A categorical K\"unneth formula for constructible Weil sheaves

We prove a K\"unneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p>0$ for various coefficients, including finite discrete rings, algebraic field extensions $E \supset \mathbf Q_\ell$, $\ell \ne p$ and their rings of integers $O_E$. We also consider a variant for ind-construtible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.


Introduction
The classical Künneth formula expresses the (co-)homology of a product of two spaces X 1 and X 2 in terms of the tensor product of the (co-)homology of the individual factors.For two topological spaces, for example, one has under suitable finiteness hypothesis an isomorphism on singular cohomology with rational coefficients.Such cohomology groups are naturally morphism groups in the derived categories of sheaves on these spaces.So one may ask whether the Künneth formula can be extended to a categorical level, that is, whether it is possible to relate the derived categories of sheaves on X 1 and X 2 to those on their product X 1 × X 2 .Statements in this direction are referred to as categorical Künneth formulas and are known in different contexts: for example, for the respective derived categories of topological sheaves, for D-modules on varieties in characteristic 0 and for quasi-coherent sheaves, see [GKRV22,Section A.2].
In addition to (1.1) above, categorical Künneth formulas require decomposing a sheaf on X 1 × X 2 into exterior products M 1 ⊠ M 2 , with M 1 , M 2 being sheaves on X 1 , X 2 , respectively.For varieties in characteristic p > 0, an analogous decomposition for constructible (pro-)étale sheaves fails in general, and so does a categorical Künneth formula in this context, see Example 1.4 below.The main result of the manuscript at hand (see Theorem 1.3) shows how to rectify the failure by adding equivariance data under partial Frobenius morphisms, that is, one arrives at a categorical Künneth formula for constructible Weil sheaves.Our work relies on the analogous result [Dri80, Theorem 2.1] for étale fundamental groups known as Drinfeld's lemma, see Section 5.3 for details and references.
Let X be a scheme over a finite field F q , where q is a p-power.Fix an algebraic closure F/F q , and denote by X F the base change.The partial (q-)Frobenius φ X := Frob X × id F defines an endomorphism of X F .
Definition 1.1.The Weil-proétale site X Weil proét is the following site: Objects are pairs (U, ϕ) consisting of U ∈ (X F ) proét , the proétale site of X F [BS15], equipped with an endomorphism ϕ : U → U of F-schemes covering φ X .Morphisms are given by equivariant maps.A family {(U i , ϕ i ) → (U, ϕ)} of morphisms is a cover if the family {U i → U } is a cover in (X F ) proét .
*The second named author T.R. is funded by the European Research Council (ERC) under Horizon Europe (grant agreement nº 101040935), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124 and the LOEWE professorship in Algebra.J.S. was supported by Deutsche Forschungsgemeinschaft (DFG), EXC 2044-390685587, Mathematik Münster: Dynamik-Geometrie-Struktur.
The Weil-proétale site sits in the sequence of sites given by the functors U ← (U, ϕ) and (U F , φ U ) ← U in the opposite direction.The maps (1.2) commute over * proét , the proétale site of the point.Thus, for any condensed ring Λ viewed as a sheaf of rings on * proét , we get pullback functors on derived categories of proétale Λ-sheaves D(X, Λ) → D X Weil , Λ → D(X F , Λ).
In analogy with the definition of lisse and constructible sheaves (as recalled in Definition 3.1), we introduce the categories of lisse and constructible Weil sheaves D lis X Weil , Λ ⊂ D cons X Weil , Λ as the full subcategories of D X Weil , Λ that are dualizable, resp.that are Zariski locally on X dualizable along a constructible stratification.These categories are equivalent to the corresponding categories of sheaves on the prestack X F /φ X , that is, equivalent to the homotopy fixed points of the induced φ * X -action: Proposition 1.2 (Proposition 4.4, Proposition 4.11).The pullback of sheaves along (X F ) proét → X Weil proét induces an equivalence of Λ * -linear symmetric monoidal stable ∞-categories for • ∈ {∅, lis, cons}.
Thus, objects in D • X Weil , Λ are pairs (M, α) with M ∈ D • (X F , Λ) and α : M ∼ = φ * X M .On the abelian level, we recover the classical approach [Del80, Definition 1.1.10].If Λ is a finite discrete ring, then every Weil descent datum on constructible Λ-sheaves is effective so that D cons X Weil , Λ ∼ = D cons (X, Λ), see Proposition 4.16.However, the categories are not equivalent if Λ = Z, Z ℓ , Q ℓ , say.This relates to the difference between continuous representations of Galois groups such as Ẑ versus Weil groups such as Z.
For several F q -schemes X 1 , . . ., X n , a similar process is carried out for their product X := X 1 × Fq . . .× Fq X n equipped with the partial Frobenii φ Xi : X F → X F , see Section 4.2.Generalizing Proposition 1.2, there is an equivalence of Λ * -linear symmetric monoidal stable ∞-categories (1.3) for • ∈ {∅, lis, cons}.The category on the left is defined using the Weil-proétale site X Weil 1 × . . .× X Weil n proét consisting of objects (U, ϕ 1 , . . ., ϕ n ) with U ∈ (X F ) proét and pairwise commuting endomorphisms ϕ i : U → U covering the partial Frobenii φ Xi : X F → X F for all i = 1, . . ., n.The category on the right is the category of simultaneous homotopy fixed points, see Section 2.2.For constructible Weil sheaves, (1.3) relies on decompositions of partial Frobenius invariant cycles in X F , see Proposition 4.8.
The following result is referred to as the categorical Künneth formula for constructible Weil sheaves (or, derived Drinfeld's lemma): Theorem 1.3 (Theorem 5.2, Remark 5.3).Let F q be a finite field of characteristic p > 0. Let X 1 , . . ., X n be finite type F q -schemes.Let Λ be either a finite discrete ring of prime-to-p-torsion, or an algebraic field extension E ⊃ Q ℓ , ℓ = p, or its ring of integers O E .
Then the external tensor product of sheaves (M 1 , . . ., M n ) → M 1 ⊠ . . .⊠ M n induces an equivalence and likewise for the categories of lisse Weil sheaves if, in the case Λ = E, one assumes the schemes X 1 , . . ., X n to be geometrically unibranch (for example, normal).
This statement can also be recast as the symmetric monoidality of the functor sending a Weil prestack X Weil , which is defined on R-points by , to its ∞-category of constructible sheaves (Theorem 5.6).
The tensor product of ∞-categories (see Section 2) is formed using the natural Λ * -linear structures on the categories.We have an analogous equivalence for the categories of lisse Weil sheaves with coefficients Λ in finite discrete p-torsion rings like Z/p m , m ≥ 1, see Theorem 5.2.As the following example shows, the use of Weil sheaves is necessary for the essential surjectivity to hold.This behavior is mentioned in the first arXiv version of [GKRV22, Equation (0.8)] which is one of the main motivations for our work.
F be the affine line so that X F = A 2 F with coordinates denoted by x 1 and x 2 .Then U defines a finite étale cover with Galois group Z/p.Let M ∈ D lis (A 2 F , Λ) be the sheaf in degree 0 associated with some non-trivial character Z/p → Λ × * .For λ, µ ∈ F not differing by a scalar in F × p , the fibers U | {x1=λ} , U | {x1=µ} are not isomorphic over A 1 F by Artin-Schreier theory.Hence, M ≃ φ * Xi M and one can show that M ≃ M 1 ⊠ M 2 for any M i ∈ D(A 1 F , Λ).If Λ as above is p-torsion free, then the full faithfulness of (1.4) is a direct consequence of the Künneth formula for X i,F , i = 1, . . ., n.For Λ = Z/p m , we use Artin-Schreier theory instead.It would be interesting to see whether the lisse p-torsion case can be extended to constructible sheaves.In both cases, the essential surjectivity relies on a variant of Drinfeld's lemma for Weil group representations, see Theorem 5.9, together with a characterization of partial-Frobenius stable algebraic cycles (Proposition 4.8) as well as a decomposition argument for representations of a product of abstract groups (Proposition 5.12).
With a view towards [Laf18], we consider Weil sheaves whose underlying sheaf is ind-constructible, but where the action of the partial Frobenii do not necessarily preserve the constructible pieces.For finite type F q -schemes X 1 , . . ., X n and Λ as in Theorem 1.3, we consider the category of simultaneous homotopy fixed points for • ∈ {indlis, indcons}.Then the external tensor product induces a fully faithful functor Unlike the case of lisse or constructible sheaves, the functor is not essentially surjective as one can add freely actions by the partial Frobenii, see Remark 6.6.However, we can identify a large class of objects in the essential image of (1.5).When combined with the smoothness results of Xue [Xue20c, Theorem 4.2.3],we obtain, for example, that the compactly supported cohomology of moduli stacks of shtukas over global function fields lies in the essential image of (1.5), see Section 6.2 for details.
Remark 1.5.Another motivation for this work is our (T.R. and J.S.) ongoing project aiming for a motivic refinement of [Laf18].In this project, we will need a motivic variant of Drinfeld's lemma.Since triangulated categories of motives such as DM(X, Q) carry t-structures only conditionally, we need a Drinfeld lemma to be a statement about triangulated categories.In conjunction with the conjecture relating Weil-étale motivic cohomology to Weil-étale cohomology [Kah03,Gei04,Lic05], our results suggest to look for a Drinfeld lemma for constructible Weil motives.

Recollections on ∞-categories
Throughout this section, Λ denotes a unital, commutative ring.We briefly collect some notation pertaining to ∞categories from [Lur17,Lur09].As in [Lur09, Section 5.5.3],Pr L denotes the ∞-category of presentable ∞-categories with colimit-preserving functors.It contains the subcategory Pr St ⊂ Pr L consisting of stable ∞-categories.
2.1.Monoidal aspects.The category Pr L carries the Lurie tensor product [Lur17, Section 4.8.1].This tensor product induces one on the full subcategory Pr St ⊂ Pr L consisting of stable ∞-categories [Lur17, Proposition 4.8.2.18].For our commutative ring Λ, the ∞-category Mod Λ of chain complexes of Λ-modules, up to quasi-isomorphism, is a commutative monoid in Pr St with respect to this tensor product.This structure includes, in particular, the existence of a functor Mod Λ × Mod Λ → Mod Λ which, after passing to the homotopy categories is the classical derived tensor product on the unbounded derived category of Λ-modules.We define Pr St Λ to be the category of modules, in Pr St , over Mod Λ .Noting that modules over Mod Λ are in particular modules over Sp, the ∞-category of spectra, Pr St Λ can be described as the ∞-category consisting of stable presentable ∞-categories together with a Λ-linear structure, such that functors are continuous and Λ-linear.Therefore Pr St Λ carries a symmetric monoidal structure, whose unit is Mod Λ .We will also denote by Pr St ω the category of compactly generated presentable with functors that send compact objects to compact objects (equivaletly, those whose right adjoint is continuous).
In order to express monoidal properties of ∞-categories consisting, say, of bounded complexes, recall from [Lur17, Corollary 4.8.1.4joint with Lemma 5.3.2.11] or [BZFN10, Proposition 4.4] the symmetric monoidal structure on the ∞-category Cat Ex ∞ (Idem) of idempotent complete stable ∞-categories and exact functors: it is characterized by that is, the compact objects in the Lurie tensor product of the Ind-completions.With respect to these monoidal structures, the Ind-completion functor (taking values in compactly generated presentable ∞-categories with the Lurie tensor product) and the functor forgetting the compact generatedness ∞ (Idem), so that we can consider its category of modules, denoted as Cat Ex ∞,Λ (Idem).This category inherits a symmetric monoidal structure denoted by D 1 ⊗ PerfΛ D 2 .Any stable ∞-category D is canonically enriched over the category of spectra Sp.We write Hom D (−, −) for the mapping spectrum.Any category in Pr St Λ is canonically enriched over Mod Λ , so that we refer to Hom D (−, −) ∈ Mod Λ as the mapping complex.For example, for M, N ∈ Mod Λ , then Hom ModΛ (M, N ) is commonly also denoted by RHom(M, N ).Its n-th cohomology is the Hom-group Hom(M, N [n]) in the classical derived category.
2.2.Fixed points of ∞-categories.A basic structure in Drinfeld's lemma is the equivariance datum for the partial Frobenii.In this section, we assemble some abstract results where such ∞-categorical constructions are carried out.
Because of these facts, we will usually not specify where the limit above is formed.Note that all functors Cat Ex ∞ (Idem) except for the forgetful functor marked ( * ) preserve limits, see [Lur17, Corollary 4.2.3.3] and [Lur09, Proposition 5.5.3.13] for the rightmost two functors.To give a concrete example of that failure in our situation, note that Fix(D, id D ) = Fun(BZ, D), that is, objects are pairs (M, α) consisting of some M ∈ D and some automorphism α : M ∼ = M .Now consider D = Vect fd Λ , the (abelian) category of finite-dimensional vector spaces over a field Λ.The natural functor is fully faithful, but not essentially surjective: given an automorphism α of an infinite-dimensional vector space M , there need not be a filtration M = M i by finite-dimensional subspaces M i that is compatible with α.
Fixed point categories inherit t-structures as follows: Lemma 2.2.Let φ : D → D be a functor in Cat Ex ∞ (Idem).Suppose D carries a t-structure such that φ is texact.Then Fix(D, φ) carries a unique t-structure such that the evaluation functor is t-exact.There is a natural equivalence Proof.Let us abbreviate D := Fix(D, φ).For • being either "≤ 0" or "≥ 0", we put D • := Fix(D • , φ), which is a (non-stable) ∞-category.This is clearly the only choice for a t-structure making ev a t-exact functor.It satisfies the claim about the hearts of the t-structure by definition.
We need to show that it is a t-structure.Being a limit of full subcategories, the categories D • are full subcategories of D. Since φ, being t-exact, commutes with τ ≤0 D and τ ≥0 D , these two functors also yield truncation functors for D.
Then there is a canonical equivalence Proof.The categories Fix( D i , φ i ) are compactly generated: the forgetful functor U : Fix( D i , φ i ) → D i = Ind(D i ) preserves colimits, so its left adjoint L preserves compact objects.Moreover, U is conservative, so that the objects L(d i ), for d i ∈ D i , form a family of compact generators.Then, we use that any compactly generated category in Pr St Λ is dualizable [Lur18, Remark D.7.7.6 (1)] so that tensoring with it preserves limits.

Lisse and constructible sheaves
In order to state and prove the categorical Künneth formula for Weil sheaves, we use the framework for lisse and constructible sheaves provided by [HRS23].For the convenience of the reader, we collect here some basics of the formalism.
Throughout, Λ denotes a condensed ring, for example any T1-topological ring such as discrete rings, algebraic extensions E/Q ℓ or their ring of integers O E .In the synopsis below, we refer to the latter choices of Λ as the standard coefficient rings.We write Λ * for the underlying ring.Let D(X, Λ) be the derived category of sheaves of Λ-modules on the proétale site X proét .Definition 3.1 ([HRS23, Definition 3.3, Definition 8.1]).For every scheme X and every condensed ring Λ, there are the full subcategories (3.1)By definition, the left hand category of lisse sheaves consists of the dualizable objects in the right-most category.An object (henceforth referred to as a sheaf) M in the right hand category is constructible, if on any affine U ⊂ X there is a finite stratification into constructible locally closed subschemes U i ⊂ U such that M | Ui is lisse, that is, dualizable.Finally, an ind-lisse (respectively, ind-constructible) sheaf is a filtered colimit, in the category D(X, Λ), of lisse (respectively, constructible) sheaves.The corresponding full subcategories of D(X, Λ) are denoted by For the standard coefficient rings Λ above and quasi-compact quasi-separated (qcqs) schemes X, that definition of lisse and constructible sheaves agrees with the classical ones, see [HRS23,§7] for details.
The categories enjoy the following properties: the category D(X, Λ) is an object in Pr St Λ * .The functor restricts to a functor Perf Λ * → D lis (X, Λ), and the categories D lis (X, Λ) ⊂ D cons (X, Λ) are objects in Cat Ex ∞,Λ (Idem).In particular, all categories listed in (3.1) are stable idempotent complete Λ * -linear ∞-categories.(According to [HS21, Theorem 2.2], it also satisfies v-descent, but we will not need this in this paper.)The functor X → D indcons (X, Λ), resp.X → D indlis (X, Λ) satisfies hyperdescent for quasi-compact étale, resp.finite étale covers, see [HRS23,Corollary 8.7].(iv) If Λ = colim Λ i is a filtered colimit of condensed rings and X is qcqs, then the natural functors colim D lis (X, Λ i ) are equivalences [HRS23, Proposition 5.2].(v) If X is qcqs, then any constructible sheaf is bounded with respect to the t-structure on D(X, Λ) [HRS23,Corollary 4.11].(vi) For X locally Noetherian (and much more generally), the t-structure on D(X, Λ) restricts to one on D lis (X, Λ) and D cons (X, Λ) provided that Λ is t-admissible in the sense of [HRS23, Definition 6.1].Here, t-admissibility is a combination of an algebraic and a topological condition: first, Λ * needs to be regular coherent (for example, any regular Noetherian ring of finite Krull dimension, but Z/ℓ 2 is excluded).The topological condition on the condensed structure of Λ is satisfied for all the standard coefficient rings listed above, see [HRS23, Theorem 6.2].(vii) For X locally Noetherian (and again more generally), a sheaf is lisse if and only if it is proétale locally the constant sheaf associated to a perfect complex of Λ * -modules, see [HRS23, Theorem 4.13].(viii) Let X be a qcqs scheme.If the Λ-cohomological dimension is uniformly bounded for all proétale affines U = lim i U i over X, then Ind(D cons (X, Λ)) = D indcons (X, Λ) and likewise for ind-lisse sheaves.If X is of finite type over F q or a separably closed field, this condition holds for any of the above standard rings.For discrete p-torsion rings, algebraic extensions E/Q p and their ring of integers O E , this holds for arbitrary qcqs schemes in characteristic p. See [HRS23, Lemma 8.6, Proposition 8.2].
For schemes X 1 , . . ., X n over a fixed base scheme S (for example, the spectrum of a field) and a condensed ring Λ, we denote the external product in the usual way: Here p i : X := X 1 × S . . .× S X n → X i are the projections.This functor induces the functor in Pr St Λ * .Here we regard D(X i , Λ) as objects in Pr St Λ * , like in (i) in the synopsis above.The external tensor product of constructible sheaves is again constructible, and hence induces a functor in Cat Ex ∞,Λ * (Idem) and likewise for the categories of ind-constructible, resp.(ind-)lisse sheaves.

Weil sheaves
In this section, we introduce the categories consisting of lisse, resp.constructible, resp.all Weil sheaves.These are the categories featuring in the categorical Künneth formula (Theorem 1.3).Throughout this section, X is a scheme over a finite field F q of characteristic p > 0. Unless the contrary is mentioned, we impose no conditions on X.Moreover, Λ is a condensed ring.We fix an algebraic closure F of F q , and denote by X F := X × Fq Spec F the base change.Denote by φ X (resp.φ F ) the endomorphism of X F that is the q-Frobenius on X (resp.Spec F) and the identity on the other factor.Let be the categories of lisse, resp.constructible, resp.all proétale sheaves of Λ-modules on X F (Definition 3.1).These categories are objects in Cat Ex ∞,Λ * (Idem), that is, Λ * -linear stable idempotent complete symmetric monoidal ∞categories where Λ * = Γ( * , Λ) is the underlying ring.
4.1.The Weil-proétale site.The Weil-étale topology for schemes over finite field is introduced in [Lic05], see also [Gei04].Our approach for the proétale topology is slightly different: Definition 4.1.The Weil-proétale site of X, denoted by X Weil proét , is the following site: Objects in X Weil proét are pairs (U, ϕ) consisting of U ∈ (X F ) proét equipped with an endomorphism ϕ : U → U of F-schemes such that the map U → X F intertwines ϕ and φ X .Morphisms in X Weil proét are given by equivariant maps, and a family Note that X Weil proét admits small limits formed componentwise as lim(U i , ϕ i ) = (lim U i , lim ϕ i ).In particular, there are limit-preserving maps of sites given by the functors (in the opposite direction) U ← (U, ϕ) and (U F , φ U ) ← U .We denote by D(X Weil , Λ) the unbounded derived category of sheaves of Λ X -modules on X Weil proét .The maps of sites (4.1) induce functors whose composition is the usual pullback functor along X F → X.
Remark 4.2.The functor D(X, Λ) → D(X Weil , Λ) is not an equivalence in general.This relates to the difference between continuous representations Galois versus Weil groups.See, however, Proposition 4.16 for filtered colimits of finite discrete rings Λ.
We have the following basic functoriality: Let j : U → X be a weakly étale morphism and consider the corresponding object (U F , φ U ) of X Weil proét .Then the slice site (X Weil proét ) /(U F ,φU ) is equivalent to U Weil proét .This gives a functor (X proét ) op → Pr St Λ , U → D(U Weil , Λ) which is a hypercomplete sheaf of Λ * -linear presentable stable categories.Also, we obtain an adjunction j !: D(U Weil , Λ) ⇄ D(X Weil , Λ) : j * that is compatible with the ((j F ) ! , (j F ) * )-adunction under (4.2).The category D(X Weil , Λ) is equivalent to the category of φ X -equivariant sheaves on X F , as we will now explain.
For each i ≥ 0, consider the object (X i , Φ i ) ∈ X Weil proét with X i = Z i+1 × X F the countably disjoint union of X F , the map X i → X F given by projection and the endomorphism Φ i : By pullback, we get a limit-preserving map of sites Lemma 4.3.For each i ≥ 0, the map (4.3) induces an equivalence on the associated 1-topoi.
Proof.As universal homeomorphisms induce equivalences on proétale 1-topoi [BS15, Lemma 5.4.2],we may assume that X is perfect.In this case, the sites (4.3) are equivalent because φ X is an isomorphism.Explicitly, an inverse is given by sending an object U ∈ (X i−1 ) proét to the object Weil sheaves admit the following presentation as the φ * X -fixed points of D(X F , Λ), see Definition 2.1: Proposition 4.4.The last functor in (4.2) induces an equivalence (4.4) Remark 4.5.Objects in (4.4) are pairs (M, α) where M ∈ D(X F , Λ) and α is an isomorphism M ∼ = φ * X M .Note that the composition φ X • φ F is the absolute q-Frobenius of X F .In particular, it induces the identity on proétale topoi, see [BS15, Lemma 5.4.2].Therefore, replacing φ * X by φ * F in (4.4) yields an equivalent category.Proof of Proposition 4.4.The structural morphism (X 0 , Φ 0 ) → (X F , φ X ) is a cover in X Weil proét .Its Čech nerve has objects (X i , Φ i ) ∈ X Weil proét , i ≥ 0 as above.By descent, there is an equivalence Under Lemma 4.3, the cosimplicial 1-topos associated with (X Weil proét ) /(X•,Φ•) is equivalent to the cosimplicial 1-topos associated with the action of φ * X on (X F ) proét .The equivalence (4.5) then becomes D X Weil , Λ ) is equivalent to the homotopy fixed points of D(X F , Λ) with respect to the action of φ * X , which is our claim.4.2.Weil sheaves on products.The discussion of the previous section generalizes to products of schemes as follows.Let X 1 , . . ., X n be schemes over F q , and denote by X := X 1 × Fq . . .× Fq X n their product.For every 1 ≤ i ≤ n, we have a morphism φ Xi : X i,F → X i,F as in the previous section.We use the notation φ Xi to also denote the corresponding map on X F = X 1,F × F . . .× F X n,F which is φ Xi on the i-th factor and the identity on the other factors.
We define the site (X Weil 1 × . . .× X Weil n ) proét whose underlying category consists of tuples (U, ϕ 1 , . . ., ϕ n ) with U ∈ (X F ) proét and pairwise commuting endomomorphisms ϕ i : U → U such that the following diagram commutes for all 1 ≤ i ≤ n.As before, we denote by D X Weil 1 ×. ..×XWeil n , Λ the corresponding derived category of Λ-sheaves.Using a similar reasoning as in the previous section, we can identify this category of sheaves with the homotopy fixed points Roughly speaking, objects in the category D(X Weil 1 × . . .× X Weil n , Λ) are given by tuples (M, α 1 , . . ., α n ) with M ∈ D(X F , Λ) and with pairwise commuting equivalences α i : M ∼ = φ * Xi M .That is, equipped with a collection of equivalences φ * Xj (α i ) • α j ≃ φ * Xi (α j ) • α i for all i, j satisfying higher coherence conditions.4.3.Partial-Frobenius stability.For schemes X 1 , . . ., X n over F q , we denote by X := X 1 × Fq • • • × Fq X n their product together with the partial Frobenii Frob Xi : X → X, 1 ≤ i ≤ n.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 : 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 fixed i, the subset Z is Frob Xj -invariant for all j = i.This remark, which also applies to Spec F, 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 [Dri87, Proposition 1.1] for coherent sheaves: Lemma 4.7.Let X be a qcqs F q -scheme, and let k/F q be a separably closed field.Denote by p : X k → X the projection.Then Z → p −1 (Z) induces a bijection {constructible subsets in X } ↔ {partial-Frobenius invariant, constructible subsets in X k }.
Proof.The injectivity is clear because p is surjective.It remains to check the surjectivity.Without loss of generality we may assume that k is 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 subsets Z and for X affine over F q .By Noetherian approximation (Lemma 4.9), we reduce further to the case where X is of finite type over F q and still affine.Now we choose a locally closed embedding X → P n Fq into projective space.A closed subset Z ′ ⊂ X k is φ k -invariant if and only if its closure inside P n k is so.Hence, it is enough to consider the case where X = P n Fq is the projective space.Let Z ′ be a closed Frob kinvariant subset of X k .When viewed as a reduced subscheme, the isomorphism φ k restricts to an isomorphism of The following proposition generalizes the results [Lau04, Lemma 9.2.1] and [Laf18, Lemme 8.12] in the case of curves.
Proposition 4.8.Let X 1 , . . ., X n be qcqs F q -schemes, and denote X = X 1 × Fq . . .× Fq X n .Then any partial-Frobenius invariant constructible closed subset Z ⊂ X is a finite set-theoretic union of subsets of the form In particular, any partial-Frobenius invariant constructible open subscheme U ⊂ X is a finite union of constructible open subschemes of the form Proof.By induction, we may assume n = 2.By Noetherian approximation (Lemma 4.9), 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 over F q , and locally closed subschemes are equipped with their reduced subscheme structure.Let Z ⊂ X 1 ×X 2 be a partial-Frobenius invariant closed subscheme.The complement U = X 1 × X 2 \ Z is also partial-Frobenius invariant.
In the proof, we can replace X 1 (and likewise X 2 ) by a stratification in the following sense: Suppose 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 × X 2 → X i are flat, and that the topological space underlying the schemetheoretic 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 2 is zero dimensional reduces to Lemma 4.7).In the induction step, we may assume, using the above stratification argument, that both X i are irreducible with generic point η i .We let η i be a geometric generic point over η i , and denote by p i : X 1 × X 2 → X i the two projections.Both p i are 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 , we are done.We can therefore replace X i by p i (U ) and assume that both p i : U → X i are surjective.
The base change U × X2 η 2 is a φ η 2 -invariant subset of X 1 × η 2 .By Lemma 4.7, it is thus of the form U 1 × η 2 for some open subset U 1 ⊂ X 1 .There is an inclusion (of open subschemes of X 1 × η 2 ): U × X2 η 2 ⊂ U 1 × η 2 .It becomes a set-theoretic equality, and therefore an isomorphism of schemes, after base change along η 2 → η 2 .By faithfully flat descent, this implies that the two mentioned subsets of X 1 × η 2 agree.We claim U 1 = X 1 .Since the projection U → X 2 is surjective, in particular its image contains η 2 , so that U 1 is a non-empty subset, and therefore open dense in the irreducible scheme X We claim that there is a non-empty open subset A 2 ⊂ X 2 such that The underlying topological space of V = X 1 ×X 2 \U is Noetherian and thus has finitely many irreducible components V j .The closure of the projection p 2 (V j ) ⊂ X 2 does not contain η 2 , since X 1 × η 2 ⊂ U .Thus, A 2 := 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 have Z ∩X 1 ×A 2 = ∅, so that we may replace X 2 by 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 4.9.Let X 1 , . . ., X n be qcqs F q -schemes, and denote X = X 1 × Fq . . .× Fq X n .Let X i = lim j X ij be a cofiltered limit of finite type F q -schemes with affine transition maps, and write X = lim j X j , X j := X 1j × Fq . . .× Fq X nj (see [Sta17, Tag 01ZA] for the existence of such presentations).Let Z ⊂ X be a constructible closed subset.Then the intersection 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 m Xij (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 m Xij (Z j ) × Xj X = Frob m Xi (Z) as sets for all m ∈ Z.Thus, Z ′ = Z ′ j × Xj X which, also, is constructible closed because X → X j is affine.4.4.Lisse and constructible Weil sheaves.In this subsection, we define the subcategories of lisse and constructible Weil sheaves and establish a presentation similar to (4.4).Let X 1 , . . ., X n be schemes over F q , and denote Both categories are idempotent complete stable Γ(X, Λ)-linear symmetric monoidal ∞-categories.
From the presentation (4.6), we get that a Weil sheaf M is lisse if and only if the underlying object M F ∈ D(X F , Λ) is lisse.So (4.6) restricts to an equivalence The same is true for constructible Weil sheaves by the following proposition: , Λ is constructible if and only if the underlying sheaf M F ∈ D(X F , Λ) is constructible.Consequently, (4.6) restricts to an equivalence , Λ such that M F is constructible.We may assume that all X i are affine.We claim that there is a finite subdivision X F = ⊔X α into constructible locally closed subsets such that M F | Xα is lisse and such that each X α is partial Frobenius invariant.
Assuming the claim we finish the argument as follows.By Proposition 4.8, any open stratum U = X j0 ⊂ X F is a finite union of subsets of the form U 1,F × F . . .× F U n,F and the restriction of M to each of them is lisse.In particular, the complement X F \U is defined over F q and arises as a finite union of schemes of the form X ′ = X ′ 1 × Fq . . .× Fq X ′ n for suitable qcqs schemes X ′ i over F q .Intersecting each X ′ F with the remaining strata ⊔ j =j0 X j , we conclude by induction on the number of strata.
It remains to prove the claim.We start with any finite subdivision X F = ⊔X ′ j into constructible locally closed subsets such that M F | X ′ j is lisse.Pick an open stratum X ′ j0 , and set This is a constructible open subset of X F by Lemma 4.9 applied to its closed complement.Furthermore, M F | Xj 0 is lisse by its partial Frobenius equivariance, noting that φ * Xi induces equivalences on proétale topoi to treat the negative powers in (4.9).As before, X F \X j0 is defined over F q .So replacing X ′ j , j = j 0 by X ′ j ∩ (X F \X j0 ), the claim follows by induction on the number of strata.
In the case of a single factor X = X 1 , the preceding discussion implies For a Noetherian scheme X, let π 1 (X) be the étale fundamental groupoid of X as defined in [SGA03, Exposé V, §7 and §9].Its objects are geometric points of X, and its morphisms are isomorphisms of fiber functors on the finite étale site of X.This is an essentially small category.The automorphism group in π 1 (X) at a geometric point x → X is profinite.It is denoted π 1 (X, x) and called the étale fundamental group of (X, x).If X is connected, then the natural map Bπ 1 (X, x) → π 1 (X) is an equivalence for any x → X.If X is the disjoint sum of schemes X i , i ∈ I, then π 1 (X) is the disjoint sum of the π 1 (X i ), i ∈ I.In this case, if x → X factors through X i , then π 1 (X, x) = π 1 (X i , x).Definition 4.12.Let X 1 , . . ., X n be Noetherian schemes over F q , and write X = X 1 × Fq . ..×Fq X n .The Frobenius-Weil groupoid is the stacky quotient where we use that the partial Frobenii φ Xi induce automorphisms on the finite étale site of X F .
For n = 1, we denote FWeil(X) = Weil(X).Even if X is connected, its base change X F might be disconnected in which case the action of φ X permutes some connected components.Therefore, fixing a geometric point of X F is inconvenient, and the reason for us to work with fundmental groupoids as opposed to fundmental groups.The automorphism groups in Weil(X) carry the structure of locally profinite groups: indeed, if X is connected, then Weil(X) is, for any choice of a geometric point x → X F , equivalent to the classifying space of the Weil group Weil(X, x) from [Del80, Définition 1.1.10].Recall that this group sits in an exact sequence of topological groups where π 1 (X F , x) carries its profinite topology and Z the discrete topology.The topology on the morphism groups in Weil(X) obtained in this way is independent from the choice of x → X F .The image of Weil(X, x) → Z is the subgroup mZ where m is the degree of the largest finite subfield in Γ(X, O X ).In particular, we have m = 1 if X F is connected.Let us add that if x → X F is fixed under φ X , then the action of φ X on π 1 (X F , x) corresponds by virtue of the formula φ * X = (φ * F ) −1 to the action of the geometric Frobenius, that is, the inverse of the q-Frobenius in Weil(F/F q ).
Likewise, for every n ≥ 1, the stabilizers of the Frobenius-Weil groupoid are related to the partial Frobenius-Weil groups introduced in [Dri87, Proposition 6.1] and [Laf18, Remarque 8.18].In particular, there is an exact sequence 1 → π 1 (X F , x) → FWeil(X, x) → Z n , for each geometric point x → X F .This gives FWeil(X) the structure of a locally profinite groupoid.
Let Λ be either of the following coherent topological rings: a coherent discrete ring, an algebraic field extension E ⊃ Q ℓ for some prime ℓ, or its ring of integers O E ⊃ Z ℓ .For a topological groupoid W , we will denote by Rep Λ (W ) the category of continuous representations of W with values in finitely presented Λ-modules and by Rep fp Λ (W ) ⊂ Rep Λ (W ) its full subcategory of representations on finite projective Λ-modules.Here finitely presented Λ-modules M carry the quotient topology induced from the choice of any surjection Λ n → M , n ≥ 0 and the product topology on Λ n .Lemma 4.13.In the situation above, the category Rep Λ (W ) is Λ * -linear and abelian.In particular, its full subcategory Rep fp Λ (W ) is Λ * -linear and additive.Proof.Let W disc be the discrete groupoid underlying W , and denote by Rep Λ (W disc ) the category of W discrepresentations on finitely presented Λ-modules.Evidently, this category is Λ * -linear.It is abelian since Λ is coherent (Synopsis 3.2 (vi), see also [HRS23,Lemma 6.5].We claim that Rep Λ (W ) ⊂ Rep Λ (W disc ) is a Λ * -linear full abelian subcategory.If Λ is discrete (and coherent), then every finitely presented Λ-module carries the discrete topology and the claim is immediate, see also [Sta17,Tag 0A2H].For Λ = E, O E , one checks that every map of finitely presented Λ-modules is continuous, every surjective map is a topological quotient and every injective map is a closed embedding.For the latter, we use that every finitely presented Λ-module can be written as a countable filtered colimit of compact Hausdorff spaces along injections, and that every injection of compact Hausdorff spaces is a closed embedding.This implies the claim.
Definition 4.14.For an integer n ≥ 0, we write D {−n,n} lis (X, Λ) for the full subcategory of D lis (X, Λ) of objects M such that M and its dual M ∨ lie in degrees [−n, n] with respect to the t-structure on D(X, Λ).Lemma 4.15.In the situation above, there is a natural functor that is fully faithful.Moreover, the following properties hold if Λ is either finite discrete or Λ = O E for E ⊃ Q ℓ finite: (1) An object M lies in the essential image of (4.14) if and only if its underlying sheaf M F is locally on (X F ) proét isomorphic to N ⊗ Λ * Λ X F for some finitely presented Λ * -module N .
Proof.There is a canonical equivalence of topological groupoids π 1 (X that are compatible with the action of φ Xi for all i = 1, . . ., n.So we obtain the fully faithful functor (4.14) by passing to fixed points, see (4.13), (4.7) and Lemma 2.2 (see also Remark 2.3).

Part (1) describes the essential image of Rep
So if Λ is finite discrete or profinite, then the first functor in (4.15) is an equivalence, and we are done.Part (2) is immediate from (1), noting that an object in the essential image of (4.15) is lisse if and only if its underlying module is finite projective.Likewise, part (3) is immediate from (1), using Synopsis 3.2 (vii).Here we need to exclude rings like Λ = Z/ℓ 2 in order to have a t-structure on lisse sheaves.
Finally, if all X i are geometrically unibranch, so is X F which follows from the characterization [Sta17, Tag 0BQ4].In this case, we get π 1 (X F ) ∼ = π proét 1 (X F ) by [BS15,Lemma 7.4.10].This finishes the proof.4.6.Weil-étale versus étale sheaves.We end this section with the following description of Weil sheaves with (ind-)finite coefficients.Note that such a simplification in terms of ordinary sheaves is not possible for Λ = Z, Z ℓ , Q ℓ , say.
Proposition 4.16.Let X be a qcqs F q -scheme.Let Λ be a finite discrete ring or a filtered colimit of such rings.Then the natural functors Proof.Throughout, we repeatedly use that filtered colimits commute with finite limits in Cat ∞ .Using compatibility of D cons with filtered colimits in Λ (Synopsis 3.2 (iv)), we may assume that Λ is finite discrete.By the comparison result with the classical bounded derived category of constructible sheaves [HRS23, Proposition 7.1], we can identify the categories D • (X, Λ), resp.D • (X F , Λ) for • ∈ {lis, cons} with full subcategories of the derived category of étale Λ-sheaves D(X ét , Λ), resp.D(X F,ét , Λ). Write X = lim X i as a cofiltered limit of finite type F q -schemes X i with affine transition maps [Sta17, Tag 01ZA].Using the continuity of étale sites [Sta17, Tag 03Q4], there are natural equivalences colim D • (X i , Λ) ) for • ∈ {lis, cons}.Hence, we can assume that X is of finite type over F q .
To show full faithfulness, we claim more generally that the natural map is fully faithful.As Λ is torsion, this is immediate from [Gei04, Corollary 5.2] applied to the inner homomorphisms between sheaves.Let us add that this induces fully faithful functors on bounded below objects, see [BS15, Proposition 5.2.6 (1)].
It remains to prove essential surjectivity.Using a stratification as in Definition 4.10, it is enough to consider the lisse case.Pick M ∈ D lis (X Weil , Λ).It is enough to show that M lies is in the essential image of (4.17), noting that the functor detects dualizability.As M is bounded, this will follow from showing that for every j ∈ Z, the cohomology sheaf H j (M ) ∈ D(X Weil , Λ) ♥ is in the essential image of (4.17).
Fix j ∈ Z.As M is lisse, the underlying sheaf H j (M ) F ∈ D(X F , Λ) ♥ is proétale-locally constant (Synopsis 3.2 (vii)) and valued in finitely presented Λ-modules.By Lemma 4.15 (1), it comes from a representation of Weil(X).Restriction of representations along Weil(X) → π 1 (X) fits into a commutative diagram where the upper horizontal arrow is an equivalence since Λ is finite.In particular, the object H j (M ) is in the essential image of the fully faithful functor (4.17).

The categorical Künneth formula
We continue with the notation of Section 4. In particular, F q denotes a finite field of characteristic p > 0. Recall from Section 2 the tensor product of Λ * -linear idempotent complete stable ∞-categories.The external tensor product of sheaves (M 1 , . . for • ∈ {lis, cons}.Throughout, we consider the following situation.In Remark 5.3 we explain the compatibility of (5.1) with certain (co-)limits in the schemes X i and coefficients Λ, which allows to relax these assumptions on X and Λ somewhat.
Situation 5.1.The schemes X 1 , . . ., X n are of finite type over F q , and Λ is the condensed ring associated with one of the following topological rings: (a) a finite discrete ring of prime-to-p-torsion; (b) the ring of integers O E of an algebraic field extension E ⊃ Q ℓ for ℓ = p (for example Zℓ ); (c) an algebraic field extension E ⊃ Q ℓ for ℓ = p (for example Qℓ ); (d) a finite discrete p-torsion ring that is flat over Z/p m for some m ≥ 1.
In the p-torsion free cases (a), (b) and (c), the full faithfulness is a direct consequence of the Künneth formula applied to the X i,F .In the p-torsion case (d), we use Artin-Schreier theory instead.It would be interesting to see whether this part can be extended to constructible sheaves using the mod-p-Riemann-Hilbert correspondence as in, say, [BL19].In all cases, the essential surjectivity relies on a variant of Drinfeld's lemma for Weil group representations.
Before turning to the proof of Theorem 5.2, we record the following compatibility of the functor (5.1) with (co-)limits.This can be used to reduce the case of an (infinite) algebraic extension E ⊃ Q ℓ in cases (b) and (c) above to the case where E ⊃ Q ℓ is finite.In the sequel we will therefore assume E is finite in these cases.Remark 5.3 can further be used to extend Theorem 5.2 to qcqs F q -schemes X i and finite discrete rings like Z/m for any integer m ≥ 1 in cases (a) and (d).
(1) Filtered colimits in Λ.First off, extension of scalars along any map of condensed rings Λ → Λ ′ induces a commutative diagram in Cat Ex ∞,Λ * (Idem): It follows from the compatibility of D cons with filtered colimits in Λ (Synopsis 3.2 (iv)) that both sides of (5.1) are compatible with filtered colimits in Λ.
(2) Finite products in Λ.Let Λ = Λ i be a finite product of condensed rings.For any scheme X, the natural map D • (X, Λ) → D • (X, Λ i ) is an equivalence for • ∈ {∅, lis, cons}, and likewise for Weil sheaves if X is defined over F q .As Λ * = Λ i, * , we see that (5.1) is compatible finite products in the coefficients.(3) Limits in X i for discrete Λ. Assume that Λ is finite discrete, see Situation 5.1 (a), (d).Let X 1 , . . ., X n be qcqs F q -schemes.Write each X i as a cofiltered limit X i = lim X ij of finite type F q -schemes X ij with affine transition maps [Sta17, Tag 01ZA].As Λ is finite discrete, we can use the continuity of étale sites as in (4.16) to show that the natural map is an equivalence for • ∈ {lis, cons}.Thus, (5.1) is compatible with cofiltered limits of finite type F q -schemes with affine transition maps.

5.1.
A formulation in terms of prestacks.Before turning to the proof, we point out a formulation of the results of the previous subsection in terms of symmetric monoidality of a certain sheaf theory.This formulation makes the connection with constructions in the geometric approaches to the Langlands program [GKRV22, Zhu21, LZ19] more manifest.Readers not familiar with prestacks and formulations of sheaf theories on them can safely skip this section.The categories of constructible, resp.lisse Λ-sheaves assemble into a lax symmetric monoidal functor Namely, as a functor it sends a scheme X to the category of constructible, resp.lisse Λ-sheaves on X, and a morphism f : X → Y to the functor f * : D • (Y, Λ) → D • (X, Λ).These are objects, resp.maps in the ∞-category Cat Ex ∞,Λ (Idem) := Mod Perf Λ (Cat Ex ∞ (Idem)), cf.Section 2.1 for notation.The lax monoidal structure is given by the external tensor product of sheaves: That is, we consider the category of schemes as symmetric monoidal with respect to the fiber product over F, and the external tensor product is natural on X and Y in the appropriate sense, see [GL19, Section 3.1], [GR17, Section III.2] for details and precise statements.This functor ⊠ often fails to be an equivalence, so D •,Λ is not symmetric monoidal.The assertion of Theorem 5.2 is that this issue is resolved by replacing sheaves with Weil sheaves.In order to formulate Theorem 5.2 as the monoidality of a certain functor, we need to replace the category of schemes by a category of objects that model Weil sheaves.We will represent these by taking the appropriate formal quotient by the partial Frobenius automorphism.Such formal quotients can be taken in the category of prestacks.
We denote by PreStk F the category of (accessible) functors from the category CAlg F of commutative algebras over F to the ∞-category Ani of Anima.The functor of taking points embeds the category of schemes fully faithfully into PreStk F .We denote by D the functor obtained by right Kan extension [Lur09, §4.3.2]along the inclusion (Sch fp F ) op ⊂ (PreStk F ) op .Concretely, [Lur18, Proposition 6.2.1.9,Proposition 6.2.3.1],given a prestack Y which can be written as a colimit of schemes Y α over some indexing category A we have a canonical equivalence (5.4) This limit is formed in Cat Ex ∞,Λ (Idem); recall from around (2.3) that the Ind-completion functor to Cat Ex ∞,Λ (Idem) → Pr St Λ does not preserve (even finite) limits.With this general sheaf theory in place, we can restrict our attention to the class of prestacks that is relevant to the derived Drinfeld lemma.Definition 5.4.Let X be a scheme over F q .The Weil prestack is defined as is an equivalence for any N i ∈ D lis (X Weil i ).Using Zariski descent for both sides, we may assume that each X i is affine.As Λ is finite discrete (see also the discussion around (4.16)), the invariance of the étale site under perfection reduces us to the case where each X i is perfect.The proof now proceeds by several reduction steps: 1) reduce to N i = Λ Xi ; 2) reduce to Λ = Z/p; 3) reduce to q = p being a prime.The last step 4) is then an easy computation.
Step 1): We may assume N i = Λ Xi .In order to show (5.10) is a quasi-isomorphism, it suffices to show this after applying τ ≤r for arbitrary r.The complexes N i are bounded (Synopsis 3.2 (v)).By shifting them appropriately, we may assume r = 0. Note that RΓ(X Weil i , N i ) ∼ = RΓ(X i , N i ), see Proposition 4.16.By right exactness of the tensor product, we have τ ≤0 ( i RΓ(X i , N i )) = i τ ≤0 RΓ(X i , N i ).By the comparison with the classical notion of constructible sheaves (for discrete coefficients, see [HRS23,Proposition 7.1] and the discussion preceding it), there is an étale covering U i → X i such that N i | Ui is perfect-constant.Let U i,• be the Čech nerve of this covering.By étale descent, we have RΓ(X i , N i ) = lim [j]∈∆ RΓ(U i,j , N i ).
For each r ∈ Z, there is some j r such that This can be seen from the spectral sequence (note that it is concentrated in degrees j ≥ 0 and degrees j ′ ≥ r for some r, since the complexes N i are bounded from below) As the tensor product in (5.10) commutes with finite limits, we may thus assume that each N i is perfect-constant.
Another dévissage reduces us to the case N i = Λ Xi , the constant sheaf itself.
Step 2): We may assume Λ = Z/p.By assumption, Λ is flat over Z/p m for some m ≥ 1.We immediately reduce to Λ = Z/p m .For any perfect affine scheme X = Spec R in characteristic p > 0, we claim that RΓ(X, Z/p m ) ⊗ Z/p m Z/p r ∼ = RΓ(X, Z/p r ).Assuming the claim, we finish the reduction step by tensoring (5.10) with the short exact sequence of Z/p m -modules 0 → Z/p m−1 → Z/p m → Z/p → 0, using that finite limits commutes with tensor products.It remains to prove the claim.The Artin-Schreier-Witt exact sequence of sheaves on X ét yields Now we use that W m (R) ⊗ Z/p m Z/p r ∼ = → W r (R) compatibly with F , which holds since R is perfect.This shows the claim, and we have accomplished Step 2).
Step 3): We may assume q is prime.Recall that q = p r is a prime power.In order to reduce to the case r = 1, let X ′ i := X i , but now regarded as a scheme over F p .We have X ′ i,F = r i=1 X i,F .The Galois group Gal(F q /F p ) is generated by the p-Frobenius, which acts by permuting the components in this disjoint union.Thus, we have D(( ).The same reasoning also applies to several factors X Weil i , so we may assume our ground field to be F p .
Step 4): Set R := i,Fp R i , R F := R ⊗ Fp F. We write φ i for the p-Frobenius on R i and also for any map on a tensor product involving R i , by taking the identity on the remaining tensor factors.By Artin-Schreier theory, we have RΓ(X Weil i , Z/p) where φ is the absolute p-Frobenius of R F .Thus, the right hand side in (5.10) is the homotopy orbits of the action of Z n+1 on R F , whose basis vectors act as φ 1 , . . ., φ n and φ.Note that φ is the composite φ , where φ F is the Frobenius on F. Thus, the previously mentioned Z n+1 -action on R F is equivalent to the one where the basis vectors act as φ 1 , . . ., φ n and φ F .We conclude our claim by using that Let X 1 , . . ., X n be Noetherian schemes over F q , and denote X = X 1 × Fq . . .× Fq X n .Recall the Frobenius-Weil groupoid FWeil(X), see Definition 4.12.The projections X F → X i,F onto the single factors induce a continuous map of locally profinite groupoids (5.11) Theorem 5.9 (Version of Drinfelds's lemma).Let Λ be as in Situation 5.1.Restriction along the map (5.11) induces an equivalence (5.12) between the abelian categories of continuous representations on finitely presented Λ-modules.
Proof.For all objects x ∈ FWeil(X), that is, all geometric points x → X F , passing to the automorphism groups induces a commutative diagram of locally profinite groups The left vertical arrow is surjective [Sta17, Tags 0BN6, 0385].Thus µ x is surjective as well and hence (5.12) is fully faithful.For essential surjectivity, it remains to show that any continuous representation FWeil(X, x) → GL(M ) on a finitely presented Λ-module M factors through µ x .The key input is Drinfeld's lemma: it implies that µ x induces an isomorphism on profinite completions.Therefore, it is enough to apply Lemma 5.10 below with H := FWeil(X, x) → Weil(X 1 ) × . . .× Weil(X n ) =: G and K := π 1 (X F , x).This completes the proof of (5.12).
The following lemma formalizes a few arguments from [Xue20b, §3.2.3], and we reproduce the proof for the convenience of the reader: Lemma 5.10 (Drinfeld, Xue).Let Λ be as in Situation 5.1.Let µ : H → G be a continuous surjection of locally profinite groups that induces an isomorphism on profinite completions.Assume that there exists a compact open normal subgroup K ⊂ H containing ker µ such that H/K is finitely generated and injects into its profinite completion.Then µ induces an equivalence between their categories of continuous representations on finitely presented Λ-modules.
Proof.The case where Λ is finite discrete is obvious, and hence so is the case Λ = O E for some finite field extension E ⊃ Q ℓ .The case Λ = E is reduced to Λ = Q ℓ .As µ is surjective, it remains to show that every continuous representation ρ : H → GL(M ) on a finite-dimensional Q ℓ -vector space factors through G, that is, ker µ ⊂ ker ρ.One shows the following properties: (1) The group ker µ is the intersection over all open subgroups in K which are normal in H.
(2) The group ker ρ∩K is a closed normal subgroup in H such that K/ ker ρ∩K ∼ = ρ(K) is topologically finitely generated.These properties imply ker µ ⊂ ker ρ ∩ K as follows: For a finite group L, let U L := ∩ ker(K → L) where the intersection is over all continuous morphisms K → L that are trivial on ker ρ ∩ K.Because of the topologically finitely generatedness in (2), this is a finite intersection so that U L is open in K. Also, it is normal in H, and hence ker µ ⊂ U L by (1).On the other hand, it is evident that ker ρ ∩ K = ∩ L U L because K is profinite.
For the proof of (1) observe that ker µ agrees with the kernel of H → H ∧ ∼ = G ∧ by our assumption on the profinite completions.Using ker µ ⊂ K and the injection H/K → (H/K) ∧ implies (1).
For (2) it is evident that ker ρ ∩ K is a closed normal subgroup in H. Since K is compact, its image ρ(K) is a closed subgroup of the ℓ-adic Lie group GL(M ), hence an ℓ-adic Lie group itself.The final assertion follows from [Ser64, théorème 2].
For the overall goal of proving essential surjectivity in Theorem 5.2, we need to investigate how representations of product groups factorize into external tensor products of representations.In view of Lemma 4.13 and its proof, it is enough to consider representations of abstract groups, disregarding the topology.This is done in the next section.Proof.For full faithfulness, it is enough to consider the case • = indcons.Using Lemma 2.5, it remains to show that the functor (6.4) is fully faithful.In view of (6.1), this is immediate from the Künneth formula for constructible Λ-sheaves as explained in Section 5.2.To identify objects in the essential image, we note that the fully faithful functors (6.3) and (6.4) induce a Cartesian diagram (see Lemma 2.5): / / D • (X F , Λ), (6.5) for • ∈ {indlis, indcons}.Thus, it is enough to show that the object M F underlying 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 (6.4), the split constructible case reduces to the split lisse case, see also the proof of Theorem 5.2 in Section 5.5.So assume • = indlis and let M F ∈ D(X F , Λ) be split lisse.As each cohomology sheaf H j (M F ), j ∈ Z is at least ind-lisse (see also [HRS23,Remark 8.4]), an induction on the cohomological length of M F reduces us to show that H j (M F ) lies in the essential image.By definition, being split lisse implies that the action of π proét 1 (X F ) on H j (M F ) factors through π proét 1 (X 1,F ) × . . .× π proét 1 (X n,F ).Then the arguments of Section 5.5 show that H j (M F ) lies is in the essential image of the lower horizontal arrow in (6.5).We leave the details to the reader.Remark 6.6.The functor (6.3) 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 (for example, M as in Example 1.4), then F (M ) will not be either.This is, however, the only obstacle for essential surjectivity: as noted in the proof of Theorem 6.5, the diagram (6.5) is Cartesian.6.2.Cohomology of shtuka spaces.Finally, let us mention a key application of Theorem 6.5.Let X be a smooth projective geometrically connected curve over F q .Let N ⊂ X be a finite subscheme, and denote its complement by Y = X\N .Let E ⊃ Q ℓ , ℓ = p be an algebraic field extension containing a fixed square root of q.Let O E be its ring of integers and denote by k E the residue field.Let Λ be any of the topological rings E, O E , k E .Let G be a split (for simplicity) reductive group over F q .We denote by G the Langlands dual group of G considered as a split reductive group over Λ.
In the seminal works [Dri80,Laf02] (G = GL n ) and [Laf18,LZ19] (general reductive G) on the Langlands correspondence over global function fields, the construction of the Weil(Y )-action on automorphic forms of level N is realized using the cohomology sheaves of moduli stacks of shtukas, defined in [Var04]  (6.8) Next, using the finiteness [Xue20b] and smoothness [Xue20c, Theorem 4.2.3]results, the classical Drinfeld's lemma (Theorem 5.9) applies to give objects H N,I (W ) ∈ Rep cts Λ Weil(Y ) I .The construction of the natural transformation (6.6) encodes the functoriality and fusion satisfied by the objects {H N,I (W )} for varying I and W .
However, in order to analyze construction (6.6) further, it is desirable to upgrade the natural transformation functors (6.6) to the derived level.Namely, to have construction for the complexes {H I (W )} I,W and not just for their cohomology sheaves, compare with [Zhu21].Such an upgrade is possible using the derived version of Drinfeld's lemma, as given in the following proposition.A further study of this construction will appear in future work of the first named author (T.H.).In particular, the action of π 1 (X I F ) on H N,I (W ) factors through the product π 1 (X F ) I .So it is ind-(split lisse) in the sense of Definition 6.4, and we are done by Theorem 6.5.Remark 6.8.One can upgrade the above construction in a homotopy coherent way to show that the whole complex H N,I (W ) lies in D indcons (Y Weil ) I , Λ .If N = ∅ so that H N,I (W ) is known to be bounded, then Proposition 6.7 implies that H N,I (W ) lies in the essential image of (6.9).

5. 4 .
Factorizing representations.In this subsection, let Λ be a Dedekind domain [Sta17, Tag 034X].Thus, any submodule N of a finite projective Λ-module M is again finite projective.Given any group W , we write Rep fp Λ (W ) for the category of W -representations on finite projective Λ-modules.As in [CR06, Sections 73.8, 75], we say that such a W -representation M is fp-simple if any subrepresentation 0 = N ⊂ M has maximal rank.By induction on the rank, every non-zero representation in Rep fp Λ (W ) admits a non-zero fp-simple subrepresentation.The proof of the following lemma is left to the reader.It parallels [CR06, Theorem 75.6].Lemma 5.11.A representation M ∈ Rep fp Λ (W ) is fp-simple if and only if M ⊗ Λ Frac(Λ) is fp-simple (hence, simple).
and [Laf18, Section 2].As explained in [LZ19, GKRV22, Zhu21], the output of the geometric construction of Lafforgue can be encoded as a natural transformation H N,I : Rep fp Λ G I → Rep cts Λ Weil(Y ) I , I ∈ FinSet (6.6) of functors FinSet → Cat from the category of finite sets to the category of 1-categories.Here the functor Rep fp Λ G • assigns to a finite set I the category of algebraic representations of G I on finite free Λ-modules, and Rep cts Λ Weil(Y ) • the category of continuous representations of Weil(Y ) I in Λ-modules.In both cases, the transition maps are given by restriction of representations.Let us recall some elements of its construction.For a finite set I, [Var04] and [Laf18, Section 2] define the ind-algebraic stack Cht N,I classifying I-legged G-shtukas on X with full level-N -structure.The morphism sending a G-shtuka to its legs p N,I : Cht N,I → Y I , (6.7) is locally of finite presentation.For every W ∈ Rep fp Λ G I , there is the normalized Satake sheaf F N,I,W on Cht N,I , see [Laf18, Définition 2.14].Base changing to F and taking compactly supported cohomology, we obtain the object H N,I (W ) def = (p N,I,F ) ! (F N,I,W,F ) ∈ D indcons Y I F , Λ , see [Laf18, Définition 4.7] and [Xue20a, Definition 2.5.1].Under the normalization of the Satake sheaves, the degree 0 cohomology sheaf H N,I (W ) def = H 0 (H I (W )) ∈ D indcons Y I F , Λ ♥ corresponds to the middle degree compactly supported intersection cohomology of Cht N,I .Using the symmetries of the moduli stacks of shtukas, the sheaf H N,I (W ) is endowed with a partial Frobenius equivariant structure [Laf02, §6].So we obtain objects H N,I (W ) ∈ D indcons (Y Weil ) I , Λ ♥ .
φ * * Xi , see Remark 2.3.Explicitly, for n = 2, this is the homotopy limit of the diagram The Weil sheaf M is called lisse if it is dualizable.(Here dualizability refers to the symmetric monoidal structure on D X Weil The Weil sheaf M is called constructible if for any open affine U i ⊂ X i there exists a finite subdivision into constructible locally closed subschemes U ij ⊆ U i such that each restriction M | U Relation with the Weil groupoid.In this subsection, we relate lisse Weil sheaves with representations of the Weil groupoid.Throughout, we work with étale fundamental groups as opposed to their proétale variants in order to have Drinfeld's lemma available, see Section 5.3.The two concepts differ in general, but agree for geometrically unibranch (for example, normal) Noetherian schemes, see [BS15, Lemma 7.4.10].