,N), but any such M is a direct factor of some multiple middle convolution of N and its dual, so its !(M) lies in some !(N)^⊗r ⊗ (!(N)^∨)^⊗s, so is ι-pure of weight zero.

Once we have (4) and (6), the final conclusion is obvious.

Given a finite extension field E/k, and a Q^×_ℓ -valued character ρ of the group G(E), we have the Kummer sheaf L_ρ on G/E, defined by pushing out the Lang torsor 1 – Frob_E : G/E → G/E, whose structural group is G(E), by ρ. For L/E a finite extension, and ρ_L := ρ ◦ Norm_L/E, the Kummer sheaf L_ρL on G/L is the pullback of L_ρ on G/E. Pulled back to G/k, the Kummer sheaves L_ρ are just the characters of finite order of the tame fundamental group π^tame_i(G/k); here we view this group as the inverse limit, over finite extensions E/k, of the groups G(E), with transition maps provided by the norm.

We say that a character ρ of some G(E) is good for an object N in P_arith if this becomes true after extension of scalars to G/k, i.e., if the “forget supports” map is an isomorphism j_!(N ⊗ L_ρ) ≅ Rj (N ⊗ L_ρ). As an immediate corollary of the previous theorem, applied to N ⊗ L_ρ on G/E, we get the following.

Corollary 4.2. Suppose N in P_arith is ι-pure of weight zero and arithmetically semisimple. If a character ρ of some G(E) is good for N, then for every M in <N>_arith, H⁰_c(G/k, M ⊗L_ρ) is ι-pure of weight zero for Frob_E, and the construction

M H⁰_c(G/k, M ⊗L_ρ)

is a fibre functor on <N>_arith, on which Frob_E acts and is ι-pure of weight zero.