Quasi-coherent Hecke category and Demazure Descent
aa r X i v : . [ m a t h . R T ] O c t QUASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT
SERGEY ARKHIPOV AND TINA KANSTRUP
To Boris Feigin on the occasion of his 60-th birthdaywith gratitude and admiration
Abstract.
Let G be a reductive algebraic group with a Borel subgroup B . We definethe quasi-coherent Hecke category for the pair ( G, B ) . For any regular Noetherian G -scheme X we construct a monoidal action of the Hecke category on the derived categoryof B -equivariant quasi-coherent sheaves on X . Using the action we define the DemazureDescent Data on the latter category and prove that the Descent category is equivalent tothe derived category of G -equivariant sheaves on X . Introduction
The present paper is the second one in the series devoted to the study of DemazureDescent. In [AK] we introduced the notion of Demazure Descent Data on a triangulatedcategory. A category with Demazure Descent Data is a higher analog for a representationof the degenerate Hecke algebra (see [HLS]).Such representations arise naturally from geometry. Let X be a compact real manifoldacted on by a compact simple Lie group G with a fixed maximal torus T . Harada et al con-structed a natural action of the degenerate Hecke algebra of the corresponding type on the T -equivariant K-groups of X . They showed that the G -equivariant K-groups are identifiednaturally with the invariants for the action. In a way we categorify the construction from[HLS] in our paper.Let G be a reductive algebraic group with a fixed Borel subgroup B . Recall that the(finite) Hecke algebra is defined classically as the algebra of B ( F q ) -biequivariant functionson the group G ( F q ) with values in C . The multiplication is provided by convolution. Itturns out that the stack B \ G/B is a universal geometric tool to produce "algebras", both inthe usual and in the categorical sense. In particular, the categories of constructible sheavesand D-modules on B \ G/B were studied in [Tan]. They were used as a natural source offinite Hecke algebra actions on categories of geometric origin.In the present paper, we consider the quasi-coherent Hecke category QCHecke ( G, B ) ,the derived category of B -biequivariant quasi-coherent sheaves on G . For technical reasonswe prefer to work with the equivalent category - the derived category of G -equivariantquasi-coherent sheaves on G/B × G/B .Let us outline the structure of the paper. We recall the standard definitions and factsrelated to equivariant quasi-coherent sheaves on a scheme in Section 2. We introduce themonoidal category QCHecke ( G, B ) in Section 3. In Section 4 we recall the definitions of Demazure Descent Data on a triangulated category and of the corresponding Descentcategory.Let X be a regular Noetherian scheme acted on by a reductive algebraic group G . Sec-tion 5 is devoted to the construction of Demazure Descent Data on the derived categoryof B -equivariant quasicoherent sheaves on X in terms of the natural monoidal action ofQCHecke ( G, B ) on the category by convolution. Theorem:
For w ∈ W let X w denote the closure of the corresponding G orbit in G/B × G/B . Then the functors of convolution with the structure sheaves O X w define DemazureDescent Data on D (QCoh B ( X )) .In the last Section, we study the corresponding Descent category. We prove the centralresult of the paper: Theorem:
Desc ( D (QCoh B ( X )) , D w , w ∈ W ) is equivalent to D (QCoh G ( X )) .1.1. Acknowledgements.
The authors are grateful to H.H. Andersen, C. Dodd, V. Ginzburg,M. Harada and R. Rouquier for many stimulating discussions. The project started in thesummer of 2012 when the first named author visited IHES. S.A. is grateful to IHES forperfect working conditions. Part of the work was done while the second author visited R.Bezrukavnikov at MIT in the Fall 2013. T.K. would like to express her deepest gratitudeto R. Bezrukavnikov for all that he taught her during her stay and for useful commentson previous versions of the text. T.K. would also like to thank MIT for perfect workingconditions.Both authors’ research was supported in part by center of excellence grants "Centrefor Quantum Geometry of Moduli Spaces" and by FNU grant "Algebraic Groups andApplications".1.2.
Conventions.
In the present paper, we work over an algebraically closed field k ofcharacteristic zero. An algebraic group means an affine algebraic group scheme over k . Allschemes are supposed to be Noetherian, of finite Krull dimension over k .2. Equivariant quasi-coherent sheaves on a scheme
Below we collect the main facts about equivariant quasi-coherent sheaves to be used later.In this Section, K denotes a not necessarily reductive algebraic group.Let X be a K -scheme. Denote the action (resp., the projection) map K × X → X byac (resp., by p ). Recall that a K -equivariance structure on a quasi-coherent sheaf M on X is given by an isomorphism θ between p ∗ ( M ) and ac ∗ ( M ) such that the further pull-backsof θ to K × K × X satisfy the standard cocycle condition (see e.g. [Bri], Section 2, for theprecise formulation).The category of quasi-coherent sheaves on X (resp., of K -equivariant quasi-coherentsheaves on X ) is denoted by QCoh( X ) (resp., by QCoh K ( X ) ). We have the forgetfulfunctor Oblv : QCoh K ( X ) → QCoh( X ) . UASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT 3
Below we always assume that X is good enough, and any K -equivariant quasi-coherentsheaf has a uniformly bounded resolution by K -equivariant quasi-coherent sheaves locallyfree on X .Let f : X → Y be a K -equivariant map of K -schemes. The functors of push-forwardand pull-back are extended naturally to the categories of equivariant sheaves: f ∗ : QCoh K ( Y ) → QCoh K ( X ) , ( M, θ ) ( f ∗ M, f ∗ θ ◦ canonical isomorphisms ) ,f ∗ : QCoh K ( X ) → QCoh K ( Y ) , ( M, θ ) ( f ∗ M, (Id × f ) ∗ θ ◦ canonical isomorphisms ) . Notice that both f ∗ and f ∗ commute with Oblv .Let
K, H be algebraic groups acting on a scheme X so that the actions commute. Assumethat X admits an H -equivariant quotient q : X → X/K which is a locally trivial principal K -bundle. Denote the quotient scheme X/K by Y . Lemma 2.1.
The inverse image functor provides an equivalence of Abelian categories
QCoh H ( Y ) → QCoh H × K ( X ) .Proof. See [Bri], discussion in Section 2. (cid:3)
Let X be a K -scheme. It is known that the forgetful functor QCoh K ( X ) → QCoh( X ) has an exact right adjoint functor denoted by Av K , and for any M ∈ QCoh( X ) the naturalmap M → Oblv ◦ Av K ( M ) is an embedding. It follows that the category QCoh K ( X ) hasenough injective objects (see [Bez], Section 2). Moreover, Varagnolo and Vasserot state thatunder very mild restrictions on X , any unbounded complex in QCoh K ( X ) has a K-injectiveresolution and a K-flat resolution (see [VV], 1.5.6 and further paragraphs).To avoid further restrictions on X , we work in the unbounded derived category D QCoh K ( X ) .The functor Oblv is exact and extends to the functor D QCoh K ( X ) → D QCoh( X ) .Let f : X → Y be a K -morphism of normal Noetherian quasi-projective K -schemes.Consider the derived functors Lf ∗ : D QCoh K ( Y ) → D QCoh K ( X ) and Rf ∗ : D QCoh K ( X ) → D QCoh K ( Y ) . It is known that the functors Rf ∗ and Oblv (resp., Lf ∗ and Oblv ) commute (see e.g. [VV],Lemma 1.5.9 and the discussion immediately after it).Derived tensor products in D QCoh K ( Y ) and D QCoh K ( X ) are denoted by L ⊗ Y and L ⊗ X respectively. We have the projection formula as follows: Proposition 2.2.
For N ∈ D QCoh K ( Y ) and M ∈ D QCoh K ( X ) we have a canonicalisomorphism Rf ∗ N L ⊗ Y M ≃ Rf ∗ ( N L ⊗ X Lf ∗ M ) .Proof. See [VV] section 1.5.8. (cid:3)
SERGEY ARKHIPOV AND TINA KANSTRUP
Recall the equivariant flat base change theorem. Let g : Z → Y be a flat K -morphism,where Z is a normal quasi-projective K -scheme. Consider the Cartesian squre Z × Y X f ′ (cid:15) (cid:15) g ′ / / X f (cid:15) (cid:15) Z g / / Y Proposition 2.3.
The standard adjunction map provides an isomorphism of functors Lg ∗ Rf ∗ ≃ Rf ′∗ Lg ′∗ . Convolution and the quasi-coherent Hecke category
Convolution monoidal structure.
Let
Z, Y and X be K -schemes. Consider theprojections Z × Y × X pr x x ♣♣♣♣♣♣♣♣♣♣♣ pr (cid:15) (cid:15) pr ' ' ◆◆◆◆◆◆◆◆◆◆◆ Z × Y Z × X Y × X The group K acts on each of the four schemes in the diagram diagonally, and the projectionsare K -equivariant.The convolution product ∗ is defined as follows: ∗ : D (QCoh K ( Z × Y )) × D (QCoh K ( Y × X )) → D (QCoh K ( Z × X )) ,M ∗ M := R pr ∗ ( L pr ∗ M L ⊗ Z × Y × X L pr ∗ M ) . Suppose that Z = Y = X is regular. In this case, the convolution product becomes amonoidal structure ∗ : D (QCoh K ( X × X )) × D (QCoh K ( X × X )) → D (QCoh K ( X × X )) in a weak sense: the associativity constraint ( M ∗ M ) ∗ M ˜ → M ∗ ( M ∗ M ) is not specified.In the same way, if Z = Y is regular, the convolution product induces a monoidal action(in the same weak sense) of the monoidal category D (QCoh K ( Y × Y )) on the category D (QCoh K ( Y × X )) .3.2. Comparing two convolutions.
The key technical statement in the proof of asso-ciativity of the monoidal structure above as well as of several more specific Lemmas belowrequires the following setup.Suppose we have the K -schemes X , X and X , Z , Z ′ and Z . We are given the K -equivariant flat maps p : Z → X , p : Z → X , q : Z → X , q : Z → X , and α : Z → Z ′ → Z such that p ′ = p ◦ α and p ′ = p ◦ α are also flat. UASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT 5
Consider the standard projectionspr : Z × X Z → Z , pr : Z × X Z → Z , pr : Z × X Z → X × X , pr ′ : Z ′ × X Z → Z ′ , pr ′ : Z ′ × X Z → Z ′ , pr ′ : Z ′ × X Z → X × X . We introduce the convolution products ∗ : D (QCoh K ( Z ′ )) × D (QCoh K ( Z )) → D (QCoh K ( X × X )) ,M ∗ M := R pr ∗ ( L pr ∗ ( Rα ∗ ( M )) L ⊗ Z × X Z L pr ∗ ( M )) and ∗ ′ : D (QCoh K ( Z ′ )) × D (QCoh K ( Z )) → D (QCoh K ( X × X )) ,M ∗ ′ M := R pr ′ ∗ ( L (pr ′ ) ∗ ( M ) L ⊗ Z ′ × X Z L (pr ′ ) ∗ ( M )) . Lemma 3.1.
The convolutions ∗ and ∗ ′ are canonically isomorphic.Proof. Denote the base change of the map α via pr by β : Z ′ × X Z → Z × X Z .The argument in the proof is standard and combines the base change via pr for the Rα ∗ and the projection formula for the map β . (cid:3) Remark . A typical special case in which Lemma 3.1 is applied is as follows. Take X = X = X = X . For a flat surjective K -equivariant map X → Y consider Z ′ = X × Y X, Z = Z = Z = X × X. Lemma 3.1 implies that convolution operations defined via X × Y X × X and via X × X × X coincide. Remark . In particular the unit object in the monoidal category D (QCoh K ( X × X )) isgiven by the structure sheaf of the diagonal in X × X denoted by O X ∆ .3.3. Convolution and correspondences.
Let
X, Y , . . . Y n be regular K -schemes. Sup-pose we are given flat surjective maps φ i : X → Y i , i = 1 , . . . , n . Denote the fiber product X × Y i X ⊂ X × X by α i : X i → X × X . Consider the iterated fibered product Z i ,...,i k := X i × X . . . × X X i k = X × Y i X . . . × Y ik X ⊂ X k +1 . We have the map provided by the projections to the first and last factors α i ,...,i k : Z i ,...,i k → X × X. Denote the image of the map by X i ,...,i k ⊂ X × X . All the defined schemes are actednaturally by K and all the defined maps are K -equivariant.Consider the sheaves M i := α i ∗ ( O X i ) . The category D (QCoh K ( X × X )) acts on thecategory D (QCoh K ( X )) by convolution. Denote the functor of convolution with M i by D i . Lemma 3.4.
The functor D i : D (QCoh K ( X )) → D (QCoh K ( X )) is isomorphic to thefunctor Rφ ∗ i ◦ Lφ i ∗ . SERGEY ARKHIPOV AND TINA KANSTRUP
Proof.
Denote the two projections X i → X by q ,i and q ,i . Let N ∈ D (QCoh K ( X )) . ApplyLemma 3.1 as suggested in Remark 3.2. Notice that M i ∗ N f → Rq ,i ∗ Lq ∗ ,i ( N ) . Applying flatbase change we obtain the statement of the Lemma. (cid:3) Corollary . Each functor D i is isomorphic to a comonad. Suppose additionally that themaps φ i : X → Y i are rational. Then the comonads D i are coprojectors, i.e. the coproductmaps D i → D i ◦ D i are isomorphisms of functors.Our goal is to describe the composition of the functors D i ◦ . . . ◦ D i n explicitly. Denote Rα i ,...,i k ∗ ( O Z i ,...,ik ) ∈ D (QCoh K ( X × X )) by M i ,...,i k . Lemma 3.6.
We have a natural isomorphism of objects in D (QCoh K ( X × X )) M i ∗ . . . ∗ M i k f → M i ,...,i k . Proof.
We proceed by induction. Like in the proof of the previous Lemma, consider thetwo projections q ,i and q ,i : X × Y i X × X → X × X and notice that M i ∗ . . . ∗ M i k f → M i ∗ M i ,...,i k f → Rq ,i ∗ Lq ∗ ,i ( M i ,...,i k ) . Recall that Z i ,...,i k = X × Y i Z i ,...,i k . Applying base change, we get that M i ∗ . . . ∗ M i k isisomorphic to the direct image of O Z i ,...,ik for the composition of maps Z i ,...,i k = X i × X Z i ,...,i k → X i × X → X × X × X pr → X × X. The latter map coincides with α i ,...,i k . (cid:3) Corollary . Suppose additionally that the map α i ,...,i k : Z i ,...,i k → X i ,...,i k is rational.Then the convolution product M i ∗ . . . ∗ M i k is isomorphic to the structure sheaf of X i ,...,i k .3.4. Quasi-coherent Hecke category.
Fix a reductive algebraic group K with an alge-braic subgroup H . Consider the K -scheme Y = K/H . Definition 3.8.
The monoidal category ( D (QCoh K ( K/H × K/H )) , ∗ ) is called the quasi-coherent Hecke category and it is denoted by QCHecke ( K, H ) .Notice that for a K -scheme X we have D (QCoh H ( X )) ≃ D (QCoh K ( K/H × X )) . Taking Z = Y = K/H in the setting 3.1 we get the monoidal actionQCHecke ( K, H ) × D (QCoh H ( X )) → D (QCoh H ( X )) . UASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT 7 Demazure Descent
Notations.
From now on G is a reductive algebraic group. Let T be a Cartan sub-group of G and let X (resp., Y ) be the weight (resp., the coroot) lattice of G . Choose aBorel subgroup T ⊂ B ⊂ G .Denote the set of roots for G by Φ = Φ + ⊔ Φ − . Let { α , . . . , α n } be the set of simpleroots. The Weyl group W = Norm ( T ) /T of the fixed maximal torus acts naturally onthe lattices X and Y and on the R -vector spaces spanned by them, by reflections in roothyperplanes. The simple reflection corresponding to a simple root α i is denoted by s i .For an element w ∈ W denote the length of a minimal expression of w via the generators { s i } by ℓ ( w ) . The unique longest element in W is denoted by w . We have a partialordering on W called the Bruhat ordering: w ′ ≤ w if there exists a reduced expression for w ′ that can be obtained from a reduced expression for w by deleting a number of simplereflections.The monoid Br + with generators { T w , w ∈ W } and relations T w T w = T w w if ℓ ( w ) + ℓ ( w ) = ℓ ( w w ) in W is called the braid monoid of G .4.2. Demazure Descent.
Fix a reductive algebric group G with the Weyl group W andthe braid monoid Br + . Definition 4.1.
A weak braid monoid action on a triangulated category C is a collectionof triangulated functors D w : C → C , w ∈ W satisfying braid monoid relations, i.e. for all w , w ∈ W there exist isomorphisms offunctors D w ◦ D w ≃ D w w , if ℓ ( w w ) = ℓ ( w ) + ℓ ( w ) . Notice that we neither fix the braid relations isomorphisms nor impose any additionalrelations on them.Recall that a comonad structure ( ǫ, η ) on an endofunctor D : C → C is a structure of aco-associative coalgebra on D in the monoidal category of endofunctors. Here ǫ (resp., η )denotes the counit (resp., the coproduct) morphism for the comonad. A comonad D is aco-projector if the coproduct map D → D ◦ D is an isomorphism. Definition 4.2.
Demazure Descent Data on a triangulated category C is a weak braidmonoid action { D w } together with a co-projector structure ( ǫ k , η k ) on the functor D s k forevery simple reflection s k .4.3. The descent category.
Consider a triangulated category C with a fixed DemazureDescent Data { D w , w ∈ W } . Definition 4.3.
The Descent category
Desc ( C , D w , w ∈ W ) is the full subcategory in C with objects M such that for all k the cones of all counit maps D s k ( M ) ǫ k → M are isomorphicto . SERGEY ARKHIPOV AND TINA KANSTRUP
Remark . Suppose that C has functorial cones. Then Desc ( C , D w , w ∈ W ) is a full trian-gulated subcategory in C since it is the intersection of kernels of the functors Cone ( D s k → Id ) . However, one can prove this statement not using functoriality of cones. Namely, an ob-ject M ∈ C is an object in Desc ( C , D w , w ∈ W ) if and only if all counit maps D s k ( M ) → M are isomorphisms, and it is easy to see that this condition is stable under taking shifts andcones. Lemma 4.5.
An object M ∈ Desc ( C , D w , w ∈ W ) is naturally a comodule over each D s k .Proof. By definition the comonad maps η : D s k → D s k , ǫ : D s k → Idmake the following diagrams commutative D s k ( M ) ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ η M (cid:15) (cid:15) Id ◦ D s k ( M ) D s k ( M ) ǫ Dsk ( M ) o o D s k ( M ) ǫ M / / MD s k ( M ) D sk ( ǫ M ) O O ǫ Dsk ( M ) / / D s k ( M ) ǫ M O O Notice that the rerequirement that Cone ( D s k ( M ) → M ) ≃ is equivalent to invertibilityof the counit map ǫ M . By functoriality D s k ( ǫ M ) is an isomorphism with inverse D s k ( ǫ − M ) .By the first diagram ǫ D sk M is inverse to η M . We get the commutative diagram M ǫ − M / / ǫ − M (cid:15) (cid:15) D s k ( M ) η M (cid:15) (cid:15) D s k ( M ) D sk ( ǫ − M ) / / D s k ( M ) The counit axiom is verified similarly. Thus, M is a D s k -comodule with the coaction ǫ − M . (cid:3) Remark . Recall that in the usual Descent setting either in Algebraic Geometry or inabstract Category Theory (Barr-Beck Theorem) Descent Data includes a pair of adjointfunctors and their composition which is a comonad. By definition, the Descent category forsuch data is the category of comodules over this comonad. Our definition of
Desc ( C , D w , w ∈ W ) for Demazure Descent Data formally is not about comodules, yet the previous Lemmademonstrates that every object of Desc ( C , D w , w ∈ W ) is naturally equipped with structuresof a comodule over each D k and any morphism in Desc ( C , D w , w ∈ W ) is a morphism of D k -comodules. UASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT 9 Demazure Descent for D (QCoh G ( G/B × X )) Let X be a scheme equipped with an action of a reductive algebraic group G . Forevery element of the Weyl group w ∈ W we construct a functor D w acting on the cat-egory D (QCoh G ( G/B × X )) . The functor is defined in terms of the monoidal action of QCHecke(
G, B ) .5.1. Bott-Samelson varieties.
Recall that the orbits for the the diagonal action of G on G/B × G/B are in bijection with elements of the Weyl group [CG, Theorem 3.1.9]. For w ∈ W let X w be the closure of the corresponding orbit. The structure sheaves of the orbitclosures O X w , w ∈ W , ar objects of the category QCHecke(
G, B ) .We will use an explicit description of the orbit closures. For a simple root α k denotethe standard parabolic subgroup corresponding to α k and containing B by P k . We have X s k = G/B × G/P k G/B . To simplify the notations below we write X k for X s k . Noticethat the subscheme X k ⊂ G/B × G/B is regular and each of the projections to the factors p , p : X s k → G/B is flat. Thus we are in the setting of 3.3.For an element w ∈ W , fix a reduced expression w = s i . . . s i k where k = ℓ ( w ) . Theiterated fibered product Z i ,...,i k := X i × G/B . . . × G/B X i k = G/B × G/P i G/B . . . × G/P ik G/B ⊂ ( G/B ) k +1 is called the Bott-Samelson variety for w (and for the fixed reduced expression of it).We have the map provided by the projections to the first and last factors α i ,...,i k : Z i ,...,i k → G/B × G/B.
The main geometric result we use is the following theorem due to Cline, Parshall and Scott.
Theorem 5.1. [CPS] (1) The image of the map α i ,...,i k coincides with X w .(2) The map α i ,...,i k : Z i ,...,i k → X w is birational, in particular Rα i ,...,i k ∗ ( O Z i ,...,ik ) = O X w . Demazure functors.
Consider the functor D w : D (QCoh G ( G/B × X )) → D (QCoh G ( G/B × X )) , D w ( M ) := O X w ∗ M. Below we prove that the functors D w , w ∈ W, form Demazure Descent Datum on thecategory D (QCoh G ( G/B × X )) . Remark . The monoidal structure on the category
QCHecke(
G, B ) is defined in the weaksense, in particular, we can not talk about a co-associative coalgebra in this category. Thusthe two parts of the definition of Demazure Descent Data are to be treated separately.Braid relations between the objects O X w , w ∈ W are checked in D (QCoh G ( G/B × X )) , butco-projector structures on X X si are defined using different considerations.Consider the projection p i : G/B × X → G/P i × X and the corresponding inverse anddirect image functors Lp ∗ i , Rp i ∗ : D (QCoh G ( G/B × X )) ←− −→ D (QCoh G ( G/P i × X )) . The endofunctor D i of the category D (QCoh G ( G/B × X )) given by their compositioncarries a canonical strucutre of a comonad (see for example [Mac, section VII.6]). Sincethe map is rational, the projection formula implies that the comonad is co-projector: thecoproduct map D i → D i ◦ D i is an isomorphism. Lemma 5.3.
The functors D s i and D i are isomorphic.Proof. The statement follows immediately from Lemma 3.4. (cid:3)
Notice that the associativity for the monoidal action of
QCHecke(
G, B ) on the category D (QCoh G ( G/B × X )) implies that all relations up to a non-specified isomorphism can bechecked in the Hecke category. Proposition 5.4.
Let w = s k · · · s k n be a reduced expression. Then O X k ∗ . . . ∗ O X kn and O X w are isomorphic as objects in QCHecke(
G, B ) .Proof. By Lemma 3.6, the convolution product O X k ∗ . . . ∗ O X kn is isomorphic to thedirect image of the structure sheaf of the Bott-Samelson variety Rα i ,...,i k ∗ ( O Z i ,...,ik ) . ByTheorem 5.1, the latter object is isomorphic to the structure sheaf of X w . (cid:3) Now we are prepared to prove the central result of the paper.
Theorem 5.5.
The functors { D w , w ∈ W } form Demazure Descent Data on the category D (QCoh G ( G/B × X )) .Proof of Theorem 5.5. We have proved that each of the functors D i is a comonad and thecoproduct maps are isomorphisms of functors.It remains to show that for all w , w ∈ W with ℓ ( w w ) = ℓ ( w ) + ℓ ( w ) we have D w ◦ D w ≃ D w w . Fix reduced expressions for the Weyl gorup elements w = s k · · · s k n and w = s j · · · s j m .Since ℓ ( w w ) = ℓ ( w ) + ℓ ( w ) the expression w w = s k · · · s k n s j · · · s j m . For any M ∈ D (QCoh G ( G/B × X )) we obtain D w ◦ D w ( M ) f →O X w ∗ O X w ∗ M f →O X k ∗ · · · ∗ O X kn ∗ O X j ∗ · · · ∗ O X jm ∗ M f →O X w w ∗ M f → D w w ( M ) . (cid:3) Descent category
Consider the Descent category for the consturcted Demazure Descent Data on the cate-gory D (QCoh G ( G/B × X )) . Theorem 6.1.
The descent category
Desc ( D (QCoh B ( X )) , D w , w ∈ W ) is equivalent to D (QCoh G ( X )) . UASI-COHERENT HECKE CATEGORY AND DEMAZURE DESCENT 11
Notice that the inverse image functors Lp ∗ k : D (QCoh G ( G/P i × X )) → D (QCoh G ( G/B × X )) are fully faithful by projection formula since the maps are rational. The same is truefor Lp ∗ : D (QCoh G ( × X )) → D (QCoh G ( G/B × X )) . Here p denotes the projection G/B × X → X . Lemma 6.2.
An object M in D (QCoh G ( G/B × X )) belongs to the essential image of Lp ∗ k if and only if the coaction map M → D k ( M ) is an isomorphism.Proof. We identified the functor D k with the composition Lp ∗ k Rp k ∗ . Thus, M ≃ D k ( M ) implies that M belongs to the essential image of Lp ∗ k . Assume that M = Lp ∗ k ( N ) for some N ∈ D (QCoh G ( G/P k × X )) . By projection formula, the adjunction map Id → Rp k ∗ p ∗ k isan isomorphism. Thus we have D k ( M ) ≃ Lp ∗ k Rp k ∗ Lp ∗ k ( N ) ≃ Lp ∗ k ( N ) = M. (cid:3) Remark . The same argument shows that an object M in D (QCoh G ( G/B × X )) belongsto the essential image of Lp ∗ if and only if D w ( M ) is isomorphic to M . Proof of Theorem 6.1.
Let M ∈ Desc ( D (QCoh B ( X )) , D w , w ∈ W ) . For every simple root α k the object D k ( M ) is isomorphic to M . Choose a reduced expression s k · · · s k n for w .We have D k ◦ · · · ◦ D k n ( M ) ≃ D w ( M ) . It follows that M belongs to the essential image of D (QCoh G ( X )) .In particular, the descent category Desc ( D (QCoh B ( X )) , D w , w ∈ W ) is a full subcate-gory in the essential image of the functor Lp ∗ .To prove the other embedding, notice that the map p factors as G/B × X → G/P k × X → X. It follows that the essential image of Lp ∗ is a full subcategory in the essential image of Lp ∗ k for all k . This completes the proof of the Theorem. (cid:3) References [AK] S. Arkhipov, T. Kanstrup,
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