K$K$ ‐Motives and Koszul duality

We construct an ungraded version of Beilinson–Ginzburg–Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of K$K$ ‐theory. For this, we introduce and study categories DKS(X)$\hbox{DK}_\mathcal {S}(X)$ of S$\mathcal {S}$ ‐constructible K$K$ ‐motivic sheaves on varieties X$X$ with an affine stratification. We show that there is a natural and geometric functor, called Beilinson realisation, from S$\mathcal {S}$ ‐constructible mixed sheaves DSmix(X)$\hbox{D}^{mix}_\mathcal {S}(X)$ to DKS(X)$\hbox{DK}_\mathcal {S}(X)$ . We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible K$K$ ‐motivic sheaves on Langlands dual flag varieties.


INTRODUCTION
Let ⊃ be a split reductive group with a Borel subgroup and = ∕ be the flag variety. Denote by ∨ ⊃ ∨ the Langlands dual group with a Borel subgroup and by ∨ = ∨ ∕ ∨ the Langlands dual flag variety. In this article we prove an ungraded and -theoretic version of Beilinson-Ginzburg-Soergel's Koszul duality.
Theorem. There is a commutative diagram of functors where the horizontal arrows are equivalences Let us explain the ingredients of this diagram. In [6,30] and [7], Beilinson-Ginzburg-Soergel consider a category D ( ) ( ) of ( )-constructible mixed sheaves on , which is a graded version of the ( )-constructible derived category D ( ) ( ) of sheaves on the flag manifold (ℂ) and, equivalently, the bounded derived category of category  of the Lie algebra = Lie( (ℂ)). In particular, there is an autoequivalence (1) of D ( ) ( ) called Tate twist, shifting the grading, and a functor called Betti realisation, forgetting the grading.
Most remarkably, Beilinson-Ginzburg-Soergel construct a triangulated equivalence, called Koszul duality,K os ∶ D ( ) ( ) → D ( ∨ ) ( ∨ ) mapping projective perverse sheaves to intersection cohomology complexes and intertwining the Tate twist (1) with the shift twist (1) [2]. Although very desirable, there is no geometric construction of Koszul duality yet. All six functors commute with Tate twists and hence some new geometric constructions have to enter the picture.
The main idea of this article is to construct an ungraded version of Koszul duality and to thereby fill in the bottom right corner of the above diagram. This is achieved by a -theoretic point of view and the observation that Betti realisation forgets the Tate twist (1) whereas passage to -theory forgets the shift twist (1) [2].
To make this observation more precise, we use Soergel-Wendt's very satisfying construction of D ( ) ( ) as a full subcategory of the category of motivic sheaves DM( ∕ , ℚ), see [33]. We define the category DK ( ∨ ) ( ∨ ) as a full subcategory of the category of -motivic sheaves DK( ∨ ∕ , ℚ) analogously. -motivic sheaves can be thought of as a -theoretic cousin of the category of constructible sheaves computing algebraic -theory instead of Betti cohomology.
There is a functor, which we call Beilinson realisation, expressing Beilinson's realisation that, rationally, motivic cohomology is a graded refinement of algebraic -theory, see [8]. There is a natural isomorphism, called Bott isomorphism, ℚ ≅ ℚ(1) [2] in DK ( ∨ ) ( ∨ ) and hence forgets the shift by (1) [2]. The construction of the ungraded Koszul duality functor is just a copy of Soergel's construction ofKos. The functor send projective perverse sheaves to so-called intersection -theory complexes, a -theoretic version of intersection cohomology complexes. Both sides admit a combinatorial description in terms of the homotopy category of Soergel modules. The commutativity of the diagram is hence immediate. We proceed as follows. In the second section we recall Cisinski-Déglise's [16] construction ofmotivic sheaves and motivic sheaves. In the third section we consider the categories D  ( ) and DK  ( ) for varieties with an affine stratification . We study natural -structures and weight structures on these categories and recall Soergel's Erweiterungssatz. In the fourth and last section we return to the flag variety. We recall Soergel's construction of the Koszul duality functor and prove the theorem mentioned above. In the Appendix we collect some useful facts about weight structures and -structures.
(1) The case of modular coefficients is work in progress joint with Shane Kelly and building on [18] and [19]. (2) The construction of an equivariant (both in the sense of Borel and Bredon) version ofmotivic sheaves is work in progress. This will allow to consider an ungraded version of Bezrukavnikov-Vilonen's equivariant/monodromic Koszul duality, see [11], and open new pathways to Soergel's conjecture on Koszul duality for real groups, see [31] and [10], as well as the quantum -theoretic geometric Satake, see [15] Moreover, the author is considering a motivic Springer correspondence involving the affine Hecke algebra which generalises [17,[26][27][28] and [20] and provides a derived version of Lusztig's comparison between the graded affine Hecke algebra and the affine Hecke algebra, see [24].

MOTIVIC SHEAVES D'APRÈS CISINSKI-DÉGLISE
Motivic sheaves can be viewed as an amalgamation of the topological notion of sheaves on manifolds and the algebro-geometric concept of motivic cohomology. In this section, we give an overview of the most important properties of motivic sheaves and -motivic sheaves, a lesser known variant. We then discuss the Beilinson realisation functor, concrete constructions and Tate motives.

Background on motivic sheaves
To every quasi-projective variety over a perfect field , one can associate a ℚ-linear, tensortriangulated category of motivic sheaves DM( ). There are various equivalent constructions and names for the category DM( ), see [16,Introduction]. For = ( ), the category of motivic sheaves DM( ) coincides with Voevodsky's triangulated category of mixed motives over , see [36].
The motive of the projective line ℙ 1 splits as a direct sum and induces an autoequivalence (1) = − ⊗ ℚ(1) called Tate twist that commutes with all six functors. Homomorphisms between motivic sheaves are governed by algebraic cycles. For smooth, there is a natural isomorphism with Bloch's higher Chow groups, see [12]. For = 2 , these are the usual Chow groups CH ( ) ℚ of codimension-algebraic cycles on up to rational equivalence. For cellular varieties, such as flag varieties, considered in this article these Chow groups will just coincide with the usual Borel-Moore homology of their complex points.
For each prime invertible in , there is an -adic realisation functor to the category of -adic sheaves and for = ℂ a Betti realisation functor to the category of sheaves on (ℂ) equipped with the metric topology, see [3] and [2]. Both types of realisation functors are compatible with the six functors and induce the cycle class maps For our purposes, all varieties are defined over ℤ and all sheaves we consider are of geometric origin. We will hence take the freedom to treat -adic sheaves on ∕ and sheaves on (ℂ) interchangeably using [5].

Background on -motivic sheaves
Very similarly to the system of categories of motivic sheaves DM( ) there is a system of -motivic sheaves DK( ) associated to quasi-projective varieties over a perfect field , see [16, section 13.3]. -motivic sheaves are also equipped with a full six-functor formalism and Tate twist with the same properties. A main difference is that the -motive of ∶ ℙ 1 → ( ) splits into two isomorphic copies This implies that the shift twist functor (1) [2] acts as the identity on DK( ), a phenomenon known as Bott periodicity.
As the name suggests, homomorphisms between -motivic sheaves are governed by -theory. For smooth, there is natural isomorphism with the rational higher algebraic -theory of .

The Beilinson realisation functor
There is a very close relationship between motivic cohomology (which is isomorphic to higher Chow groups) and algebraic -theory, first observed by Beilinson [8]. The rational -theory of a smooth variety naturally decomposes in eigenspaces of the Adams operations turning • ( ) into a bigraded ring, see [37,IV.5]. By [12], there are natural isomorphisms ( ) ( ) ≅ CH ( , ) ℚ such that for = 0 the decomposition in (1) yields the Chern character isomorphism Hence, rational motivic cohomology can be regarded as a graded refinement of algebraic -theory. These results admit a relative version. Namely, there is functor which we call Beilinson realisation. It is compatible with the six functors and Tate twists and it induces for all , ∈ DM( ) an isomorphism that specialises to the decomposition (1) in Adams eigenspaces. This way, motivic sheaves can be regarded as a graded refinement of -motivic sheaves where (1) [2] is the shift of grading functor and forgets the grading.

Construction of ( -)motivic sheaves
We sketch Cisinski-Déglise's construction of the categories of -motivic sheaves DK( ) and motivic sheaves DM( ), [16,. First, one considers the ring spectrum ℚ, representing rational homotopy invarianttheory in the stable motivic homotopy category SH(X). This allows to consider the category of -motivic sheaves over , as the homotopy category of modules over ℚ, . The system of categories DK( ) forms a so-called motivic triangulated category (see [16,Definition 2.4.45]) which entails a six-functor formalism with all desired properties.
As shown by Riou [29], the spectrum admits an Adam's decomposition similar to (1) Denoting Б, ∶= (0) , Cisinski-Déglise define the category of Beilinson motives over , as the homotopy category of modules over Б, . The system of categories DM Б ( ) forms a motivic triangulated category and is shown to be equivalent to other definitions of motivic sheaves, see [16,Chapter 16]. We hence write DM( ) = DM Б ( ) and refer to objects as motivic sheaves.
The inclusion map Б, → ℚ, induces a forgetful functor ∶ DK( ) → DM( ) whose left adjoint ,ℚ is naturally isomorphic to Б, as an Б, -module and there is an isomorphism of ring spectra For ∈ DM( ) this yields the following simple formula: which implies (2). Using [

Tate motives over affine spaces of finite fields
Let = be a finite field. We denote by the categories of mixed Tate ( -)motives (observe that ℚ( ) ≅ ℚ[−2 ] in DK and is hence not needed as a generator). Since the rational higher -theory and the rational higher Chow groups of a finite field vanish, these categories of become semi-simple and one can easily show: There are equivalences of tensor-triangulated categories with the bounded derived categories of (graded) finite-dimensional vector spaces over ℚ. Here we let ℚ( ) ∈ DMT( , ℚ) correspond to ℚ sitting in grading degree − and cohomological degree 0 by convention.
For quasi-projective varieties ∕ the categories DM( , ℚ) and DK( , ℚ) are equipped with a weight structure , see [22] and [14]. This weight structure descends to Tate motives and assigns the weight 2 − to ℚ( )[ ] such that We observe that the -structure and weight structure on DKT( ) coincide. As explained in Remark A.3(1), -structures and weight structures usually behave very differently. Our case just happens to be quite degenerate, since DKT( ) is semi-simple.
We observe that the induced functor Rather, is compatible with the weight structures: As explained in [33,Section 3.4], the interplay of the -structure and weight structure on DMT( , ℚ) can be seen as toy case of Koszul duality. So Proposition 2.2 gives a subtle first hint that should be related to Koszul duality!

Constructible motivic sheaves
Let = be a finite field. Let ∕ be a variety with a cell decomposition (also called affine stratification), that is, where  is some finite set and each ∶ → is a locally closed subvariety isomorphic to for some ⩾ 0. In this situation, Soergel-Wendt [33] make the following definition: the full subcategory of motivic sheaves which restrict to mixed Tate motives on the strata.
For this category to be well-behaved, so for example closed under Verdier duality, Soergel-Wendt impose the following technical condition on the stratification. We will abbreviate D  ( ) = MTDer  ( , ℚ) and speak of -constructible motivic sheaves, and assume that  is Whitney-Tate from now on.
We can now copy their definition in the context of -motives.

Definition 3.3. The category of -constructible -motivic sheaves is
the full subcategory of the category of -motivic sheaves DK( , ℚ) of objects which restrict to Tate motives on the strata.
Since the functor ∶ DM( ) → DK( ) commutes with the six operations, we see that it descends to a functor and observe that the Whitney-Tate condition with respect to DM( ) implies the one for DK( ).
In order to be closed under the six functors, we need to restrict us to morphisms of varieties which are compatible with their affine stratification in the following sense. (1) for all ∈  ′ the inverse image −1 ( ) is a union of strata; (2) for each mapping into ′ , the induced map ∶ → ′ is a surjective affine map.

Weight structures
The categories DM g ( ) and DK g ( ) of objects of geometric origin naturally come with a weight structure, called Chow weight structure, whose hearts are generated by objects of the form * ℚ for smooth projective maps ∶ → , see [22]. We will still define the weight structures on the subcategories D  ( ) and DK  ( ) by hand using the gluing formalism described in Appendix A. We note that our definition coincides with the restriction of the Chow weight structures. (2) ! , * are left -exact.

Pointwise purity and the weight complex functor
In general, objects in the heart of a weight structure on a category  admit no positive extension, that is, Hom  ( , [ ]) = 0 for all , ∈  =0 and > 0. We will show that under a certain pointwise purity assumption there are also no negative extensions for objects in D  ( ) =0 and DK  ( ) =0 . Pointwise pure objects in D  ( ) are under some assumptions in fact sums of (appropriately shifted and twisted) intersection complexes, that is, simple perverse motives. See [33,Corollary 11.11].
Pointwise pure objects are very special since they have no non-trivial extensions amongst each other. Proof. The statement for D  ( ) follows from the one of DK  ( ) using . For DK  ( ), we observe that the pointwise purity implies that , ∈ DK  ( ) =0 ∩ DK  ( ) =0 , where by DK  ( ) =0 we denote the heart of the bottom = 0 perverse -structure on DK  ( ). Hence the statement for negative follows from the axioms of the -structure and the statement for positive from the axioms of the weight structure. □ Pointwise purity allows us to consider the category D  ( ) and DK  ( ) as homotopy categories of their weight zero objects. Proof. We prove the statement for D  ( ), the case of DK  ( ) is done in the same way. The pointwise purity assumption and Proposition 3.10 show that there are no non-trivial extensions in D  ( ) between objects in D  ( ) =0 . Trivially, the same holds true in K (D  ( ) =0 ). Since the weight complex functor restricts to the inclusion D  ( ) =0 → K(D  ( )) an inductive argument ('dévissage') shows that the functor is indeed fully faithful, where we use that D  ( ) is generated by D  ( ) =0 as a triangulated category.
The compatibility with follows since is weight exact. □ We note that there is a different way of proving the last theorem using a formalism called 'tilting', see [33] and [32]. We prefer the weight complex functor, since it also exists without the pointwise purity assumption. The weight complex functor even exists for all motivic sheaves and -motivic sheaves of geometric origin, where the heart of the weight structure is the category Chow motives, see [13].

Erweiterungssatz
The Erweiterungssatz as first stated in [30] and reproven in a more general setting in [21] allows a combinatorial description of pointwise pure weight zero sheaves on in terms of certain modules over the cohomology ring of . In the case of being the flag variety, these modules are called Soergel modules. In [33] a motivic version is considered, which easily extends to -motives.    Proof. All statements are direct consequences of the discussion in Section 2.3. □ We remark that motivic cohomology is bigraded (higher Chow groups) and -theory graded (higher -groups). In our particular setup (affine stratification, finite field base, rational coefficients) all the higher groups vanish. We hence see one grading less.
Under a certain technical assumption the functors ℍ and are fully faithful on pointwise pure objects.

FLAG VARIETIES AND KOSZUL DUALITY
We discuss the particular case of flag varieties and Koszul duality.

Flag varieties
Let ⊃ ⊃ be a split reductive group over with a Borel subgroup and maximal torus.

Translation functors and pointwise purity
We recall the inductive construction of pointwise pure objects in D ( ∨ ) ( ∨ ), see [33,Section 6]. First of all, the object ,! ℚ is pointwise pure, where ∈ denotes the identity. For a simple reflection ∈ we denote by ∨ = ∨ ∪ ∨ ∨ the minimal parabolic and the smooth proper morphism (in fact the map is a projective bundle) The functor = * , * is called translation functor. It clearly preserves pointwise pure objects. For an arbitrary ∈ with = ( ) we choose a shortest expression = 1 ⋯ . Then the object 1 ⋯ ,! ℚ is called a Bott-Samelson motive. It is pointwise pure, has support ∨ and a unique indecomposable direct summand, which we will denote by , with support ∨ . In fact all pointwise pure objects are sums of shifts twits of the motives and the objects  = [ ( )] are simple perverse motives (intersection cohomology complexes) by the decomposition theorem! Subsumed, we get where by ≅, ⊕, we denote closure under isomorphism, finite direct sum and direct summands. We see that all objects in D ( ∨ ) ( ∨ ) =0 are pointwise pure.
We observe that exactly the same construction works for DK ( ∨ ) ( ∨ ). We denote  = ( ) and obtain Here, the objects  may be taken as a definition for intersection -theory complexes  =  . Furthermore, the weight complex functor induces equivalences of categories, see Theorem 3.11, compatible with the functor in the obvious way.

Soergel modules I
The categories D ( ∨ ) ( ∨ ) =0 and DK ( ∨ ) ( ∨ ) =0 can be described combinatorially in terms of Soergel modules, using the functors ℍ and and the Erweiterungssatz, see Section 3.4. We recall the explicit description of ℍ( ∨ ) and ( ∨ ). Recall that ( ) = Hom( ∨ , ) denotes the character lattice. Then there are natural isomorphisms where S( ( ) ⊗ ℚ) denotes the symmetric algebra, and S( ( ∨ ) ⊗ ℚ) + the ideal of invariants of positive degree under the action of . In [30] it is shown that the Bott-Samelson motives

Projective perverse sheaves
We describe the 'Koszul dual' of the last sections. This is the 'classical story'. First, there is a functor, called Betti realization functor,

Soergel modules II
Soergel shows in [30] that categories of projective perverse objects D ( ) ( ) =0 and D ( ) ( ) =0 can be described in terms of Soergel modules as well. First, Soergel's Endomorphismensatz states that there is an isomorphism of graded algebras In [30] this statement is originally proven representation-theoretically for category . There is also a topological proof, due to Bezrukavnikov-Riche, see [9]. Then, Soergel's Struktursatz shows that the functorŝ

Koszul duality
The existence of the following Koszul duality functorKos for D ( ) ( ) was first conjectured by  and proven by Soergel in [30] using the combinatorial descriptions in terms of Soergel modules from above. The very elegant formulation using motivic sheaves is due to Soergel-Wendt, [33]. The functorKos can be constructed as the composition Under this equivalence the projective perverse motivic sheaf is sent to the intersection complex  . It also intertwines the grading shifts ( ) and ( ) [2 ]. For further properties we refer to [7].
We can now consider the ungraded version of Koszul duality in exactly the same way, namely, we have equivalences Under this equivalence the projective perverse sheaf  is sent to the intersection -theory complex  . The functor Kos inherits all the nice properties ofKos. Combining everything, we hence obtain the quite satisfying commutative diagram
The full subcategory  =0 =  ⩽0 ∩  ⩾0 is called the heart of the weight structure. The weight (1) The standard example of a bounded -structure is the bounded derived category D () of an abelian category , where we set The standard example of a bounded weight structure is the bounded homotopy category of chain complexes K () of an additive category , where we set and by ≅ we denote closure under isomorphism. This already showcases an important distinction between -structures and weight structures. While the heart of a -structure is abelian, the heart of a weight structure is only additive in general, and behaves more like the subcategory of projectives or injectives in an abelian category. (2) We use the cohomological convention for weight and -structures. One can easily translate to the homological convention, by setting  ⩽0 =  ⩾0 and  ⩾0 =  ⩽0 .
Proposition A.4. Let  be a triangulated category with a -structure or weight structure. The categories  =  ⩽0 ,  ⩾0 ,  ⩽0 and  ⩾0 are extension stable. That is, for any distinguished triangle in  with , ∈ , also ∈ .
We will use standard terminology for exactness of functors. ). We say that is -exact ( -exact) if is both left and right -exact (or -exact).

A.2
Gluing As explained in [5], -structures can be glued together. In fact the axiomatic setup required to perform such a gluing also works for weight structures. But there is subtle and essential difference in the definition of the gluing of -structures and weight structures, exchanging * and ! functors.
Definition A.6 [5,Section 1.4.3]. We call sequence of triangulated functors and categories a gluing datum if the following properties are fulfilled.
(3) One has * * = 0.  Proof. The statement for -structures is [ Weight complex and realisation functors It is often possible to realise a triangulated category with -structure as the derived category of its heart. Similarly, one can often realise a triangulated category with a weight structure as the homotopy category of chain complexes of its heart. We recall some statements from the literature. Theorem A.8. Let  be an 'enhanced' triangulated category, meaning that either (1) (Derivator)  = (pt), where is a strong stable derivator.
Assume that  is equipped with a -structure. Then there is a triangulated functor called realisation functor D ( =0 ) →  restricting to the inclusion of the heart  =0 → .
Assume that  is equipped with a bounded weight structure. Then there is a triangulated functor called weight complex functor restricting to the inclusion of the heart  =0 → K ( =0 ).

Proof.
For the statement about -structures, we refer to [35] for derivators, [23] for ∞-categories and [4] for -categories. For the statement about weight structures, we refer to [13] for -categories and [1,34] for ∞-categories. In fact, the derivator assumption implies the -category assumption by [25]. □ There are different assumptions under which the above functors can be shown to be fully faithful. We refer to the references in the proof above. Furthermore, it can be shown that realisation and weight complex functors are compatible with 'enhanced' exact triangulated functors between 'enhanced' triangulated categories. We note that the categories of motives and the six operations between them are all 'enhanced'.

A C K N O W L E D G E M E N T S
We warmly thank Shane Kelly for explaining -motives to us. We are grateful for discussions with Wolfgang Soergel and Geordie Williamson. We thank the referee for helpful comments. This publication was written while the author was a guest at the Max Planck Institute for Mathematics in Bonn.

J O U R N A L I N F O R M AT I O N
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