The topic of this post is a curious functor, discovered by Deligne, on the category of sheaves over the affine line, which is a “sheafification” of the Fourier transform for functions.

Recall that the classical Fourier transform is an almost-involution of the Hilbert space ${L^2(\mathbb{R})}$. We shall now discuss the Fourier-Deligne transform, which is an almost-involution of the bounded derived category of ${l}$-adic sheaves, ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$. The Fourier transform is defined by multiplying a function with a character (which depends on a parameter) and integrating. Analogously, the Fourier-Deligne transform will twist an element of ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$ by a character depending on a parameter, and then take the cohomology.

More precisely, consider the following: let $G$ be a LCA group, $G^*$ its dual. We have a canonical character on $\phi: G \times G^* \to \mathbb{C}^*$ given by evaluation. To construct the Fourier transform $L^2(G) \to L^2(G^*)$, we start with a function $f: G \to \mathbb{C}$. We pull back to $G \times G^*$, multiply by the evaluation character $\phi$ defined above, and integrate along fibers to give a function on $G^*$.

Everything we’ve done here has a sheaf-theoretic analog, however: pulling back a function corresponds to the functorial pull-back of sheaves, multiplication by a character corresponds to tensoring with a suitable line bundle, and integration along fibers corresponds to the lower shriek push-forward. Much of the classical formalism goes over to the sheaf-theoretic case. One can prove an “inversion formula” analogous to the Fourier inversion formula (with a Tate twist).

Why should we care? Well, Laumon interpreted the Fourier transform as a suitable “deformation” of the cohomology of a suitable sheaf on the affine line, and used it to give a simplified proof of the main results of Weil II, without using scary things like vanishing cycles and Picard-Lefschetz theory. The Fourier transform also behaves very well with respect to perverse sheaves: it is an auto-equivalence of the category of perverse sheaves, because of the careful way in which it is calibrated. Its careful use can be used to simplify some of the arguments in BBD that also rely on other scary things.

4.1. The Artin-Schreier sheaf

The “twisting” mechanism we shall need for this will be the Artin-Schreier sheaf.

Let ${Z}$ be any scheme, and let ${H}$ be a group.

Definition 31 An ${H}$-torsor is a sheaf ${\mathcal{F}}$ of sets on the étale site of ${Z}$, together with an action

$\displaystyle H \times \mathcal{F} \rightarrow \mathcal{F},$

which is locally trivial: there is an étale cover of ${Z}$ on which ${\mathcal{F}}$ becomes ${H}$-isomorphic to the trivial torsor ${H \times H \rightarrow H}$ (where ${H}$ is considered as a constant sheaf).

As usual, it follows that torsors are classified by ${H^1}$ (in the étale topology!), which is defined by cocycles and coboundaries in the non-abelian case (by the spectral sequence, this is also the derived functor ${H^1}$ for an abelian sheaf). When ${H}$ is finite, an ${H}$-torsor corresponds to a finite, continuous ${\pi_1(Z, \overline{z})}$-set with an ${H}$-action, because an ${H}$-torsor will then be a finite étale cover (and indeed a Galois cover, because the ${H}$-action furnishes the necessary set of automorphisms).

Let ${\mathcal{F}}$ be an ${H}$-torsor, and let ${\rho: H \rightarrow GL(V)}$ be a continuous representation of ${H}$ on a finite-dimensional ${\overline{\mathbb{Q}_l}}$-vector space ${V}$: by this, we assume that ${\rho}$ takes its image in matrices involving elements in a finite extension of ${\mathbb{Q}_l}$; of course this is automatic when ${H}$ is finite. Then we can obtain a smooth ${\overline{\mathbb{Q}_l}}$-sheaf ${\mathcal{F} \times_H V}$. One way to obtain this is to do the same “gluing” procedure of trivial torsors given by a cocycle, but to do it with trivial ${\overline{\mathbb{Q}_l}}$-sheaves (the cocycle is fed into ${\rho}$). Thus:

Proposition 32 A representation ${\rho: H \rightarrow \mathrm{GL}(V)}$ (satisfying the above hypotheses) leads to a covariant functor from ${H}$-torsors to smooth ${\overline{\mathbb{Q}_l}}$-sheaves.

From the definition via cocycles, the following is also clear:

Proposition 33 Given representations ${V, W}$ of ${H}$, we have for any ${H}$-torsor ${\mathcal{F}}$ canonical isomorphisms: $\mathcal{F} \times_H V) \otimes (\mathcal{F} \times_H W) \simeq \mathcal{F} \times_H (V \otimes_{\overline{\mathbb{Q}_l}} W) (\mathcal{F} \times_H V) \oplus (\mathcal{F} \times_H W) \simeq \mathcal{F} \times_H (V \oplus_{\overline{\mathbb{Q}_l}} W)$

Note further that this feature commutes with pull-back. That is, if ${\mathcal{F}}$ is an ${H}$-torsor on ${Y}$ and ${f: X \rightarrow Y}$ is a morphism, we can define place on the pull-back ${f^* \mathcal{F}}$ a natural structure of ${H}$-torsor. Then, for any ${H}$-representation ${V}$,

$\displaystyle f^* \mathcal{F} \times_H V \simeq f^* (\mathcal{F} \times_H V).$

It follows that stalks commute with this functor ${\times_H V}$.

Finally,

Lemma 34 Let ${\mathcal{F}, \mathcal{F}'}$ be torsors over the groups ${H, H'}$. Suppose given a map ${H \rightarrow H'}$. If ${\mathcal{F} \rightarrow \mathcal{F}'}$ is a morphism of torsors (equivariant with respect to these groups), and ${V}$ is a representation of ${H'}$, then there is a map

$\displaystyle \mathcal{F} \times_{H} V \rightarrow \mathcal{F}' \times_{H'} V.$

This is an isomorphism.

To define the Artin-Schreier sheaf (which is a smooth ${\overline{\mathbb{Q}_l}}$-sheaf on the affine line ${\mathbb{A}^1_{\kappa}}$), we start with a simple example of a torsor. Let ${G_0 \rightarrow \mathrm{Spec} \kappa}$ be a smooth, geometrically connected commutative algebraic group. Let ${G = G_0 \times_{\kappa}\overline{\kappa}}$. There is a Frobenius endomorphism ${F_0:G_0 \rightarrow G_0}$ over ${\mathrm{Spec} {\kappa}}$, which is given on the associated algebras by raising to the power ${q}$.

The base-change to ${\kappa}$, ${F: G \rightarrow G}$ (the so-called relative Frobenius) is then given in affine coordinates is given by ${(x, y, \dots) \mapsto (x^q, y^q, \dots)}$ (for ${q = |\kappa|}$). This is a morphism of algebraic groups, because the group law was defined over ${\kappa}$. We can consider the morphism ${F_0-1: G \rightarrow G}$, which sends ${x \mapsto F_0x - x}$. Call this the Lang isogeny ${L_0}$.

Theorem 35 (Lang) The Lang isogeny ${L_0}$ is a finite, surjective, étale morphism of algebraic groups.

Surjectivity is actually true even without commutativity, although then the Lang morphism is not a morphism of algebraic groups.

Proof: Indeed, ${L}$ is étale (since the Frobenius induces the zero map on tangent spaces). The image is an open subgroup, which must be the entire group. Thus it is surjective, as well. We need to see, at last, that ${L}$ is finite.

But any surjection ${f: H \rightarrow H'}$ of smooth algebraic groups over an algebraically closed field with finite kernel is proper: in fact, it is faithfully flat (by generic flatness and a translation argument), so a quotient map. Since for any ${S \subset H}$, we have ${f^{-1}(f(S)) = \bigcup_{h \in \ker f} h S}$, it follows that ${f(S)}$ is closed for any closed ${ S \subset H}$ (we have used ${f}$‘s being a quotient map). Thus ${f}$ is proper, and since it is quasi-finite, Chevalley’s theorem implies that ${f}$ is finite. Note that ${L_0}$ is geometrically connected: that is, the base-change to ${\kappa}$ remains a connected étale cover of ${G}$ (which is itself isomorphic to ${G}$!). It follows that the action of the geometric fundamental group ${\pi_1(G, \overline{x})}$ (for ${\overline{x}}$ some geometric point) on the étale cover is nontrivial, or the pull-back to ${G}$ would be split.

One way to interpret this result is that the non-abelian Galois cohomology $H^1$ of an affine algebraic group over a finite field is trivial; consequently, given an exact sequence of algebraic groups $1 \to G' \to G \to G'' \to 1$ over $\overline{\mathbb{F}_q}$, the sequence of $\mathbb{F}_q$-points is also exact.

Now ${G_0(\kappa)}$ is a finite group, which clearly acts on ${G}$ by multiplication.

Proposition 36 The Lang isogeny ${L_0: G_0 \rightarrow G_0}$ makes ${G_0}$ into a torsor over the finite group ${G_0(\kappa)}$.

Proof: We only need to check that the map is locally trivial, in the étale topology. But this follows because the geometric fibers of ${L_0}$ (which are each isomorphic to the kernel) are ${G_0}$-torsors. In fact, we find that for any surjection of algebraic groups ${G \rightarrow G'}$ with finite kernel, ${G}$ becomes a torsor (over ${G'}$) for the kernel.

In general, note that if ${\mathcal{F}}$ is a locally constant constructible sheaf of sets on the étale site of some scheme ${X}$ (corresponding to a finite étale cover) with action of a finite group ${A}$, then ${\mathcal{F}}$ is an ${A}$-torsor if and only if the geometric stalks ${\mathcal{F}_{\overline{x}}}$ are ${A}$-torsors. This follows (and we sketch the argument) because if ${\mathcal{F}_{\overline{x}}}$ is an ${A}$-torsor, we can find elements ${x_1, \dots, x_n}$ in some étale neighborhood of ${x}$ which are permuted amongst themselves by ${A}$ and which fill all the sections, locally: this implies trivialty.

Let ${x \in G_0}$ be a ${\kappa}$-rational point. We know that the fundamental group ${\pi_1(G_0, \overline{x})}$ acts on the torsor ${L_0}$, inducing isomorphisms of the étale cover. Let us determine the action of the arithmetic Frobenius ${F_x^{-1} \in \mathrm{Gal}(\overline{\kappa}/\kappa)}$, which maps to an element of ${\pi_1(G_0, \overline{x})}$, on this torsor: it must be translation by some element of ${G_0(\kappa)}$, as we have already remarked. We have a cartesian diagram

We need to analyze the action of ${F_x^{-1}}$ on ${L_0^{-1}(x)}$. This is the Galois action, and corresponds to the morphism on rings ${a \rightarrow a^{\kappa}}$: in fact, the Galois action is of this form (raising to a power) for any scheme finite étale over ${\mathrm{Spec} \kappa}$. It follows that on the geometric fiber, this corresponds in coordinates to ${y \mapsto y^q}$. However, since we are in the fiber over ${x}$, this corresponds to multiplication by ${x}$ on that fiber. It follows by naturality of the fundamental group that the induced automorphism of the cover ${G_0 \stackrel{L_0}{\rightarrow} G_{0}}$ is simply translation by ${x}$.

We have proved:

Proposition 37 Let ${G_0}$ be a smooth linear algebraic group over ${\kappa}$. If ${x \in G_0}$ is a ${\kappa}$-rational point, then the arithmetic Frobenius ${F_x^{-1}}$ at ${x}$ induces (through ${\pi_1}$-action) the automorphism of ${L_0}$ given by translation by ${x}$.

Definition 38 Consider ${\mathbb{A}^1_{\kappa}}$ as a group scheme over ${\mathrm{Spec} \kappa}$ and the Lang isogeny ${L_0: \mathbb{A}^1_{\kappa} \rightarrow \mathbb{A}^1_{\kappa}}$ (which is just ${x \mapsto x^q - x}$), which becomes a torsor over ${\kappa = \mathbb{A}^1_{\kappa}(\kappa)}$. Given a character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$, we define the Artin-Schreier sheaf $\mathcal{L}_\chi$ by applying ${\chi}$ to the ${\kappa}$-torsor ${L_0}$.

The Artin-Schreier sheaf is thus a smooth ${\overline{\mathbb{Q}_l}}$-sheaf of rank one on ${\mathbb{A}^1_{\kappa}}$. We note that ${\mathcal{L}_{\chi} \otimes \mathcal{L}_{\chi'} \simeq \mathcal{L}_{\chi \chi'}}$, because of what we have discussed above.

We now wish to study the local action of the Galois group.

Proposition 39 Let ${x \in \mathbb{A}^1_{\kappa}(\kappa) = \kappa}$. Then the action of the geometric Frobenius on ${(\mathcal{L}_{\chi})_{\overline{x}}}$ is given by multiplication by ${\chi(x)^{-1}}$.

Proof: In fact, we recall that the operation of obtaining a ${\overline{\mathbb{Q}_l}}$-sheaf from a torsor commutes with pull-back. As a result, it commutes with taking stalks, and the Galois equivariance is preserved. But the ordinary Frobenius acts on the Lang torsor ${L_0: \mathbb{A}^1_{\kappa} \rightarrow \mathbb{A}^1_{\kappa}}$ by translation by ${x}$. This corresponds after forming ${\overline{\mathbb{Q}_l}}$-sheaves to multiplying by ${\chi(x)}$. Since the geometric Frobenius is the inverse, we are done.

Next, we wish to discuss how the Artin-Schreier sheaf behaves with respect to a change of base field.

Proposition 40 Let ${\kappa'}$ be a finite extension of ${\kappa}$, and let ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$ be a character. Then the pull-back of ${\mathcal{L}_{\chi}}$ via ${\mathbb{A}^1_{\kappa'} \rightarrow \mathbb{A}^1_{\kappa}}$ is the ${\overline{\mathbb{Q}_l}}$-sheaf ${\mathcal{L}_{\chi \circ \mathrm{Tr}_{\kappa'/\kappa}}}$ on ${\mathbb{A}^1_{\kappa'}}$.

Note that the sheaf ${\mathcal{L}_{\chi \circ \mathrm{Tr}}}$ is obtained from a different torsor (in fact, a ${\kappa'}$-torsor) over ${\mathbb{A}^1_{\kappa'}}$, not the old torsor over ${\mathbb{A}^1_{\kappa}}$.

Proof: The pull-back of ${L_0: \mathbb{A}^1_{\kappa} \rightarrow \mathbb{A}^1_{\kappa}}$ is the torsor ${\mathbb{A}^1_{\kappa'} \rightarrow \mathbb{A}^1_{\kappa'}}$ given in geometric affine coordinates as ${x \mapsto x^q - x}$, as before. This is still a Galois cover, with covering group ${\kappa}$, and this with the character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$ gives the pull-back of ${\mathcal{L}_{\chi}}$.

Suppose ${q' = |\kappa'| = q^n}$.

There is a Lang map ${L_1: \mathbb{A}^1_{\kappa'} \rightarrow \mathbb{A}^1_{\kappa'}}$ given by ${x \mapsto x^{q^n} - x}$. This can be expressed as the composite

$\displaystyle \mathbb{A}^1_{\kappa'} \stackrel{x \mapsto x + x^q + \dots + x^{q^{n-1}}}{\rightarrow} \mathbb{A}^1_{\kappa'} \stackrel{L_0 \times_\kappa \kappa'}{\rightarrow} \mathbb{A}^1_{\kappa'}.$

The map ${\mathbb{A}^1_{\kappa'} \rightarrow \mathbb{A}^1_{\kappa'}}$, ${x \mapsto x + x^q + \dots + x^{q^{n-1}}}$ is a morphism of torsors over ${\kappa'}$ and ${\kappa}$ respectively, equivariant with respect to the trace map ${\mathrm{Tr}: \kappa' \rightarrow \kappa}$. Consequently, “tensoring” the first one over ${\overline{\mathbb{Q}_l}}$ with respect to ${\chi}$ is the same as tensoring the second (necessarily over the pull-back by ${\mathrm{Tr}}$).

Motivated by this result, we shall regard a character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$ as a family of characters on each finite extension ${\kappa'}$, by means of the trace. So a character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$ will be tacitly used to give characters ${\chi: \kappa' \rightarrow \overline{\mathbb{Q}_l}^*}$ for each ${\kappa'/\kappa}$, which will not of course agree; however, the usage should not cause confusion. The different line bundles we get are all compatible.

4.2. The Fourier-Deligne transform

We shall use the category ${\mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$, which is the bounded derived category of ${\overline{\mathbb{Q}_l}}$-sheaves on the affine line ${\mathbb{A}^1_{\kappa}}$.

Fix a nontrivial character ${\chi: \kappa \rightarrow \overline{\mathbb{Q}_l}^*}$.

Definition 41 Consider the two projections ${\pi_1, \pi_2: \mathbb{A}^2_{\kappa} \rightarrow \mathbb{A}^1_{\kappa}}$, the multiplication map ${m: \mathbb{A}^2_{\kappa} \rightarrow \mathbb{A}^1_{\kappa}}$ (in geometric coordinates, ${(x, y) \mapsto xy}$), We define the functor ${\mathbf{F}(\chi): \mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l}) \rightarrow \mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$ as follows. Given ${K_0 \in \mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$, we set

$\displaystyle \mathbf{F}(\chi)(K_0) = \mathbf{R} \pi_{2!} ( \pi_1^* K_0 \otimes m^* \mathcal{L}(\chi))[1].$

The definition is formally analogous to the familiar Fourier transform on ${\mathbb{R}}$, but let us motivate it further with a discussion of the “function-sheaf correspondence.” Given a Weil sheaf ${\mathcal{F}_0}$ on ${X_0}$, we define a function ${T_{\mathcal{F}_0}: X_0(\kappa) \rightarrow \overline{\mathbb{Q}_l}}$ via:

$\displaystyle T_{\mathcal{F}_0}(x) = \mathrm{Tr} F_x|_{\mathcal{F}_{\overline{x}}} .$

This is defined on the ${\kappa}$-rational points of the affine line, that is, the elements of ${\kappa}$. We can extend it to a complex of sheaves by taking the alternating trace of the Frobenius on the cohomology, and in this way we can associate a function ${T_{K_0}}$ to any ${K_0 }$ in the derived category.

We find:

Theorem 42

1. If ${ f_0: Y_0 \rightarrow X_0}$ is a morphism of ${\kappa}$-schemes, then ${T_{f^* K_0} = T_{K_0} \circ f}$.
2. If ${h_0: X_0 \rightarrow Z_0}$ is a compactifiable morphism, then ${T_{\mathbf{R} h_{0!} K_0} = (h_0)_! T_{K_0}}$, where the operator ${h_{0!} }$ is defined on functions by ${h_{0!} \phi(z) = \sum_{x \in X_0(\kappa), h_{0!}(x) = z} \phi(x)}$.

Proof: The first result is obvious. The second is a restatement of the Grothendieck trace formula(!).

Let ${X_0 = \mathbb{A}^1_{\kappa}}$ now, so the ${\kappa}$-rational points are just ${\kappa}$. Let ${K_0 \in \mathbf{D}^b_c(\mathbb{A}^1_{\kappa}, \overline{\mathbb{Q}_l})}$. The claim is that ${T_{\mathbf{F}(\chi)(K_0)}}$ is almost the discrete Fourier transform of ${T_{K_0}}$.

To see this, will use the above result (that is, essentially the trace formula), and the following computation of the stalks of the Fourier transform:

With this in mind, we want to prove the promised claim about the function ${T_{\mathbf{F}(\chi)(K_0)}}$. Namely, given ${x \in \kappa = \mathbb{A}^1_{\kappa}(\kappa)}$, then we find by that

$\displaystyle T_{\mathbf{F}(\chi)(K_0)} = - \sum_{y \in \kappa} T_{K_0}(y)\chi(xy) \ \ \ \ \ (10)$

This is because we must take a sum over the ${\kappa}$-rational points in the fiber, and we know  how the Frobenius acts on the ${\mathcal{L}_{\chi}}$ part. The minus sign comes from the shift involved in defining the functor ${\mathcal{F}}$.

4.3. The inversion formula

The Fourier transform of functions on ${\mathbb{R}}$ has the property that a similarly defined operator is its inverse. We want to show the same thing for the Fourier-Deligne transform. Namely:

Theorem 43 (Inversion) There is a natural isomorphism ${\mathbf{F}_{\chi^{-1}} \circ \mathbf{F}_{\chi}(K_0) \simeq K_0(-1)}$.

I don’t really want to prove this here. I’ll try to explain it next time if I can figure out how to interpret the argument properly as a “sheafification” of the classical one.