Drinfeld’s lemma and the Global Function Field Lanlgands Cor-
The Langlands Correspondence over function fields
An important symmetry of the space of unbranched automorphic functions is the effect of Hecke algebras. The connection to the arithmetic of the function field comes in through the equivalence Dlis(𝑋 ,Qℓ) ≃D𝑏(Rep(𝜋𝑒𝑡´.
Drinfeld’s lemma and the Künneth formula for Weil sheaves
Our first main result is a generalization of Drinfeld's lemma to the entire derived category of constructible disks. For Lisse discs, this holds ifΛ is a torsion ring or an extension of Zℓ, and always holds for Lisse if𝑋.
Application to the moduli of bundles and shtukas
Suppose that for each𝜑: 𝑠 → 𝑠′operator Note that ( . 1)𝑅 is given by the composition of the lower functors in the commutative diagram below.
The Categorical Trace in the Categorical Local Lanlgands Program 12
Convolution Patterns and the Iwahori-Hecke category
The abstract results in the second part of the thesis are motivated by the understanding of unipotent representations. As we did with the function arc dictionary, we can relate this monoid category to the category of unipotent representations by taking the "track of Frobenius".
The Categorical trace construction
These definitions and its implication for the study of the category of unipotent representations are the subject of a joint project with Xinwen Zhu and are beyond the scope of this thesis. As in (Lurie, 2009, Section 5.5.3), PrL denotes the ∞ category of presentable∞ categories with colimit-preserving functors.
Monoidal aspects
It contains the subcategory PrSt ⊂ PrL consisting of stable∞ categories. and exact functors: it is characterized by 𝐷1⊗𝐷. 4.1), that is, the compact objects in the Lurie tensor product of the Ind completions. Each category in PrStΛ is canonically enriched by ModΛ, so we refer to Hom𝐷(−,−) ∈ ModΛ as the mapping complex.
Fixed points of infinity categories
In particular, the object H𝑗(𝑀) is in the essential image of the fully faithful functor equation (6.17). By the full fidelity of Equation (8.4), the split settable case reduces to the split lisse case, see also the proof of 7.1.2 in Section 7.6. Then letℎ𝑜𝑟 𝑖 𝑧=𝑎𝑙 𝑙and let𝑣 𝑒𝑟 𝑡 consist of the following class of morphisms: 𝑓 : 𝑋 →𝑌 →𝑌 belongs to 𝑣 (if the 𝑣) :D (𝑌) → D (𝑋) allows a continuous left adjoint. (𝑓) 𝐿 and for each diagram as(11.2), the induced Beck-Chevalley map is an equivalence.
By Lemma 12.3.1 the functor between trace categories can be realized as a limit of functors between Hochschild complexes, so to get full fidelity it suffices to have full fidelity at the level of the functors𝐴⊗𝑛⊗Λ𝐹. Then, via the construction of Section 10.4, the object 𝑄 admits the structure of a (𝑋×𝑌 𝑋)-bimodule in Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧.
Lisse and constructible sheaves
Constructible Weil Sheaves
The Weil-proétale site
This yields a functor (𝑋 . pro´et) → PrStΛ, 𝑈 ↦→ D(𝑈Weil,Λ), a hypercomplete bundle of Λ∗-linearly presentable stable categories. According to Lemma 6.1.3, the cosimplical 1-topos associated with (𝑋Weil . pro´et)/(𝑋•,Φ•) is equivalent to the cosimplical 1-topos associated with the action of𝜙∗.
Weil sheaves on products
Partial-Frobenius stability
Then any partially Frobenius invariant constructible closed subset𝑍 ⊂ 𝑋 is a finite set-theoretic union of subsets of the form𝑍. In particular, every partially Frobenius invariant constructible open subscheme𝑈 ⊂ 𝑋 is a finite union of constructible open subschemes of the form𝑈.
Lisse and constructible Weil sheaves
We note that every Frob𝑋𝑖 induces a homeomorphism on the underlying topological space of 𝑋 such that 𝑍′ is well defined. From the presentation (6.6) we get that a Weil sheaf 𝑀 is lisse if and only if the underlying object 𝑀.
Relation with the Weil groupoid
For the topological groupoid𝑊, we will denote by RepΛ(𝑊) the category of continuous representations of 𝑊 with values in finitely represented Λ-modules, and by RepfΛ.p(𝑊) ⊂RepΛ(𝑊) its entire subcategory of representations on finite projective Λ-modules. The Λ-modules 𝑀 finally introduced here carry a quotient topology induced from choosing an arbitrary surjection Λ 𝑛 → 𝑀, 𝑛 ≥ 0 and a product topology on Λ 𝑛.
Weil-étale versus étale sheaves
Before proceeding to the proof of Theorem 7.1.2, we note the following compatibility of the functor equation (7.1) with (co-)limits. In terms of the duality onLincatΛ, the functor 𝐹𝑜 is equivalent to the left adjoint of 𝐹∨.
The categorical Künneth formula
The main result
In Remark 7.1.3 we explain the compatibility of equation (7.1) with (joint) bounds on the schemes 𝑋𝑖and coefficients Λ, which allows to relax somewhat these assumptions on 𝑋 and Λ. Thus, equation (7.1) is compatible with cofiltered bounds of finite F𝑞-type schemes with affine transition maps.
A formulation in terms of prestacks
We denote by PreStkF the category of (accessible) functors from the category CAlgF of commutative algebras overF to the ∞-category Ani of Anima. To check its restriction to the (symmetric monoidal) subcategory SchfpWeilis symmetric monoidal, it suffices to show that the slack monoidal maps are indeed isomorphisms.
Full faithfulness
To show that equation (7.9) is a quasi-isomorphism, it suffices to show this by applying 𝜏≤𝑟 to any 𝑟. Thus, the right-hand side in (7.9) is the homotopic orbit of the action Z𝑛+1 on 𝑅.
Drinfeld’s lemma
The case where Λ is finitely discrete is obvious, and so is the case Λ = O𝐸 for a finite field extension 𝐸 ⊃ Qℓ. The group ker𝜇 is the intersection of all open subgroups in 𝐾 that are normal in 𝐻.
Factorizing representations
Using (Bourbaki, 2012, Section 13.4 Corollaire) applied to the group algebras, it is enough to show that𝑀. Since Λ is perfect, any finite-dimensional representation is semi-simple if and only if it is absolutely semi-simple, see (Bourbaki, 2012, Section 13.1).
Essential surjectivity
The composition of the correspondences𝑌 ← 𝑆 → 𝑋 and 𝑍 ← 𝑇 → 𝑌 is given by the correspondence 𝑍 ← 𝑊 → 𝑋 formed by. Then assumption (2) of the statement implies that the essential images of the functor⊠𝑅◦𝜂★◦𝑚† on D (𝑋.
Ind-constructible Weil sheaves
Ind-constructible Weil sheaves
A sheaf 𝑀 is called a split lisse if it is a lisse and the action 𝜋pro´et. Then the arguments of Section 7.6 show that H𝑗(𝑀 . F) lies in the essential image of the lower horizontal arrow in equation (8.5).
Cohomology of shtuka spaces
For Λ ∈ {𝐸 ,O𝐸, 𝑘𝐸} and any 𝑊 ∈ RepΛ(𝐺b𝐼), the shtuka cohomology (8.8) lies in the essential image of the fully faithful functor. FunRAd(𝑆,LincatΛ)) the subcategory of Fun(𝑆,LincatΛ) consisting of functors 𝐹: 𝑆 → Lincat such that for all arrows 𝑠 → 𝑠′ in 𝑆 the functor 𝐹(allow a continuously) →(allow a ) right adjunct (resp. a left adjunct) and morphisms𝛼: 𝐹 → 𝐹′.
Dual categories
Relative tensor product
Then the lower horizontal functor of the above diagram is fully plausible, with the essential image generated under colimits by the image 𝑞†◦𝛿★. As in the proof of Lemma 13.5.2, it suffices to set up the following commutative diagram.
The formalism of correspondences
Category of correspondences
To emphasize that such a morphism in Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧 is a correspondence rather than an actual map, we sometimes write it as 𝑋. In particular, it makes sense to talk about associative and commutative algebra objects in Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧.
Monads and modules in the category of correspondences
Every object 𝑋 ∈ C with diagonal map Δ𝑋 : 𝑋 → 𝑋 × 𝑋 and structural map𝜋𝑋 : 𝑋 →pt belonging toℎ𝑜 𝑖 𝑧) with multiplication given byΔ𝑋 and unit given by𝜋𝑋. In addition, if 𝑋 and𝑌 are two objects satisfying the above properties and 𝑓 : 𝑋 → 𝑋 induced by 𝑓.
Monads and Segal objects
Roughly speaking, the functor Sp from (1) sends 𝑋 ← 𝑍 → 𝑋 to the object𝑍 ∈ C with the slack structure given by horizontal arrow𝑍 ×𝑋 𝑍 → 𝑍× 𝑍.
The action on bimodules
Under the same assumption as in Proposition 13.2.1 and given 𝜙𝑋, 𝜙𝑌 as above, there is a canonical factorization. On the other hand, the monadicity of the simple objects in Lemma 13.3.2 can be used to compare the ordinary trace and the geometric trace in other situations.
Sheaf theories
The notion of a sheaf theory
Let Λ𝑋 ∈ D (𝑋) denote the unitary object with respect to this symmetric monoidal structure corresponding to the functor. It follows that if the sheaf theory D is symmetric monoidal (rather than just loosely symmetric monoidal), then D (𝑋) is self-dual.
Additional base change theorems
In many examples of a sheaf theory, the functor 11.4 is fully faithful and allows a continuous right adjoint.
Category of cohomological correspondences
Consider the extended simplicial category associated with Cech's nerve 𝑞. Using Lemma 13.3.3, we identify the Czech nerve of this map with the relative Hochschild complex. 13.23), where the horizontal arrows to the left are obtained by the corresponding horizontal arrows to the right in (13.22) by passing to the right adjoint.
The categorical trace
Hochschild homology
So if Then the Hochschild homology of (𝐴, 𝐹) is nothing but the trace𝐹 in the sense of (12.1), considered as an endomorphism of 𝐴 in Morita(R).
Functoriality of Hochschild homology
Suppose that𝑀 is fully dualizable, and suppose𝛼 also admits continuous right adjoint 𝛼𝑅, and suppose𝛼𝑅 is also a 𝐴-𝐵-bimodule homomorphism, then Tr(𝑀 , 𝛼) admits continuous right adjoint.(𝑝) Then 𝛼𝑋 admits a continuous right adjoint (𝛼𝑋 )𝑅 if and only if 𝑋 is a compact object in 𝐹𝐴, in which case (𝛼𝑋)𝑅 is automatically a left 𝐴-module homomorphism since it is a right Gaitsgory and Rozenblyum, 2017a, Lemma 9.3.6.).
Categorical traces of rigid monoidal categories
Under the assumptions of Proposition 13.3.1, the cosimplicial categories obtained from the simple categories D (HH(𝑋 .. 1, 𝑄)•) by going to the right adjoint satisfy the Beck-Chevalley conditions. The left triangle is actually commutative, as we are in the case as in Lemma 13.5.3, and the right square is commutative as the natural function D (𝑅) ⊗D (𝑋′.
Geometric traces in sheaf theory
Geometric Hochschild homology
AsDis a weakly symmetric monoidal functor, D (𝐴) is an algebra object in LincatΛandD (𝑀) is aD (𝐴)-module object in LincatΛ, with multiplication and action maps given by. Then the geometric trace Trgeo(D (𝐴),D (𝐹)) is the geometric realization of this simple object in LincatΛ.
Fixed point objects and geometric traces of convolution categories
With the universal property P (Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧), the functor D : Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧 → LincatΛ expands to a continuous functor D : P (Corr(C)𝑣 𝑒𝑟 𝑡;ℎ𝑜𝑟 𝑖 𝑧 → LincatΛ). ℎ𝑜𝑟 𝑖 𝑧) → LincatΛ. The essential image is generated under colimits with the image (id𝑊1× 𝑓 ×id𝑊2)†◦ (id𝑊1×Δ𝑋 ×id𝑊2)★.
The geometric trace and relative resolutions
1 is in C𝑝𝑟 𝑜 𝑝, the top two squares of the diagram can be treated as in the co-face mapping case. Then to show that𝛿★is fully faithful, it suffices to show that the natural map.
Comparison between geometric and ordinary traces
It remains to show that the two monads are identified which will imply the proposition. We note that the commutativity of the triangle follows from Assumption (2), and the commutativity of the square under the triangle follows from Assumptions.
Functoriality of categorical traces in geometric setting
For the middle parallelogram, first consider the commutative diagram. with horizontal morphisms induced by the correspondence𝑌 ← 𝑋 → 𝑋×𝑋and vertical morphisms induced by 𝑌′← 𝑋′→ 𝑋′×𝑋′. 1) are rigid, using Lemma 12.3.1 we obtain the commutativity of the parallelogram. We would like to relate Tr(D (𝑀), 𝛼) to the above functor under certain assumptions. where the horizontal arrows are the right adjunct of naturals.
