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Research ArticleDemazure Descent and Representations of Reductive Groups

Sergey Arkhipov1 and Tina Kanstrup2

1 Matematisk Institut Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark2 Centre for Quantum Geometry of Moduli Spaces Aarhus Universitet Ny Munkegade 8000 Aarhus C Denmark

Correspondence should be addressed to Tina Kanstrup tinaqgmaudk

Received 1 November 2013 Accepted 21 January 2014 Published 25 May 2014

Academic Editor Sorin Dascalescu

Copyright copy 2014 S Arkhipov and T KanstrupThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We introduce the notion of Demazure descent data on a triangulated categoryC and define the descent category for such data Weillustrate the definition by our basic example Let 119866 be a reductive algebraic group with a Borel subgroup 119861 Demazure functorsform Demazure descent data on 119863

119887(Rep (119861)) and the descent category is equivalent to 119863

119887(Rep (119866))

1 Motivation

Thepresent paper is the first one in a series devoted to variouscases of categorical descent Philosophically our interest inthe subject grew out of attempts to understand the mainconstruction from the recent paper by Ben-Zvi andNadler [1]in plain terms that would not involve higher category theory

11 Beilinson-Bernstein Localization and Derived DescentLet 119866 be a reductive algebraic group with the Lie algebrag Denote the Flag variety of 119866 by Fl A major part ofGeometric Representation Theory originated in the seminalwork of Beılinson and Bernstein [2] devoted to investigationof the globalization functor D-mod(Fl) rarr 119880(g)-mod Thisfunctor turns out to be fully faithful and provides geometricand topological tools to investigate a wide class of 119880(g)-modules in particular the ones from the famous category OVarious generalizations of this result lead to the investigationof the categories of twistedD-modules on the Flag variety andon the base affine space for 119866 and of their derived categories

Ben-Zvi and Nadler define a certain comonad acting ona higher categorical version for the derived category of D-modules on the base affine space In fact the functor is builtinto the higher categorical treatment of Beilinson-Bernsteinlocalization-globalization construction

Using the heavy machinery of Barr-Beck-Lurie descentthe authors argue that the derived category of 119880(g)-modulesis equivalent to the category of D-modules equivariant with

respect to this comonad Thus the global sections functorbecomes equivariant with respect to the actionThe comonadis called theHecke comonad It provides a categorification forthe classical action of theWeyl group on various homologicaland K-theoretic invariants of the Flag variety

Notice that the descent construction fails to work on thelevel of the usual triangulated categories Ideally one wouldlike to replace it by a categorical action of the Weyl group orrather of the Braid group on categories of D-modules relatedto the Flag variety One would need to define a notion ofldquoinvariantsrdquo with respect to such action

12 Descent in Equivariant K-Theory Another source ofinspiration for the present paper which is in a way closerto our work is a recent article of Harada et al [3] Given acompact space 119883 with an action of a compact reductive Liegroup 119866 the authors express the 119866-equivariant K-theory of119883 via the 119879-equivariant one Here 119879 denotes a fixed maximaltorus in 119866 Harada et al show that the natural action of theWeyl group119882 on119870119879(119883) extends to an action of a degenerateHecke ring generated by divided difference operators whichwas introduced earlier in the context of Schubert calculus byDemazure The operators are called Demazure operators

Themain result in the paper [3] states that the ring119870119866(119883)

is isomorphic to the subring of 119870119879(119883) annihilated by theaugmentation ideal in the degenerate Hecke algebra In otherwords a 119879-equivariant class is 119866-equivariant if and only if itis killed by the Demazure operators

Hindawi Publishing CorporationAlgebraVolume 2014 Article ID 823467 6 pageshttpdxdoiorg1011552014823467

2 Algebra

In the present paper we define a notion of Demazuredescent on a triangulated category C Thus Demazure oper-ators are replaced by Demazure functors These functorssatisfy a categorified version of degenerate Hecke algebrarelations and form a Demazure descent data onC We definethe descent category for such data Demazure descent issupposed to be a technique replacing the naive notion ofWeylgroup invariants on the categorical level

We provide the first example of Demazure descentConsider a reductive algebraic group 119866 and fix a Borelsubgroup 119861 sub 119866 Categorifying the construction form [3]we consider Demazure functors 119863119904119894

acting on the derivedcategory of 119861-modules We prove that the functors form aDemazure descent data and identify the descent categorywith the derived category of 119866-modules

2 The Setting

21 Root Data Let 119866 be a reductive algebraic group over analgebraically closed field 119896 of characteristic zero Let 119879 be aCartan subgroup of 119866 and let (119868 119883 119884) be the correspondingroot data where 119868 is the set of vertices of theDynkin diagram119883 is the weight lattice of 119866 and 119884 is the coroot lattice of119866 Choose a Borel subgroup 119879 sub 119861 sub 119866 Denote the setof roots for 119866 by Φ = Φ

+⊔ Φminus Let 1205721 120572119899 be the set

of simple roots The Weyl group 119882 = Norm(119879)119879 of thefixed maximal torus acts naturally on the lattices 119883 and 119884

and on the R-vector spaces spanned by them by reflectionsin root hyperplanes The simple reflection corresponding toan 120572119894 is denoted by 119904119894 The elements 1199041 119904119899 form a set ofgenerators for 119882 For 119908 isin 119882 denote the length of a minimalexpression of 119908 via the generators by ℓ(119908) We have a partialordering on 119882 called the Bruhat ordering 1199081015840 le 119908 if thereexists a reduced expression for 119908

1015840 that can be obtained froma reduced expression for 119908 by deleting a number of simplereflections

The monoid Br+ with generators 119879119908 119908 isin 119882 andrelations

11987911990811198791199082

= 11987911990811199082if ℓ (1199081) + ℓ (1199082) = ℓ (11990811199082) in 119882 (1)

is called the braid monoid of 119866

22 Categories of Representations For an algebraic group 119867we denote the Hopf algebra of polynomial functions on119867 byO(119867) Let Rep(119867) be the category of O(119867)-comodules Thisis an Abelian tensor category

Let 119875119894 be the parabolic subgroup of119866 containing 119861whoseLevi subgroup has the root system 120572119894 minus120572119894 Using the naturalHopf algebramapsO(119866) rarr O(119861) andO(119875119894) rarr O(119861)we canget restriction functors

Res119894 Rep (119875119894) 997888rarr Rep (119861)

Res Rep (119866) 997888rarr Rep (119861)

(2)

The restriction functors are exact and naturally commutewith taking tensor product of representations Let 119867 be asubgroup of 119866 and119872 isin Rep(119866) Define the119867-invariant part

of 119872 to be 119872119867

= HomRep(119867)(119896119872) Consider the inductionfunctors

Ind119894 Rep (119861) 997888rarr Rep (119875119894) 119872 997891997888rarr (O (119875119894) otimes 119872)119861

Ind Rep (119861) 997888rarr Rep (119866) 119872 997891997888rarr (O (119866) otimes 119872)119861

(3)

Set Δ 119894 = Res119894 ∘ Ind119894 Rep(119861) and Δ = Res ∘ Ind

Rep(119861) Notice thatΔ 119894 andΔ are left exact since the inductionfunctors are left exact

23 The Derived Categories For an algebraic group 119867 theregular comodule O(119867) is injective in Rep(119867) moreover forany119872 isin Rep(119867) the coactionmap119872 rarr O(119867)otimes119872 providesan embedding of 119872 into an injective object In particularRep(119867) has enough injectives The algebraic De RhamcomplexΩ

∙(119867) provides an injective resolution for the trivial

comodule of the length equal to the dimension of119867 For any119872 isin Rep(119867) the complex Ω

∙(119867) otimes 119872 provides an injective

resolution for 119872 of the same lengthConsider now the bounded derived categories

119863119887(Rep(119861)) 119863

119887(Rep(119875119894)) and 119863

119887(Rep(119866)) Let 119871 119894 and 119871

be the derived functors of Res119894 and Res respectively Denotethe right derived functors of Ind119894 and Ind by 119868119894 and 119868respectively Let119863119894 = 119871 119894 ∘ 119868119894 and119863 = 119871∘119868 be the right derivedfunctors of Δ 119894 and Δ respectively

Proposition 1 (a) The functors 119871 119894 and 119871 are left adjoint to 119868119894

and 119868 respectively(b) For119872 isin 119863

119887(Rep (119861)) and119873 isin 119863

119887(Rep (119875119894)) (resp for

119872 isin 119863119887(Rep (119861)) and 119873 isin 119863

119887(Rep (119866))) we have the tensor

identities119868119894 (119872 otimes 119871 119894 (119873)) ≃ 119868119894 (119872) otimes 119873

(119903119890119904119901 119868 (119872 otimes 119871 (119873)) ≃ 119868 (119872) otimes 119873)

(4)

(c) The functors 119868119894 and 119868 take the trivial O(119861)-comoduleto the trivial O(119875119894)-comodule (resp to the trivial O(119866)-comodule)

(d) 119863119894 and 119863 are comonads for which the comonad maps119863119894 rarr 119863

2

119894and 119863 rarr 119863

2 are isomorphisms

Proof The statements corresponding to (a) and (b) for Resand Ind (resp Res119894 and Ind119894) are Propositions 34 and 36in [4] The derived functors of a pair of adjoint functors areadjoint (b) also follows from these statement for the non-derived functors since tensoring over a field is exact

119868119894 (Id otimes 119871 119894) ≃ 119877 (Ind119894 (Id otimes Res119894)) ≃ 119877 (Ind119894 otimes Id) ≃ 119868119894 otimes Id(5)

By (a) 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868 are comonads (see [5Section VI1]) (b) and (c) imply that 119868119894 ∘ 119871 119894(119873) ≃ 119873 for119873 isin 119863

119887(Rep(119875119894)) and 119868 ∘ 119871(119873) ≃ 119873 for 119873 isin 119863

119887(Rep(119866))

Thus Id rarr 119868119894 ∘ 119871 119894 (resp Id rarr 119868 ∘ 119871) and from this we get thedesired isomorphism

119863119894 = 119871 119894 ∘ 119868119894 = 119871 119894 ∘ Id ∘ 119868119894 997888rarr 119871 119894 ∘ 119868119894 ∘ 119871 119894 ∘ 119868119894 = 1198632

119894 (6)

and likewise for 119863

Algebra 3

Remark 2 It follows that the restriction functors 119871 119894 and 119871 arefully faithful

3 Demazure Descent

Fix a root data (119868 119883 119884) of the finite type with the Weylgroup 119882 and the braid monoid Br+ Consider a triangulatedcategoryC

Definition 3 A weak braid monoid action on the categoryCis a collection of triangulated functors

119863119908 C 997888rarr C 119908 isin 119882 (7)

satisfying braid monoid relations that is for all 1199081 1199082 isin 119882

there exist isomorphisms of functors

1198631199081∘ 1198631199082

≃ 11986311990811199082 if ℓ (11990811199082) = ℓ (1199081) + ℓ (1199082) (8)

Notice that we neither fix the braid relations isomor-phisms nor impose any additional relations on them

Definition 4 Demazure descent data on the category C is aweak braidmonoid action 119863119908 such that for each simple root119904119894 the corresponding functor 119863119904119894

is a comonad for which thecomonad map 119863119904119894

rarr 1198632

119904119894is an isomorphism

Here is the central construction of the paper Consider atriangulated category C with a fixed Demazure descent data119863119908 119908 isin 119882 of the type (119868 119883 119884)

Definition 5 The descent category Desc(C 119863119908 119908 isin 119882) isthe full subcategory in C consisting of objects 119872 such thatfor all 119894 the cones of the counit maps 119863119904119894

(119872)120598

997888rarr 119872 areisomorphic to 0

Remark 6 Suppose that C has functorial cones ThenDesc(C 119863119908 119908 isin 119882) a full triangulated subcategory inC being the intersection of kernels of Cone(119863119904119894 rarr Id)However one can prove this statement not using functorialityof cones

Lemma 7 An object 119872 isin Desc(C 119863119908 119908 isin 119882) is naturallya comodule over each 119863119904119894

Proof By definition the comonad maps

120578 119863119904119894997888rarr 119863

2

119904119894 120598 119863119904119894

997888rarr Id (9)

make the following diagram commutative

Ds119894

D2s119894

120578

120598 ∘ Ds119894

Id ∘ Ds119894

(10)

For Demazure descent data we require that 120578 is an iso-morphism so 120598 ∘ 119863119904119894

is also an isomorphism Let 119872 isin

Desc(C 119863119908 119908 isin 119882) That Cone(119863119904119894(119872)120598

997888rarr 119872) is

isomorphic to 0 is equivalent to saying that 119863119904119894(119872)

120598

997888rarr 119872

is an isomorphism This gives the commutative diagramConsider

120578

120598minus1

120598minus1

Ds119894(M)

Ds119894(M) D

2s119894(M)

(120598 ∘ Ds119894)minus1

M

(11)

Thus 120598minus1 satisfies the axiom for the coaction

Remark 8 Recall that in the usual descent setting either inAlgebraic Geometry or in abstract Category Theory (Barr-Beck theorem) descent data includes a pair of adjoint functorsand their composition which is a comonad By definition thedescent category for such data is the category of comodulesover this comonad Our definition of Desc(119862119863119908 119908 isin 119882)

for Demazure descent data formally is not about comodulesyet the previous Lemma demonstrates that every object ofDesc(119862119863119908 119908 isin 119882) is naturally equipped with struc-tures of a comodule over each 119863119894 and any morphism inDesc(119862119863119908 119908 isin 119882) is a morphism of 119863119894-comodules

4 Main Theorem

We now go back to considering 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868

Proposition 9 Let 119908 isin 119882 and let 119908 = 1199041198941sdot sdot sdot 119904119894119899

be a reducedexpressionThen119863119908 = 1198631198941

∘sdot sdot sdot∘119863119894119899is independent of the choice

of reduced expression and the 119863119908rsquos form Demazure descentdata onC = 119863

119887(Rep (119861))

Lemma 10 Let 119908 = 1199041198941sdot sdot sdot 119904119894119899

be a reduced expression Then

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1198611199081015840119861 (12)

where the union is over all 1199081015840 isin 119882 which is le 119908 in the Bruhatorder

Proof The proof goes by induction on 119899 = ℓ(119908) It is truefor 119899 = 1 by definition of 119875119894 Set V = 1199041198941

sdot sdot sdot 119904119894119899minus1 Using the

hypotheses we get

1198751198941sdot sdot sdot 119875119894119899minus1

119875119894119899= ( ⋃

1199081015840leV

1198611199081015840119861) (119861 cup 119861119904119894119899

119861)

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

(1198611199081015840119861) (119861119904119894119899

119861)

(13)

Let 1199081015840 be any element in 119882 and 119904 a simple reflection Then

by [6 Corollary 283] we have (1198611199081015840119861)(119861119904119861) sube 119861119908

1015840119904119861 cup

1198611199081015840119861 Thus if 119908

1015840119904119894119899

le 1199081015840

le V then (1198611199081015840119861)(119861119904119894119899

119861) iscontained in the first union If 119908

1015840le 1199081015840119904119894119899 then we have

4 Algebra

(1198611199081015840119861)(119861119904119894119899

119861) = 1198611199081015840119904119894119899

119861 by [6 Lemma 293A and section291] Thus the product can be written as

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119904119894119899

119861

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

11990810158401015840119904119894119899leV

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

(14)

Claim The conditions 11990810158401015840119904119894119899

le V and 11990810158401015840119904119894119899

le 11990810158401015840 are

equivalent to the conditions 11990810158401015840

le 119908 and 11990810158401015840119904119894119899

le 11990810158401015840

Proof of the Claim Assume that11990810158401015840119904119894119899 le V By [7 Proposition59] this implies that 11990810158401015840 le V or 119908

10158401015840le V119904119894119899 = 119908 In both cases

we get 11990810158401015840 le 119908 since V le 119908 Assume now that 11990810158401015840 le 119908 and11990810158401015840119904119894119899

le 11990810158401015840 11990810158401015840 has a reduced expression of the form

11990810158401015840

= 1199041198941sdot sdot sdot 1199041198941198951

sdot sdot sdot 1199041198941198952sdot sdot sdot 119904119894119895

119896

sdot sdot sdot 119904119894119899 (15)

where the indicates that the term has been removed fromthe product If 119895119896 = 119899 then

11990810158401015840119904119894119899

= 1199041198941sdot sdot sdot 1199041198941198951


119896

sdot sdot sdot 119904119894119899minus1le 1199041198941

sdot sdot sdot 119904119894119899minus1= V (16)

If 119895119896 = 119899 then 11990810158401015840

le V Since 11990810158401015840119904119894119899

le 11990810158401015840 by assumption we

get 11990810158401015840119904119894119899 le VThis completes the proof of the claimIf1199081015840 le V in the first union satisfies that1199081015840119904119894119899 le 119908

1015840 then itis also contained in the second union Using the claim we get

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861 (17)

Assume that 1199081015840 le 119908 and 1199081015840le 1199081015840119904119894119899 Then 119908

1015840 has a reducedexpression of the form

1199081015840= 1199041198941


sdot sdot sdot 119904119894119895119896

sdot sdot sdot 119904119894119899 (18)

If 119895119896 = 119899 then 1199081015840

le V If 119895119896 = 119899 then 1199081015840119904119894119899

le V but since1199081015840le 1199081015840119904119894119899

we get 1199081015840 le V Hence the conditions 1199081015840le V and

1199081015840le 1199081015840119904119894119899

can be replaced by 1199081015840le 119908 and 119908

1015840le 1199081015840119904119894119899 Thus

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

= ⋃

1199081015840le119908

1198611199081015840119861

(19)

This finishes the induction step

Proof of the Proposition Let 119908 isin 119882 and let 1199041198941sdot sdot sdot 119904119894119899

=

119904119895119894sdot sdot sdot 119904119895119899

be two reduced expressions for119908 By Lemma 10 thisimplies that 1198751198941 sdot sdot sdot 119875119894119899 = 1198751198951

sdot sdot sdot 119875119895119899 By [8 Theorem 31] the 119861-

module structure ofΔ 1198941 ∘ sdot sdot sdot∘Δ 119894119899 is determined upto a naturalisomorphism by the set 1198751198941 sdot sdot sdot 119875119894119899 Hence

Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899

≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899

(20)

Hence for any choice of reduced expression we can define

Δ119908 = Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119898

(21)

Let1199081 and1199082 be elements in119882 such that ℓ(11990811199082) = ℓ(1199081) +

ℓ(1199082) Pick reduced expressions 1199041198941sdot sdot sdot 119904119894119903

and 1199041198951sdot sdot sdot 119904119895119905

for1199081 and 1199082 respectively Then 1199041198941

sdot sdot sdot 1199041198941199031199041198951

sdot sdot sdot 119904119895119905is a reduced

expression for 11990811199082 and we get braid relations for the Δ119908

Δ1199081∘ Δ1199082

= Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119903

∘ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119905

= Δ11990811199082 (22)

Define

119863119908 = 119877 (Δ119908)

= 119877 (Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119898

)

= 119877 (Δ 1198941) ∘ sdot sdot sdot ∘ 119877 (Δ 119894119898

)

= 1198631198941∘ sdot sdot sdot ∘ 119863119894119898

(23)

The braid relations for 119863119908 now follows from the braidrelations for Δ119908

1198631199081∘ 1198631199082

= 119877 (Δ1199081) ∘ 119877 (Δ1199082

) ≃ 119877 (Δ1199081∘ Δ1199082

) ≃ 119877 (Δ11990811199082)

= 11986311990811199082

(24)

Theorem 11 Desc(C 119863119908 119908 isin 119882) is equivalent to119863119887(Rep (119866))

Proof Let119872 isin 119863119887(Rep(119861)) Being able to extend119872 to an ele-

ment in119863119887(Rep(119866) is equivalent to119872 being in the image of119871

Assume119872 = 119871(119873) for some119873 isin 119863119887(Rep(119866))Then119863(119872) =

119871 ∘ 119868 ∘ 119871(119873) ≃ 119871(119873) = 119872 If119863(119872) rarr119872 then119872 ≃ 119871(119868(119872))so 119872 is in the image of 119871 Thus being in the image of 119871 isequivalent to 119863(119872) rarr 119872 being an isomorphism which isagain equivalent to119872 isin ker(119862) where 119862 = Cone(119863 rarr Id)Set 119862119894 = Cone(119863119894 rarr Id)

Claim ker(119862) = ⋂119894ker(119862119894)

Proof of Claim Assume that 119872 isin ker(119862) Then 119872 = 119871(119873)

for some 119873 isin 119863119887(Rep(119866)) But then 119872 = 119871 119894(119873|119875119894

) for all 119894so 119863119894(119872) rarr 119872 is an isomorphism for all 119894 Hence 119872 isin

⋂119894ker(119862119894) Assume that 119872 isin ⋂

119894ker(119862119894) Then all 119863119894(119872) rarr

119872 are isomorphisms Choose a reduced expression 1199041198941sdot sdot sdot 119904119894119873

Algebra 5

for the longest element in the Weyl group Then 1198751198941sdot sdot sdot 119875119894119873

=

119866 By [8] we have 119863 = 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

119863 (119872) ≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

(119872)

≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873minus1

(119872)

≃ sdot sdot sdot ≃ 1198631198941(119872) ≃ 119872

(25)

Hence

Cone (119863 (119872) 997888rarr 119872) ≃ Cone (119863 (119863 (119872)) 997888rarr 119863 (119872))

(26)

By definition of a comonad we have the following commuta-tive diagram

Id ∘ D

D

D2

120598D

120578

(27)

Since 120578 is an isomorphism so is 120598119863 and thusCone(119863(119863(119872)) rarr 119863(119872)) = 0 This shows that119872 isin ker(119862)

This completes the proof of the claimFrom the claim we get that

119863119887(Rep (119866)) = ⋂

119894

ker (119862119894) (28)

which is exactly the descent category

5 Further Directions

51 Quantum Groups Fix a root data (119868 119883 119884) of the finitetype Let UA be the Lusztig quantum group over the ring ofquantum integersA = Z[V Vminus1] Denote the quantum Borelsubalgebra by BA For a simple root 120572119894 the correspondingquantum parabolic subalgebra is denoted by P119894A

Following [9] we consider the categories of locallyfinite weight modules over UA (resp over BA respover P119894A) denoted by Rep(UA) (resp by Rep(BA) respby Rep(P119894A)) We consider the corresponding derivedcategories 119863

119887(Rep(UA)) 119863119887(Rep(BA)) and 119863

119887(Rep(P119894A))

Like in the reductive algebraic group case the restrictionfunctors

119871 119863119887(Rep (UA)) 997888rarr 119863

119887(Rep (BA))

119871 119894 119863119887(Rep (P119894A)) 997888rarr 119863

119887(Rep (BA))

(29)

are fully faithful and possess right adjoint functors denoted by119868 (resp by 119868119894) Denote the comonad 119871 119894 ∘ 119868119894 by 119863119894 AndersenPolo andWenproved that the functors119863119894 define aweak braidmonoid action on the category 119863

119887(Rep(BA)) One can easily

prove that the functors formDemazure descent dataThe cor-responding descent categoryDesc(119863119887(Rep(BA)) 1198631 119863119899)

is equivalent to 119863119887(Rep(UA))

52 Equivariant Sheaves Let119883 be an affine scheme equippedwith an action of a reductive algebraic group 119866 Fix aBorel subgroup 119861 sub 119866 Like in the main body of thepresent paper consider the minimal parabolic subgroupsin 119866 denoted by 1198751 119875119899 Denote the derived categoriesof quasicoherent sheaves on 119883 equivariant with respectto 119866 (resp 119861 resp 119875119894) by 119863

119887(119876Coh119866(119883)) (resp by

119863119887(119876Coh119861(119883)) resp by 119863

119887(119876Coh119875119894(119883))) We have the

natural functors provided by restriction of equivariance119871 119863

119887(119876Coh119866(119883)) rarr 119863

119887(119876Coh119861(119883)) and 119871 119894

119863119887(119876Coh119875119894(119883)) rarr 119863

119887(119876Coh119861(119883))These functors have the

right adjoint ones 119868 resp 1198681 119868119899 The comonads 1198631 119863119899

given by the compositions of extension and restriction ofequivariance define a Demazure descent data on the cate-gory 119863

119887(119876Coh119861(119883)) The corresponding descent category is

equivalent to 119863119887(119876Coh119866(119883))

53 Algebraic Loop Group For a simple algebraic group 119866

consider the algebraic loop group 119871119866 = Map(∙

119863119866) (respthe formal arcs group 119871

+119866 = Map(119863 119866)) Here 119863 (resp

∙

119863)denotes the formal disc (resp the formal punctured disc)Consider the affine Kac-Moody central extension

1 997888rarr G119898 997888rarr 119871119866 997888rarr 119871119866 997888rarr 1 (30)

The affine analog of the Borel subgroup 119861 sub 119866 is the Iwahorisubgroup 119868119908 sub 119871

+119866 Let 1198750 119875119899 be the standard minimal

parahoric subgroups in 119871+119866 One considers the adjoint pairs

of coinduction-restriction functors 1198680 1198710 119868119899 119871119899 between119863119887(Rep(119868119908)) and 119863

119887(Rep(119875119894)) Denote the comonads 119871 119894 ∘ 119868119894

by D119894 for 119894 = 0 119899 We claim that 1198630 119863119899 form affineDemazure descent data on 119863

119887(Rep(119868119908)) We conjecture that

the descent category is equivalent to 119863119887(Rep(119871119866)) (direct

sum of the categories over all positive integral levels)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to H H Andersen C Dodd VGinzburg M Harada and R Rouquier for many stimulatingdiscussions The project started in the summer of 2012 whenthe first named author visited IHES SergeyArkhipov is grate-ful to IHES for perfect working conditions Both authorsrsquoresearch was supported in part by center of excellence grantsldquoCentre for Quantum Geometry of Moduli Spacesrdquo and byFNU grant ldquoAlgebraic Groups and Applicationsrdquo

References

[1] D Ben-Zvi and D Nadler ldquoBeilinson-Bernstein localizationover theHarish-Chandra centerrdquo httparxivorgabs12090188

6 Algebra

[2] A Beılinson and J Bernstein ldquoLocalisation de 119892-modulesrdquoComptes Rendus des Seances de lrsquoAcademie des Sciences vol 292no 1 pp 15ndash18 1981

[3] M Harada G D Landweber and R Sjamaar ldquoDivided differ-ences and theWeyl character formula in equivariant119870-theoryrdquoMathematical Research Letters vol 17 no 3 pp 507ndash527 2010

[4] J C Jantzen Representations of Algebraic Groups AcademicPress Boston Mass USA 1987

[5] S Mac Lane Categories for the Working Mathematician vol 5Springer New York NY USA 2nd edition 1998

[6] J E Humphreys Linear Algebraic Groups Springer New YorkNY USA 4th edition 1975

[7] J E Humphreys Reflection Groups and Coxeter Groups Cam-bridge University Press Cambridge UK 1997

[8] E Cline B Parshall and L Scott ldquoInduced modules andextensions of representationsrdquo Inventiones Mathematicae vol47 no 1 pp 41ndash51 1978

[9] H H Andersen P Polo and K X Wen ldquoRepresentations ofquantum algebrasrdquo InventionesMathematicae vol 104 no 1 pp1ndash59 1991

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Algebra

In the present paper we define a notion of Demazuredescent on a triangulated category C Thus Demazure oper-ators are replaced by Demazure functors These functorssatisfy a categorified version of degenerate Hecke algebrarelations and form a Demazure descent data onC We definethe descent category for such data Demazure descent issupposed to be a technique replacing the naive notion ofWeylgroup invariants on the categorical level

We provide the first example of Demazure descentConsider a reductive algebraic group 119866 and fix a Borelsubgroup 119861 sub 119866 Categorifying the construction form [3]we consider Demazure functors 119863119904119894

acting on the derivedcategory of 119861-modules We prove that the functors form aDemazure descent data and identify the descent categorywith the derived category of 119866-modules

2 The Setting

21 Root Data Let 119866 be a reductive algebraic group over analgebraically closed field 119896 of characteristic zero Let 119879 be aCartan subgroup of 119866 and let (119868 119883 119884) be the correspondingroot data where 119868 is the set of vertices of theDynkin diagram119883 is the weight lattice of 119866 and 119884 is the coroot lattice of119866 Choose a Borel subgroup 119879 sub 119861 sub 119866 Denote the setof roots for 119866 by Φ = Φ

+⊔ Φminus Let 1205721 120572119899 be the set

of simple roots The Weyl group 119882 = Norm(119879)119879 of thefixed maximal torus acts naturally on the lattices 119883 and 119884

and on the R-vector spaces spanned by them by reflectionsin root hyperplanes The simple reflection corresponding toan 120572119894 is denoted by 119904119894 The elements 1199041 119904119899 form a set ofgenerators for 119882 For 119908 isin 119882 denote the length of a minimalexpression of 119908 via the generators by ℓ(119908) We have a partialordering on 119882 called the Bruhat ordering 1199081015840 le 119908 if thereexists a reduced expression for 119908

1015840 that can be obtained froma reduced expression for 119908 by deleting a number of simplereflections

The monoid Br+ with generators 119879119908 119908 isin 119882 andrelations

11987911990811198791199082

= 11987911990811199082if ℓ (1199081) + ℓ (1199082) = ℓ (11990811199082) in 119882 (1)

is called the braid monoid of 119866

22 Categories of Representations For an algebraic group 119867we denote the Hopf algebra of polynomial functions on119867 byO(119867) Let Rep(119867) be the category of O(119867)-comodules Thisis an Abelian tensor category

Let 119875119894 be the parabolic subgroup of119866 containing 119861whoseLevi subgroup has the root system 120572119894 minus120572119894 Using the naturalHopf algebramapsO(119866) rarr O(119861) andO(119875119894) rarr O(119861)we canget restriction functors

Res119894 Rep (119875119894) 997888rarr Rep (119861)

Res Rep (119866) 997888rarr Rep (119861)

(2)

The restriction functors are exact and naturally commutewith taking tensor product of representations Let 119867 be asubgroup of 119866 and119872 isin Rep(119866) Define the119867-invariant part

of 119872 to be 119872119867

= HomRep(119867)(119896119872) Consider the inductionfunctors

Ind119894 Rep (119861) 997888rarr Rep (119875119894) 119872 997891997888rarr (O (119875119894) otimes 119872)119861

Ind Rep (119861) 997888rarr Rep (119866) 119872 997891997888rarr (O (119866) otimes 119872)119861

(3)

Set Δ 119894 = Res119894 ∘ Ind119894 Rep(119861) and Δ = Res ∘ Ind

Rep(119861) Notice thatΔ 119894 andΔ are left exact since the inductionfunctors are left exact

23 The Derived Categories For an algebraic group 119867 theregular comodule O(119867) is injective in Rep(119867) moreover forany119872 isin Rep(119867) the coactionmap119872 rarr O(119867)otimes119872 providesan embedding of 119872 into an injective object In particularRep(119867) has enough injectives The algebraic De RhamcomplexΩ

∙(119867) provides an injective resolution for the trivial

comodule of the length equal to the dimension of119867 For any119872 isin Rep(119867) the complex Ω

∙(119867) otimes 119872 provides an injective

resolution for 119872 of the same lengthConsider now the bounded derived categories

119863119887(Rep(119861)) 119863

119887(Rep(119875119894)) and 119863

119887(Rep(119866)) Let 119871 119894 and 119871

be the derived functors of Res119894 and Res respectively Denotethe right derived functors of Ind119894 and Ind by 119868119894 and 119868respectively Let119863119894 = 119871 119894 ∘ 119868119894 and119863 = 119871∘119868 be the right derivedfunctors of Δ 119894 and Δ respectively

Proposition 1 (a) The functors 119871 119894 and 119871 are left adjoint to 119868119894

and 119868 respectively(b) For119872 isin 119863

119887(Rep (119861)) and119873 isin 119863

119887(Rep (119875119894)) (resp for

119872 isin 119863119887(Rep (119861)) and 119873 isin 119863

119887(Rep (119866))) we have the tensor

identities119868119894 (119872 otimes 119871 119894 (119873)) ≃ 119868119894 (119872) otimes 119873

(119903119890119904119901 119868 (119872 otimes 119871 (119873)) ≃ 119868 (119872) otimes 119873)

(4)

(c) The functors 119868119894 and 119868 take the trivial O(119861)-comoduleto the trivial O(119875119894)-comodule (resp to the trivial O(119866)-comodule)

(d) 119863119894 and 119863 are comonads for which the comonad maps119863119894 rarr 119863

2

119894and 119863 rarr 119863

2 are isomorphisms

Proof The statements corresponding to (a) and (b) for Resand Ind (resp Res119894 and Ind119894) are Propositions 34 and 36in [4] The derived functors of a pair of adjoint functors areadjoint (b) also follows from these statement for the non-derived functors since tensoring over a field is exact

119868119894 (Id otimes 119871 119894) ≃ 119877 (Ind119894 (Id otimes Res119894)) ≃ 119877 (Ind119894 otimes Id) ≃ 119868119894 otimes Id(5)

By (a) 119863119894 = 119871 119894 ∘ 119868119894 and 119863 = 119871 ∘ 119868 are comonads (see [5Section VI1]) (b) and (c) imply that 119868119894 ∘ 119871 119894(119873) ≃ 119873 for119873 isin 119863

119887(Rep(119875119894)) and 119868 ∘ 119871(119873) ≃ 119873 for 119873 isin 119863

119887(Rep(119866))

Thus Id rarr 119868119894 ∘ 119871 119894 (resp Id rarr 119868 ∘ 119871) and from this we get thedesired isomorphism

119863119894 = 119871 119894 ∘ 119868119894 = 119871 119894 ∘ Id ∘ 119868119894 997888rarr 119871 119894 ∘ 119868119894 ∘ 119871 119894 ∘ 119868119894 = 1198632

119894 (6)

and likewise for 119863

Algebra 3


3 Demazure Descent



119863119908 C 997888rarr C 119908 isin 119882 (7)



1198631199081∘ 1198631199082

≃ 11986311990811199082 if ℓ (11990811199082) = ℓ (1199081) + ℓ (1199082) (8)




rarr 1198632




(119872)120598





120578 119863119904119894997888rarr 119863

2

119904119894 120598 119863119904119894

997888rarr Id (9)


Ds119894

D2s119894

120578

120598 ∘ Ds119894

Id ∘ Ds119894

(10)




997888rarr 119872) is


120598

997888rarr 119872


120578

120598minus1

120598minus1

Ds119894(M)

Ds119894(M) D

2s119894(M)

(120598 ∘ Ds119894)minus1

M

(11)




4 Main Theorem






119887(Rep (119861))



1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1198611199081015840119861 (12)




hypotheses we get


119875119894119899= ( ⋃

1199081015840leV

1198611199081015840119861) (119861 cup 119861119904119894119899

119861)

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

(1198611199081015840119861) (119861119904119894119899

119861)

(13)



1015840119904119861 cup

1198611199081015840119861 Thus if 119908

1015840119904119894119899

le 1199081015840

le V then (1198611199081015840119861)(119861119904119894119899


1015840le 1199081015840119904119894119899 then we have

4 Algebra

(1198611199081015840119861)(119861119904119894119899

119861) = 1198611199081015840119904119894119899


1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119904119894119899

119861

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

11990810158401015840119904119894119899leV

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

(14)


le V and 11990810158401015840119904119894119899

le 11990810158401015840 are


le 119908 and 11990810158401015840119904119894119899

le 11990810158401015840


10158401015840le V119904119894119899 = 119908 In both cases



11990810158401015840

= 1199041198941sdot sdot sdot 1199041198941198951


119896

sdot sdot sdot 119904119894119899 (15)


11990810158401015840119904119894119899

= 1199041198941sdot sdot sdot 1199041198941198951


119896



If 119895119896 = 119899 then 11990810158401015840

le V Since 11990810158401015840119904119894119899




1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861 (17)



1199081015840= 1199041198941


sdot sdot sdot 119904119894119895119896

sdot sdot sdot 119904119894119899 (18)

If 119895119896 = 119899 then 1199081015840

le V If 119895119896 = 119899 then 1199081015840119904119894119899

le V but since1199081015840le 1199081015840119904119894119899


1199081015840le 1199081015840119904119894119899


1015840le 1199081015840119904119894119899 Thus

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

= ⋃

1199081015840le119908

1198611199081015840119861

(19)



=

119904119895119894sdot sdot sdot 119904119895119899




Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899

≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899

(20)



(21)





sdot sdot sdot 1199041198941199031199041198951



Δ1199081∘ Δ1199082

= Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119903

∘ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119905

= Δ11990811199082 (22)

Define

119863119908 = 119877 (Δ119908)

= 119877 (Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119898

)

= 119877 (Δ 1198941) ∘ sdot sdot sdot ∘ 119877 (Δ 119894119898

)

= 1198631198941∘ sdot sdot sdot ∘ 119863119894119898

(23)


1198631199081∘ 1198631199082

= 119877 (Δ1199081) ∘ 119877 (Δ1199082

) ≃ 119877 (Δ1199081∘ Δ1199082

) ≃ 119877 (Δ11990811199082)

= 11986311990811199082

(24)






Claim ker(119862) = ⋂119894ker(119862119894)





119894ker(119862119894) Then all 119863119894(119872) rarr


Algebra 5


=


119863 (119872) ≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

(119872)


(119872)

≃ sdot sdot sdot ≃ 1198631198941(119872) ≃ 119872

(25)

Hence


(26)


Id ∘ D

D

D2

120598D

120578

(27)



119863119887(Rep (119866)) = ⋂

119894

ker (119862119894) (28)





119887(Rep(UA)) 119863119887(Rep(BA)) and 119863

119887(Rep(P119894A))


119871 119863119887(Rep (UA)) 997888rarr 119863

119887(Rep (BA))

119871 119894 119863119887(Rep (P119894A)) 997888rarr 119863

119887(Rep (BA))

(29)






119887(119876Coh119866(119883)) (resp by

119863119887(119876Coh119861(119883)) resp by 119863

119887(119876Coh119875119894(119883))) We have the


119887(119876Coh119866(119883)) rarr 119863

119887(119876Coh119861(119883)) and 119871 119894

119863119887(119876Coh119875119894(119883)) rarr 119863









+119866 = Map(119863 119866)) Here 119863 (resp

∙














Acknowledgments


References


6 Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 3


3 Demazure Descent



119863119908 C 997888rarr C 119908 isin 119882 (7)



1198631199081∘ 1198631199082

≃ 11986311990811199082 if ℓ (11990811199082) = ℓ (1199081) + ℓ (1199082) (8)




rarr 1198632




(119872)120598





120578 119863119904119894997888rarr 119863

2

119904119894 120598 119863119904119894

997888rarr Id (9)


Ds119894

D2s119894

120578

120598 ∘ Ds119894

Id ∘ Ds119894

(10)




997888rarr 119872) is


120598

997888rarr 119872


120578

120598minus1

120598minus1

Ds119894(M)

Ds119894(M) D

2s119894(M)

(120598 ∘ Ds119894)minus1

M

(11)




4 Main Theorem






119887(Rep (119861))



1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1198611199081015840119861 (12)




hypotheses we get


119875119894119899= ( ⋃

1199081015840leV

1198611199081015840119861) (119861 cup 119861119904119894119899

119861)

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

(1198611199081015840119861) (119861119904119894119899

119861)

(13)



1015840119904119861 cup

1198611199081015840119861 Thus if 119908

1015840119904119894119899

le 1199081015840

le V then (1198611199081015840119861)(119861119904119894119899


1015840le 1199081015840119904119894119899 then we have

4 Algebra

(1198611199081015840119861)(119861119904119894119899

119861) = 1198611199081015840119904119894119899


1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119904119894119899

119861

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

11990810158401015840119904119894119899leV

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

(14)


le V and 11990810158401015840119904119894119899

le 11990810158401015840 are


le 119908 and 11990810158401015840119904119894119899

le 11990810158401015840


10158401015840le V119904119894119899 = 119908 In both cases



11990810158401015840

= 1199041198941sdot sdot sdot 1199041198941198951


119896

sdot sdot sdot 119904119894119899 (15)


11990810158401015840119904119894119899

= 1199041198941sdot sdot sdot 1199041198941198951


119896



If 119895119896 = 119899 then 11990810158401015840

le V Since 11990810158401015840119904119894119899




1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861 (17)



1199081015840= 1199041198941


sdot sdot sdot 119904119894119895119896

sdot sdot sdot 119904119894119899 (18)

If 119895119896 = 119899 then 1199081015840

le V If 119895119896 = 119899 then 1199081015840119904119894119899

le V but since1199081015840le 1199081015840119904119894119899


1199081015840le 1199081015840119904119894119899


1015840le 1199081015840119904119894119899 Thus

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

= ⋃

1199081015840le119908

1198611199081015840119861

(19)



=

119904119895119894sdot sdot sdot 119904119895119899




Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899

≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899

(20)



(21)





sdot sdot sdot 1199041198941199031199041198951



Δ1199081∘ Δ1199082

= Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119903

∘ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119905

= Δ11990811199082 (22)

Define

119863119908 = 119877 (Δ119908)

= 119877 (Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119898

)

= 119877 (Δ 1198941) ∘ sdot sdot sdot ∘ 119877 (Δ 119894119898

)

= 1198631198941∘ sdot sdot sdot ∘ 119863119894119898

(23)


1198631199081∘ 1198631199082

= 119877 (Δ1199081) ∘ 119877 (Δ1199082

) ≃ 119877 (Δ1199081∘ Δ1199082

) ≃ 119877 (Δ11990811199082)

= 11986311990811199082

(24)






Claim ker(119862) = ⋂119894ker(119862119894)





119894ker(119862119894) Then all 119863119894(119872) rarr


Algebra 5


=


119863 (119872) ≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

(119872)


(119872)

≃ sdot sdot sdot ≃ 1198631198941(119872) ≃ 119872

(25)

Hence


(26)


Id ∘ D

D

D2

120598D

120578

(27)



119863119887(Rep (119866)) = ⋂

119894

ker (119862119894) (28)





119887(Rep(UA)) 119863119887(Rep(BA)) and 119863

119887(Rep(P119894A))


119871 119863119887(Rep (UA)) 997888rarr 119863

119887(Rep (BA))

119871 119894 119863119887(Rep (P119894A)) 997888rarr 119863

119887(Rep (BA))

(29)






119887(119876Coh119866(119883)) (resp by

119863119887(119876Coh119861(119883)) resp by 119863

119887(119876Coh119875119894(119883))) We have the


119887(119876Coh119866(119883)) rarr 119863

119887(119876Coh119861(119883)) and 119871 119894

119863119887(119876Coh119875119894(119883)) rarr 119863









+119866 = Map(119863 119866)) Here 119863 (resp

∙














Acknowledgments


References


6 Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Algebra

(1198611199081015840119861)(119861119904119894119899

119861) = 1198611199081015840119904119894119899


1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119904119894119899

119861

= ⋃

1199081015840leV

1198611199081015840119861 cup ⋃

11990810158401015840119904119894119899leV

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

(14)


le V and 11990810158401015840119904119894119899

le 11990810158401015840 are


le 119908 and 11990810158401015840119904119894119899

le 11990810158401015840


10158401015840le V119904119894119899 = 119908 In both cases



11990810158401015840

= 1199041198941sdot sdot sdot 1199041198941198951


119896

sdot sdot sdot 119904119894119899 (15)


11990810158401015840119904119894119899

= 1199041198941sdot sdot sdot 1199041198941198951


119896



If 119895119896 = 119899 then 11990810158401015840

le V Since 11990810158401015840119904119894119899




1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840leV

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861 (17)



1199081015840= 1199041198941


sdot sdot sdot 119904119894119895119896

sdot sdot sdot 119904119894119899 (18)

If 119895119896 = 119899 then 1199081015840

le V If 119895119896 = 119899 then 1199081015840119904119894119899

le V but since1199081015840le 1199081015840119904119894119899


1199081015840le 1199081015840119904119894119899


1015840le 1199081015840119904119894119899 Thus

1198751198941sdot sdot sdot 119875119894119899

= ⋃

1199081015840le119908

1199081015840le1199081015840119904119894119899

1198611199081015840119861 cup ⋃

11990810158401015840le119908

11990810158401015840119904119894119899le11990810158401015840

11986111990810158401015840119861

= ⋃

1199081015840le119908

1198611199081015840119861

(19)



=

119904119895119894sdot sdot sdot 119904119895119899




Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119899

≃ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119899

(20)



(21)





sdot sdot sdot 1199041198941199031199041198951



Δ1199081∘ Δ1199082

= Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119903

∘ Δ 1198951∘ sdot sdot sdot ∘ Δ 119895119905

= Δ11990811199082 (22)

Define

119863119908 = 119877 (Δ119908)

= 119877 (Δ 1198941∘ sdot sdot sdot ∘ Δ 119894119898

)

= 119877 (Δ 1198941) ∘ sdot sdot sdot ∘ 119877 (Δ 119894119898

)

= 1198631198941∘ sdot sdot sdot ∘ 119863119894119898

(23)


1198631199081∘ 1198631199082

= 119877 (Δ1199081) ∘ 119877 (Δ1199082

) ≃ 119877 (Δ1199081∘ Δ1199082

) ≃ 119877 (Δ11990811199082)

= 11986311990811199082

(24)






Claim ker(119862) = ⋂119894ker(119862119894)





119894ker(119862119894) Then all 119863119894(119872) rarr


Algebra 5


=


119863 (119872) ≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

(119872)


(119872)

≃ sdot sdot sdot ≃ 1198631198941(119872) ≃ 119872

(25)

Hence


(26)


Id ∘ D

D

D2

120598D

120578

(27)



119863119887(Rep (119866)) = ⋂

119894

ker (119862119894) (28)





119887(Rep(UA)) 119863119887(Rep(BA)) and 119863

119887(Rep(P119894A))


119871 119863119887(Rep (UA)) 997888rarr 119863

119887(Rep (BA))

119871 119894 119863119887(Rep (P119894A)) 997888rarr 119863

119887(Rep (BA))

(29)






119887(119876Coh119866(119883)) (resp by

119863119887(119876Coh119861(119883)) resp by 119863

119887(119876Coh119875119894(119883))) We have the


119887(119876Coh119866(119883)) rarr 119863

119887(119876Coh119861(119883)) and 119871 119894

119863119887(119876Coh119875119894(119883)) rarr 119863









+119866 = Map(119863 119866)) Here 119863 (resp

∙














Acknowledgments


References


6 Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 5


=


119863 (119872) ≃ 1198631198941∘ sdot sdot sdot ∘ 119863119894119873

(119872)


(119872)

≃ sdot sdot sdot ≃ 1198631198941(119872) ≃ 119872

(25)

Hence


(26)


Id ∘ D

D

D2

120598D

120578

(27)



119863119887(Rep (119866)) = ⋂

119894

ker (119862119894) (28)





119887(Rep(UA)) 119863119887(Rep(BA)) and 119863

119887(Rep(P119894A))


119871 119863119887(Rep (UA)) 997888rarr 119863

119887(Rep (BA))

119871 119894 119863119887(Rep (P119894A)) 997888rarr 119863

119887(Rep (BA))

(29)






119887(119876Coh119866(119883)) (resp by

119863119887(119876Coh119861(119883)) resp by 119863

119887(119876Coh119875119894(119883))) We have the


119887(119876Coh119866(119883)) rarr 119863

119887(119876Coh119861(119883)) and 119871 119894

119863119887(119876Coh119875119894(119883)) rarr 119863









+119866 = Map(119863 119866)) Here 119863 (resp

∙














Acknowledgments


References


6 Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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