CHAPTER 3

Fibre Functors

At this point, we introduce the fibre functor suggested by Deligne. The proof that it is in fact a fibre functor is given in the Appendix.

Theorem 3.1. (Deligne) Denoting by

j₀ : G_m/k ⊂ A¹/k

the inclusion, the construction

M H⁰(A¹/k, j_0!M)

is a fibre functor on the Tannakian category P.

For any Kummer sheaf L_χ on G_m/k, the operation M M ⊗ L_χ is an autoequivalence of P with itself as Tannakian category. So we get the following corollary.

Corollary 3.2. For any Kummer sheaf L_χ on G_m/k, the construction

M H⁰(A¹/k, j_0!(M ⊗ L_χ))

is a fibre functor !_χ on the Tannakian category P.

Let us say that a Kummer sheaf L_χ on G_m/k is good for the object N of P if, denoting by j : G_m/k ⊂ P¹/k the inclusion, the canonical “forget supports” map is an isomorphism

Rj_!(N ⊗ L_χ) ≅ Rj_?(N ⊗ L_χ).

Lemma 3.3. Given a semisimple object N of P and a Kummer sheaf Lχ on G_m/k, the following conditions are equivalent.

(1) The Kummer sheaf L_χ is good for N.

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

H⁰_c(G_m/k, N ⊗ L_χ) → !χ(N) := H⁰(A¹/k, j_0!(N ⊗ L_χ))

and

!χ(N) := H⁰(A¹/k, j_0!(N ⊗ L_χ)) → H⁰(G_m/k, N ⊗ L_χ)

are both isomorphisms.

(3) The natural “forget supports” map is an isomorphism

H⁰_c(G_m/k, N ⊗ L_χ) ≅ H⁰(G_m/k, N ⊗ L_χ).

Proof. We reduce immediately to the case when N is irreducible. If N is δ_t for some point t ∈ G_m(k), then every χ is good for N, all three conditions trivially hold, and there is nothing to prove. Suppose now that N is G[1] for G an irreducible middle extension sheaf on G_m/k which is not a Kummer sheaf. Replacing N by N ⊗ L_χ, we reduce to the case when χ is the trivial character . Then (1) is the statement that the inertia groups I(0) and I(∞) acting on G have neither nonzero invariants nor coinvariants, i.e., that G^I(0) = H¹(I(0), G) = 0 and G^I(∞) = H¹(I(∞), G) = 0. We factor j as j_∞ ◦ j₀, where j₀ is the inclusion of G_m into A¹, and j_∞ is the inclusion of A¹ into P¹. We have a short exact sequence of sheaves

0 → j_!G = j_∞!j_0!G → j_∞?j_0!G → G^I(∞)→ 0,

where G^I(∞) is viewed as a punctual sheaf supported at ∞. We view this as a short exact sequence of perverse sheaves

0 → G^I(∞) → j_!G[1] = j_∞!j_0!G[1] → j_∞?j_0!G[1] → 0.

Similarly, we have a short exact sequence of perverse sheaves

0 → j_∞?j_0!G[1] → Rj_∞?j_0!G[1] → H¹(I(∞),G) → 0,

where now H¹(I(∞), G) is viewed as a punctual sheaf supported at ∞. Taking their cohomology sequences on P¹, we get short exact sequences

0 → G^I(∞) → H⁰_c(G_m/k, G[1]) → H⁰(P¹/k, j_∞? j_0!G[1]) → 0

and

0 → H⁰(P¹ /k, j_∞? j_0!G[1]) → H⁰(A¹/k, j_0!G[1]) → H¹(I(∞), G) → 0.

Splicing these together, we get a four term exact sequence

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

A similar argument, starting with Rj_∞?j_0!G[1], gives a four term exact sequence

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

These two four term exact sequences show the equivalence of (1) and (2). It is trivial that (2) implies (3). We now show that (3) implies (1).

Suppose (3) holds. Then the composition of the two maps

H⁰_c (G_m/k,N) → H⁰(A¹/k, j_0!N) → H⁰(G_m/k, N)

is an isomorphism. Therefore the first map is injective, and this implies that G^I(∞) = 0, which in turn implies that H¹(I(∞), G) = 0 (since G^I(∞) and H¹(I(∞), G) have the same dimension). And the second map is surjective, which gives the vanishing of H¹(I(0), G), and this vanishing in turn implies the vanishing of G^I(0).

As noted at the end of the last chapter, an irreducible object of P is just an irreducible perverse sheaf which lies in P, i.e., it is an irreducible perverse sheaf which is not a Kummer sheaf L_χ[1]. Similarly for the notion of a semisimple object of P; it is a direct sum of irreducible perverse sheaves, each of which lies in P (i.e., none of which is a Kummer sheaf L_χ[1]). Let us denote by P_ss the full subcategory of P consisting of semisimple objects. This is a subcategory stable by middle convolution (because given two semisimple objects M and N in P, each is a completely reducible representation of the Tannakian group G_geom,M⊕N. This group is reductive, because it has a faithful completely reducible representation, namely M ⊕ N. Then every representation of G_geom,M⊕N is completely reducible, in particular the one corresponding to M ?_mid N, which is thus a semisimple object in P. For this category P_ss, its inclusion into Perv is exact, and the Tannakian group G_geom,N attached to every N in P_ss is reductive.

We end this chapter by recording two general lemmas.

Lemma 3.4. For any perverse sheaf N on G_m/k, whether or not in P, the groups Hⁱ(A¹/k, j_0!N) vanish for i ≠ 0, and

dim H⁰(A¹/k, j_0!N) = χ(G_m/k, N) = χ_c(G_m/k, N).

Proof. Every perverse sheaf on G_m/k is a successive extension of finitely many geometrically irreducible ones, so we reduce to the case when N is geometrically irreducible. If N is punctual, some δ_a, the assertion is obvious. If N is G[1] for an irreducible middle extension sheaf, then Hⁱ(A¹/k, j_0!N) = Hⁱ⁺¹(A¹/k, j_0!G). The group H²(A¹/k,j_0!G) vanishes because an affine curve has cohomological dimension one, and the group H⁰(A¹/k, j_0!G) vanishes because G has no nonzero punctual sections on G_m. Once we have this vanishing, we have dim H⁰(A¹/k, j_0!N) = χ(A¹/k, j_0!N). Then χ(A¹/k, j_0!N) = χ_c(A¹/k, j_0!N) because χ = χ_c on a curve (and indeed quite generally, cf. [Lau-CC]). Tautologically we have χ_c(A¹/k, j_0!N) = χ_c(G_m/k, N), and again χ_c(G_m/k, N) = χ(G_m/k, N).

Lemma 3.5. For any perverse sheaf N on G_m/k, whether or not in P, the groups Hⁱ_c(G_m/k, N) vanish for i < 0 and for i > 1.

Proof. Using the long exact cohomology sequence, we reduce immediately to the case when N is irreducible. If N is punctual, the assertion is obvious. If N is a middle extension F[1], then F has no nonzero punctual sections, i.e., H^–1_c (G_m/k, F[1]) = H⁰_c(G_m/k, F) = 0; the groups Hⁱ_c(G_m/k, F[1]) = Hⁱ⁺¹_c (G_m/k, F) with i ≤ –2 or i ≥ 2 vanish trivially.