Categorifications and cyclotomic rational double affine Hecke algebras

116 Pages·2015·1.55 MB·English

by  Raphaël Rouquier

Invent.math.
DOI10.1007/s00222-015-0623-7
Categorifications and cyclotomic rational double
affine Hecke algebras
RaphaëlRouquier1 · PengShan2 ·
MichelaVaragnolo3 · EricVasserot4
Received:29May2013/Accepted:17August2015
©Springer-VerlagBerlinHeidelberg2015
Abstract Varagnolo and Vasserot conjectured an equivalence between the
categoryO forCRDAHA’sandasubcategoryofanaffineparaboliccategory
O of type A. We prove this conjecture. As applications, we prove a conjec-
ture of Rouquier on the dimension of simple modules of CRDAHA’s and a
conjecture of Chuang–Miyachi on the Koszul duality for the category O of
CRDAHA’s.
ThisresearchwaspartiallysupportedbytheANRGrantnumberANR-10-BLAN-0110,
ANR-13-BS01-0001-01andANR-12-JS01-0003.R.Rouquierwaspartiallysupportedbythe
NSFGrantDMS-1161999.
B
EricVasserot
[email protected]
RaphaëlRouquier
[email protected]
PengShan
[email protected]
MichelaVaragnolo
[email protected]
1 MathematicsDepartment,UCLA,LosAngeles,CA90095-1555,USA
2 Mathématiques,Bât.425,UniversitéParis-Sud,91405OrsayCedex,France
3 Mathématiques,UniversitédeCergy-Pontoise,95011Cergy-PontoiseCedex,France
4 InstitutdeMathématiquesdeJussieu,UniversitéParisDiderot-Paris7,
75252ParisCedex05,France
123
R.Rouquieretal.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Highestweightcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Heckealgebras,q-Schuralgebrasandcategorifications . . . . . . . . . . . . . . . . . . .
4 ThecategoryO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 ThecategoryO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 ThecategoryAandCRDAHA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Consequencesofthemaintheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 TheKazhdan–Lusztigcategory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Indexofnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Rational Double affine Hecke algebras (RDAHA for short) have been intro-
duced by Etingof and Ginzburg in 2002. They are associative algebras
associated with a complex reflection group W and a parameter c. Their rep-
resentation theory is similar to the representation theory of semi-simple Lie
algebras. In particular, they admit a category O which is analogous to the
BGGcategoryO. Thiscategoryishighestweightwiththestandardmodules
labeledbyirreduciblerepresentationsofW.RepresentationsinOareinfinite
dimensional in general, but they admit a character. An important question is
todeterminethecharactersofsimplemodules.
One of the most important family of RDAHA’s is the cyclotomic one
(CRDAHA for short), where W = G((cid:2),1,n) is the wreath product of S
n
and Z/(cid:2)Z. One reason is that the representation theory of CRDAHA’s is
closelyrelatedtotherepresentationtheoryofAriki–Koikealgebras,andthat
the latter are important in group theory. Another reason is that the category
OofCRDAHA’siscloselyrelatedtotherepresentationtheoryofaffineKac–
Moodyalgebras,seee.g.[17,43,46].Athirdreason,isthatthiscategoryhasa
veryrichstructurecalledacategoricalactionofanaffineKac–Moodyalgebra.
ThisactiononOwasconstructedpreviouslyin[41].Suchstructureshavebeen
introducedrecentlyinrepresentationtheoryandhavealreadyhadremarkable
applications,seee.g.[9,29,40].
ThestructureofO dependsheavilyontheparameterc.Forgenericvalues
of c the category is semi-simple. The most non semi-simple case (which is
alsothemostcomplicatedone)occurswhenctakesaparticularformofratio-
nal numbers, see (6.2). For these parameters Rouquier made a conjecture to
determinethecharactersofsimplemodulesinO[39].Roughlyspeaking,this
conjecturesaysthattheJordan–Höldermultiplicitiesofthestandardmodules
inOaregivenbysomeparabolicKazhdan–Lusztigpolynomials.Thisconjec-
turewasknowntobetrueintheparticularcase(cid:2) = 1[39].Motivatedbythis
conjecture,Varagnolo–Vasserotintroducedin[46]anewcategoryAwhichis
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Categorificationsandcyclotomicrationaldoubleaffine…
asubcategoryofanaffineparaboliccategoryOatanegativelevelandshould
beviewedasanaffineandhigherlevelanalogueofthecategoryofpolynomial
representationsofGL .Theyconjecturedthatthereshouldbeanequivalence
N
ofhighestweightcategoriesbetweenO andA.
In this paper we prove Varagnolo–Vasserot’s conjecture (Theorem 6.9). A
firstconsequenceisaproofofRouquier’sconjecture(Theorem7.3).Asecond
remarkableapplicationisaproofthatthecategoryOisKoszul(Theorem7.4),
yielding a proof of a conjecture of Chuang–Miyachi [8], because the affine
paraboliccategoryOisKoszulby[42].
OurproofisbasedonanextensionofRouquier’stheoryofhighestweight
coversdevelopedin[39].Basically,[39]saysthattwohighestweightcovers
ofthesamealgebraareequivalentashighestweightcategoriesiftheysatisfy
asocalled1-faithfulconditionandifthehighestweightordersonbothcovers
are compatible. Here, given a situation where the highest weight covers are
notnecessarily1-faithful,weconstructbiggerfunctorstowhichwecanapply
Rouquier’stheory(seeProposition2.20).
ThecategoryOisahighestweightcoveroverthemodulecategoryHofthe
Ariki–Koike algebra via the KZ functor introduced in [22]. It is a 0-faithful
coverandiftheparametersoftheRDAHAsatisfysometechnicalcondition,
then it is even 1-faithful. A similar functor (cid:3) : A → H was introduced in
[46]usingtheKazhdan–LusztigfusionproductontheaffinecategoryOata
negative level. A previous work of Dunkl and Griffeth [16] allows to show
withoutmuchdifficultythatthereisahighestweightorderonOwhichrefines
the linkage order on A. A difficult part of the proof consists of showing that
the functor (cid:3) is indeed a cover, meaning that it is an exact quotient functor,
andthatithasthesamefaithfulnesspropertiesastheKZfunctor.Oncethisis
done, the equivalence between O and A follows directly from the unicity of
1-faithfulcoversifthetechnicalconditiononparametersmentionedaboveis
satisfied.Toprovetheequivalencewithoutthiscondition,weneedtoreplace
KZ and (cid:3) by some other covers, see the end of the introduction for more
detailsonthis.
Akeyingredientinourproofisadeformationargument.Moreprecisely,the
highestweightcategoriesA,Oadmitdeformedversionsoveraregularlocal
ring Rofdimension2.SometechnicalresultsprovethattheKazhdan–Lusztig
tensor product can also be deformed properly, which allows us to define the
deformed version of (cid:3). Next, a theorem of Fiebig asserts that the structure
of the category O of a Kac–Moody algebra only depends on the associated
Coxetersystem[20].Inparticular,thelocalizationofAataheightoneprime
ideal p ⊂ R can be described in simpler terms. Two cases appear, either p
is subgeneric or generic. In the first case, considered in Sect. 5.7.2, the cate-
gory A reduces to an analog subcategory A inside the parabolic category O
of gl associated with a Levi subalgebra of gl with 2 blocks. The latter is
N N
123
R.Rouquieretal.
closelyrelatedtothehigherlevelSchur–WeyldualitystudiedbyBrundanand
Kleshchev in [5]. In the second case, considered in Sect. 5.7.3, the category
A reduces to the corresponding category for (cid:2) = 1, which is precisely the
Kazhdan–LusztigcategoryassociatedwithaffineLiealgebrasatnegativelev-
els.Finally,weshowthattoprovethedesiredpropertiesofthefunctor(cid:3)itis
enoughtocheckthemforthelocalizationof(cid:3)ateachheightoneprimeideal
pandthisprovesthemainresult.
Now,letussayafewwordsconcerningtheorganizationofthispaper.
Section2containssomebasicfactsonhighestweightcategoriesandsome
developmentsonthetheoryofhighestweightcoversin[39].
Section3isareminderonHeckealgebras,q-Schuralgebrasandcategori-
fications.
Section 4 contains basic facts on the parabolic category O of gl and the
N
subcategory A ⊂ O introducedin[5].Theresultsin[5]arenotenoughforus
since we need to consider a deformed category A with integral deformation
parameters.ThenewmaterialisgatheredinSect.4.7.
InSect.5weconsidertheaffineparaboliccategoryO(atanegativelevel).
The monoidal structure on O is defined later in Sect. 8. Using this monoidal
structureweconstructacategoricalactiononOinSect.5.4.Then,wedefine
the subcategory A ⊂ O in Sect. 5.5. The rest of the section is devoted to the
deformation argument and the proof that A is a highest weight cover of the
module category of a cyclotomic Hecke algebra satisfying some faithfulness
conditions.
InSect.6wefirstgiveareminderonthecategoryOofCRDAHA’s,follow-
ing[22,39].Then,weproveourmaintheoremsinSects.6.3.2,6.3.3usingthe
resultsfromSect.5.8.ThisyieldsaproofofVaragnolo–Vasserot’sconjecture
[46]. For the clarity of the exposition we separate the cases of rational and
irrationallevels,althoughbothproofsareverysimilar.
InSect.7wegivesomeapplicationsofourmaintheorem,includingproofs
forRouquier’sconjectureandChuang–Miyachi’sconjecture.
Section8isareminderontheKazhdan–Lusztigtensorproductontheaffine
categoryOatanegativelevel.Wegeneralizetheirconstructioninordertogeta
monoidalstructureonarbitraryparaboliccategories,deformedoverananalytic
two-dimensional regular local ring. Several technical results concerning the
Kazhdan–Lusztigtensorproductarepostponedtotheappendix.
Tofinish,letusexplaintherelationofthisworkwithotherrecentworks.
Thecaseofirrationallevel(provedinTheorem6.11)wasconjecturedin[46,
rem.8.10(b)],asadegenerateanalogueofthemainconjecture[46,conj.8.8].
There,itwasmentionedthatitshouldfollowfrom[5,thm.C].Inthedominant
case,thishasbeenprovedrecently[24,thm.6.9.1].
While we were writing this paper I. Losev made public several papers
with some overlaps with ours. In [31,32] he developed a general formalism
123
Categorificationsandcyclotomicrationaldoubleaffine…
of categorical actions on highest weight categories. Then, he used this for-
malism in [33] to prove that the category A is equipped with a categorical
action, induced by the categorical action on O introduced in [46] (using the
Kazhdan–Lusztigfusionproduct).ThecategoricalactiononAgivesaninde-
pendent proof of Theorem 5.37(a), (b). Finally, he proposed a combinatorial
approach to prove that A is a 1-faithful highest weight cover of the cyclo-
tomicHeckealgebraundersometechnicalconditionontheparametersofthe
CRDAHA.
A first version of our paper was announced in July 2012 and has been
presentedatseveraloccasionssincethen.There,weprovedthis1-faithfulness
forA(andtheVaragnolo–Vasserot’sconjecture)underthesameconditionon
theparametersbyadeformationargumentsimilar,butweaker,totheoneused
inthepresentpaper.
Theproofwhichwegiveinthisarticleavoidsthistechnicalconditiononthe
parameters. It uses an idea introduced later, in [33]. There, I. Losev replaces
the highest weight cover A of the cyclotomic Hecke algebra H by a highest
weightcover,byA,ofabiggeralgebrathanH,whichhasbetterproperties.
After this paper was written, B. Webster sent us a copy of a preliminary
versionofhisrecentpreprint[47]proposinganotherproofofRouquier’scon-
jecturewhichdoesnotusetheaffineparaboliccategoryO.
NotethatourconstructiondoesnotuseanycategoricalactiononA.Itonly
uses representation theoretic arguments. However, since Theorem 6.9 yields
an equivalence between A and O, we can recover a categorical action on A
fromourtheoremandthemainresultof[41].ThisisexplainedinSect.7.4.
2 Highestweightcategories
InthepaperthesymbolRwillalwaysdenoteanoetheriancommutativedomain
(with1).Wedenoteby K itsfractionfield.When R isalocalring,wedenote
byk itsresiduefieldandbymitsmaximalideal.
2.1 Ringsandmodules
For any R-module M, let M∗ = Hom (M,R) denote the dual module. An
R
S-point of R is a morphism χ : R → S of commutative rings with 1. If
χ is a morphism of local rings, we say that it is a local S-point. We write
SM = M(χ) = M ⊗ S.Ifφ isa R-modulehomomorphism,weabbreviate
R
also Sφ = φ⊗ S.
R
Let P, M be the spectrum and the maximal spectrum of R. Let P ⊂ P
1
be the subset of height 1 prime ideals. For each p ∈ P, let R denote the
p
localizationof R atp.Themaximalidealof R ism = R panditsresidue
p p p
fieldisk = Frac(R/p).
p
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R.Rouquieretal.
A closed k-point of R is a quotient R → R/m = k where m ∈ M. To
unburdenthenotationwemaywritek ∈ M.
Afiniteprojective R-algebraisan R-algebrawhichisfinitelygeneratedand
projectiveasan R-module.
Wewillmainlybeinterestedinthecasewhere Risalocalring.Inthiscase,
any projective module is free by Kaplansky’s theorem. Therefore, we’ll use
indifferentlythewordsfreeorprojective.
2.2 Categories
Given A a ring, we denote by Aop the opposite ring in which the order of
multiplication is reversed. Given C is a category, let Cop be the opposite
category.
An R-categoryC isanadditivecategoryenrichedoverthetensorcategory
of R-modules.Allthefunctors F onC areassumedtobe R-linear.Wedenote
theidentityelementintheendomorphismringEnd(F)againbyF.Wedenote
theidentityfunctoronC by1C.WesaythatC isHom-finiteiftheHomspaces
arefinitelygeneratedover R.IfthecategoryC isabelianorexact,let K (C)
0
be the Grothendieck group and write [C] = K0(C)⊗Z C. If C is additive,
it is an exact category with split exact sequences and [C] is the complexi-
fied split Grothendieck group. Let [M] denote the class of an object M of
C.
Assume now that C is abelian and has enough projectives. We say that
an object M ∈ C is projective over R if HomC(P,M) is a projective R-
module for all projective objects P of C. The full subcategory C ∩ R-proj
of objects of C projective over R is an exact subcategory and the canonical
functor Db(C ∩ R-proj) → Db(C)isfullyfaithful.Anobject X ∈ C which
isprojectiveover R isrelatively R-injectiveifExt1(Y,X) = 0forallobjects
C
Y ofC thatareprojectiveover R.
IfC isthecategory A-modoffinitelygenerated(left)modulesoverafinite
projective R-algebra A,thenanobject X ∈ C isprojectiveover R ifandonly
if it is projective as an R-module. It is relatively R-injective if in addition
the dual X∗ = Hom (X,R) is a projective right A-module. If there is no
R
risk of confusion we will say injective instead of relatively R-injective. We
put C∗ = Aop-mod. The functor Hom (•,R) : Cop → C∗ restricts to an
R
equivalenceofexactcategoriesCop∩ R-proj→∼C∗∩ R-proj.
We denote by Irr(C) the sets of isomorphism classes of simple objects
of C. Let Cproj,Cinj ⊂ C be the full subcategories of projective and of
relatively R-injectiveobjects.IfC = A-mod,weabbreviateIrr(A) = Irr(C),
A-proj = Cproj and A-inj = Cinj.
Given an S-point R → S and C = A-mod, we can form the S-category
SC = SA-mod.GivenanotherR-categoryC(cid:9)asaboveandanexact(R-linear)
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Categorificationsandcyclotomicrationaldoubleaffine…
functor F : C → C(cid:9),then F isrepresentedbyaprojectiveobject P ∈ C.We
set SF = HomSC(SP,•) : SC → SC(cid:9).
Let A be a Serre subcategory of C. The canonical embedding functor
h : A → C has a left adjoint h∗ which takes an object M in C to its
maximal quotient in C which belongs to A. It admits also a right adjoint h!
which takes an object M in C to its maximal subobject in C which belongs
to A. The functor h∗ is right exact, while h! is left exact. The functor h is
fully faithful. Hence the adjunction morphisms h∗h → 1A and 1A → h!h
are isomorphisms. By definition, the adjunction morphisms 1C → hh∗ and
hh! → 1C arerespectivelyanepimorphismandamonomorphism.
Here,andintherestofthepaper,weusethefollowingnotation:acomposi-
tionoffunctors E and F iswrittenas EF whileacompositionofmorphisms
offunctorsψ andφ iswrittenasψ ◦φ.
2.3 Highestweightcategoriesoverlocalrings
Let R be a commutative local ring. We recall and complete some basic facts
abouthighestweightcategoriesover R (cf[39,§4.1]and[11],[15,§2]).
LetC beanabelian R-categorywhichisequivalenttothecategory A-mod
offinitelygeneratedmodulesoverafiniteprojective R-algebra A.
The category C is a highest weight R-category if it is equipped with a
posetofisomorphismclassesofobjects((cid:7)(C),(cid:2))calledthestandardobjects
satisfyingthefollowingconditions:
• theobjectsof(cid:7)(C)areprojectiveover R
• given M ∈ C such that HomC(D,M) = 0 for all D ∈ (cid:7)(C), we have
M = 0
• given D ∈ (cid:7)(C), there is P ∈ Cproj and a surjection f : P (cid:3) D such
that ker f has a (finite) filtration whose successive quotients are objects
D(cid:9) ∈ (cid:7)with D(cid:9) > D
• given D ∈ (cid:7),wehaveEndC(D) = R
• given D1,D2 ∈ (cid:7)withHomC(D1,D2) (cid:11)= 0,wehave D1 ≤ D2.
Thepartialorder(cid:2)iscalledthehighestweightorderofC.Wewrite(cid:7)(C) =
{(cid:7)(λ)}λ∈(cid:9),for(cid:9)anindexingposet.Notethatif≤(cid:9)isanordercoarserthan≤
(i.e.,λ ≤ μimpliesλ ≤(cid:9) μ),thenC isalsoahighestweightcategoryrelative
totheorder≤(cid:9).
An equivalence of highest weight categories C(cid:9) −→∼ C is an equivalence
whichinducesabijection(cid:7)(C(cid:9)) −→∼ (cid:7)(C).Ahighestweightsubcategory is
afullSerresubcategoryC(cid:9) ⊂ C thatisahighestweightcategorywithposet
(cid:7)(C(cid:9)) an ideal of (cid:7)(C) (i.e., if D(cid:9) ∈ (cid:7)(C(cid:9)), D ∈ (cid:7)(C) and D(cid:9) < D, then
D(cid:9) ∈ (cid:7)(C(cid:9))).
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R.Rouquieretal.
Highestweightcategoriescomewithassociatedprojective,injective,tilting
andcostandardobjects,asdescribedinthenextproposition.
Proposition2.1 LetC beahighestweightR-category.Givenλ∈ (cid:9),thereare
indecomposableobjectsP(λ) ∈ Cproj,I(λ) ∈ Cinj,T(λ) ∈ C and∇(λ) ∈ C
(the projective, injective, tilting and costandard objects associated with λ),
uniqueuptoisomorphismsuchthat
(∇) HomC((cid:7)(μ),∇(λ)) (cid:14) δλμR and Ext1C((cid:7)(μ),∇(λ)) = 0 for all
μ ∈ (cid:9),
(P)thereisasurjection f : P(λ) (cid:3) (cid:7)(λ)suchthatker f hasafiltration
whosesuccessivequotientsare(cid:7)(μ)’swithμ > λ,
(I)thereisaninjection f : ∇(λ) (cid:11)→ I(λ)suchthatcoker f hasafiltration
whosesuccessivequotientsare∇(μ)’swithμ > λ,
(T)thereisaninjection f : (cid:7)(λ) (cid:11)→ T(λ)andasurjection g : T(λ) (cid:3)
∇(λ) such that coker f (resp. kerg) has a filtration whose successive
quotientsare(cid:7)(μ)’s(resp.∇(μ)’s)withμ < λ.
Wehavethefollowingpropertiesofthoseobjects.
• ∇(λ),(cid:7)(λ), P(λ), I(λ)andT(λ)areprojectiveover R.
• Given a commutative local R-algebra S, then SC is a highest weight
S-category on the poset (cid:9) with standard objects S(cid:7)(λ) and costandard
objects S∇(λ).If R → S isalocal S-point,thentheprojective,injective
andtiltingobjectsassociatedwithλare SP(λ),SI(λ)and ST(λ).
• C∗ is a highest weight R-category on the poset (cid:9) with standard objects
(cid:7)∗(λ) = ∇(λ)∗ and with P∗(λ) = I(λ)∗, I∗(λ) = P(λ)∗, ∇∗(λ) =
(cid:7)(λ)∗ andT∗(λ) = T(λ)∗.
Proof Note that the statements of the proposition are classical when R is a
field.
The existence of the objects ∇(λ) giving Cop the structure of a highest
weight category and satisfying the Hom and Ext conditions is given by [39,
Proposition4.19].TheunicityfollowsfromLemma2.7below.Thedescription
oftheprojective,tiltingandinjectiveobjectsofC∗ isclear.
Itisshownin[39,Proposition4.14]thatSC isahighestweightcategorywith
(cid:7)(SC) = S(cid:7)(C). We denote by P (λ), I (λ), etc. the projective, injective,
S S
etc.of SC associatedwithλ.
The existence of P(λ) is granted in the definition of highest weight cate-
gories. We show by descending induction on λ that kP(λ) (cid:14) P (λ). This is
k
clearifλismaximal,forthen P(λ) = (cid:7)(λ).WehavekP(λ) = P (λ)⊕ Q,
k
whereQisadirectsumofP (μ)’swithμ > λ.Byinduction,P (μ) = kP(μ),
k k
hence Q lifts to Q˜ ∈ Cproj, and there are maps f : Q˜ → P(λ) and
g : P(λ) → Q˜ such that k(gf) = id . Since R is local and Q˜ is a finitely
Q
˜
generatedprojective R-module,wededucethatgf isanautomorphismof Q,
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Categorificationsandcyclotomicrationaldoubleaffine…
hence Q˜ is a direct summand of P(λ), so Q˜ = 0 and kP(λ) = P (λ).
k
The unicity of P(λ) is then clear, since given M,N ∈ Cproj, we have
∼
kHomC(M,N)→HomkC(kM,kN).
Given R → Salocalpoint,theresiduefieldk(cid:9)ofSisafieldextensionofk.
SincekAisasplitk-algebra,itfollowsthatgiven P aprojectiveindecompos-
(cid:9) (cid:9)
able kA-module, then k P is a projective indecomposable k A-module. We
deducethat Pk(cid:9)(λ) (cid:14) k(cid:9)⊗k kP(λ),hence PS(λ) (cid:14) SP(λ).
Thestatementsabout I(λ)followfromthoseabout P(λ)byduality.
The statements about T(λ) are proven in the same way as those for P(λ),
usingProposition2.4(b)below. (cid:16)(cid:17)
Note that (C,(cid:7)(C)) is a highest weight R-category if and only if
(kC,k(cid:7)(C)) is a highest weight k-category and the objects of (cid:7)(C) are
projectiveover R,see[39,thm.4.15].Notealsothat(cid:7)(λ)hasauniquesimple
quotient L(λ),andIrr(C) = {L(λ)}λ∈(cid:9).
LetC(cid:7) andC∇ bethefullsubcategoriesofC consistingofthe(cid:7)-filtered
and∇-filteredobjects,i.e.,objectshavingafinitefiltrationwhosesuccessive
quotientsarestandard,costandardrespectively.Theseareexactsubcategories
of C. Note that every object of C(cid:7) has a finite projective resolution, where
the kernels of the differentials are in C(cid:7). As a consequence, the canonical
functor Db(C(cid:7)) → Db(C)isfullyfaithful.Similarly,the canonicalfunctor
Db(C∇) → Db(C)isfullyfaithful,aseveryobjectofC∇hasafiniterelatively
R-injectiveresolution.
Lemma2.2 LetC,C(cid:9) behighestweight R-categories.Anexactfunctor(cid:3) :
C → C(cid:9) whichrestrictsto an equivalence(cid:3) : C(cid:7)→∼C(cid:9)(cid:7) is an equivalence
ofhighestweightcategoriesC→∼C(cid:9).
Proof Since (cid:3) identifies the projective objects in C and C(cid:9), it induces an
equivalence of their bounded homotopy categories, hence an equivalence
Db(C) → Db(C(cid:9)).Since(cid:3)isexact,wearedone. (cid:16)(cid:17)
LetCtilt = C(cid:7)∩C∇ bethefullsubcategoryofC consistingofthetilting
objects,i.e.,th(cid:2)eobjectswhichareboth(cid:7)-filteredand∇-filtered.
Let T = T(λ). The Ringel dual of C is the category C(cid:18) =
λ∈(cid:9)
EndC(T)op-mod.Itisahighestweightcategoryontheposet(cid:9)op.Thefunctor
Hom(T,•) : C → C(cid:18) restricts to an equivalence R : C∇→∼(C(cid:18))(cid:7), called
theRingelequivalence.WehaveR(∇(λ)) = (cid:7)(cid:18)(λ),R(T(λ)) (cid:14) P(cid:18)(λ)and
R(I(λ)) (cid:14) T(cid:18)(λ)forλ ∈ (cid:9),see[39,Proposition4.26].Thehighestweight
categoryC isdetermined,uptoequivalence,byC(cid:18) andweput(C(cid:18))(cid:2) = C.
There is an equivalence of highest weight categories C→∼C(cid:18)(cid:18) such that the
composition
123
R.Rouquieretal.
Cproj→∼(C(cid:18)(cid:18))proj −R−−→1 (C(cid:18))tilt −R−−→1 Cinj
∼ ∼
is isomorphic to the Nakayama duality Hom (•,A)∗. This provides also an
A
equivalenceofhighestweightcategoriesC(cid:2)→∼C(cid:18).
Now,for M ∈ C weset
lcdC(M) = min{i; ∃μ ∈ (cid:9),Exti(M,T(μ)) (cid:11)= 0},
(2.1)
rcdC(M) = min{i; ∃μ ∈ (cid:9),Exti(T(μ),M) (cid:11)= 0}.
Lemma2.3 Assume R isafield.Letλ ∈ (cid:9).Then
min{i; ∃μ ∈ (cid:9),Exti(L(λ),T(μ)) (cid:11)= 0}
= min{i; ∃μ ∈ (cid:9),Exti(L(λ),(cid:7)(μ)) (cid:11)= 0}
= min{i; ∃M ∈ C(cid:7),Exti(L(λ),M) (cid:11)= 0}.
Proof Letc ,c andc bethequantitiesdefinedbythetermsinvolvingrespec-
1 2 3
tively T(μ)’s,(cid:7)(μ)’sand M ∈ C(cid:7) inthefirsttwoequalities.Itisclearthat
c ≥ c = c .
1 2 3
Take μ minimal such that Extc2(L(λ),(cid:7)(μ)) (cid:11)= 0. There is an exact
sequence 0 → (cid:7)(μ) → T(μ) → M → 0 where M has a filtration with
subquotients (cid:7)(ν)’s where ν < μ. We deduce that Extc2(L(λ),T(μ)) (cid:11)= 0,
hencec ≤ c . (cid:16)(cid:17)
1 2
Letusrecallafewfactsonbasechangeforhighestweightcategories.
Proposition2.4 LetC beahighestweight R-category,andlet R → S bea
local S-point.Forany M,N ∈ C thefollowingholds:
(a) if S is R-flatthen SExtd(M,N) = Extd (SM,SN)foralld ∈ N,
C SC
(b) if either M ∈ Cproj or (M ∈ C(cid:7) and N ∈ C∇), then we have SHomC
(M,N) = HomSC(SM,SN),
(c) if M is R-projective then M ∈ Cproj (resp. M ∈ Ctilt, C(cid:7), Cinj) if and
onlyifkM ∈ kCproj (resp.kM ∈ kCtilt,kC(cid:7),kCinj),
(d) ifeither(M ∈ Cproj and N is R-projective)or (M ∈ C(cid:7) and N ∈ C∇)
thenHomC(M,N)is R-projective.
Proof Part(a)is[Bourbaki,Algèbre,ch.X,§6,prop.7.c].
Thestatementsin(b),(d)areclearifMisafreeA-module,andarepreserved
undertakingdirectsummands,sotheyholdfor M ∈ Cproj.
LetM ∈ C(cid:7)andN ∈ C∇.WehaveExt1(M,N) = Ext1 (SM,SN) = 0.
C SC
Asaconsequence,ifMisanextensionofM ,M ∈ C(cid:7)andthestatements(b),
1 2
(d)holdfor M ,N,thentheyholdfor M,N.Weproceednowbydescending
i
induction on λ to prove that the statement for M = (cid:7)(λ). There is an exact
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