Homological Properties of Finite Partially Ordered Sets [PhD thesis]

138 Pages·2008·0.662 MB·English

by  Sefi Ladkani

Homological Properties of Finite Partially
Ordered Sets
Thesis submitted for the degree of
“Doctor of Philosophy”
by
Sefi Ladkani
Submitted to the Senate of the Hebrew University
January 2008

This work was carried out under the supervision of
Professor David Kazhdan

Abstract
In this work we study the homological properties of finite partially ordered sets as reflected in
theirderivedcategoriesofdiagrams. Thissubjectstandsatthejunctionoftheareasofcombina-
torics,topology,representationtheoryandhomologicalalgebra.
Background
Since their introduction by Verdier and Grothendieck in order to formulate duality in algebraic
geometry, triangulated categories in general, and derived categories in particular, have found
applicationsindiverseareasofmathematicsandmathematicalphysics.
Triangulated categories have been successfully used to relate objects of different nature,
thusformingbridgesbetweenvariousareasofalgebraandgeometry. AnexampleisBeilinson’s
result[6]ontheequivalenceofthederivedcategoryofcoherentsheavesoveraprojectivespace
(whichisofcommutativenature)andthederivedcategoryoffinitedimensionalmodulesovera
certainfinitedimensional,non-commutative,algebra. Thisresultcanbeseenasastartingpoint
ofnon-commutativegeometry.
Anotherexample,motivatedbyitsapplicationstophysics,isKontsevich’sformulation[56]
of the Homological mirror symmetry conjecture as an equivalence between a certain derived
categoryofcoherentsheavesoveranalgebraicvarietyandatriangulatedcategoryofothernature
(theFukayacategory).
The question of equivalence of two derived categories arising from objects of the same na-
ture has also attracted a growing interest. For example, the question when two algebraic va-
rieties have equivalent derived categories of sheaves has been recently studied by Bondal and
Orlov [11]. Another, earlier, example is Rickard’s result [73], characterizing when two rings
have equivalent derived categories of modules, in terms of the existence of a so-called tilting
complex.
Rickard’s result leaves something to be desired, though, as for some pairs of algebras, it is
currentlynotoriouslydifficult,andsometimesevenimpossible,todecidewhetherthereexistsa
tilting complex. Such a difficulty is apparent in the still unsolved Broue´’s conjecture asserting
derivedequivalencesbetweencertainblocksofgroupalgebras[78].
i
(cid:127)(cid:127)~~~~~~•@@@@@@(cid:31)(cid:31) •@@@@@@(cid:31)(cid:31) (cid:127)(cid:127)~~~~~~•
•@@@@@@(cid:31)(cid:31) (cid:127)(cid:127)~~~~~~• •(cid:15)(cid:15)
• •
Figure1: Hassediagramsoftwonon-isomorphic,yetuniversallyderivedequivalentposets.
The problem
Weinvestigatesimilarquestionsforderivedcategoriesarisingfromfinitepartiallyorderedsets.
Afinitepartiallyorderedset(poset)isnaturallyendowedwithastructureofatopologicalspace.
The finite spaces obtained in this way are capable of modeling several quantitative topological
properties, such as the homology and homotopy groups, of various well-behaved manifolds in
Euclideanspaces[66].
A poset can also be considered as a small category, allowing one to form the category of
functors X → A, denoted AX, from a poset X to an abelian category A. The category AX,
whose objects are also known as diagrams, is abelian and can be regarded as a category of
sheavesandsometimesofmodules,asoutlinedbelow.
First,byviewingaposetasatopologicalspace,diagramscanbeconsideredassheavesover
thatspace. Tworecentapplicationsofsheavesoverposetsincludethecomputation,byDeligne,
GoreskyandMacPherson[25],ofthecohomologyofarrangementsofsubspacesinarealaffine
space, and the definition, by Karu [49], of intersection cohomology for general polytopes, in
ordertostudytheirh-vectors.
Second,whenAisthecategoryoffinitedimensionalvectorspacesoverafieldk,diagrams
canalsobeidentifiedwithmodulesovertheincidencealgebraofX overk.
Definition. WesaythattwoposetsXandY areuniversallyderivedequivalentif,foranyabelian
categoryA,the(bounded)derivedcategoriesofdiagramsDb(AX)andDb(AY)areequivalent
(astriangulatedcategories).
TwoposetsX andY arederivedequivalent overafieldk iftheirincidencealgebrasoverk
arederivedequivalent.
Our main research aim is the study of these derived categories of diagrams, and in partic-
ular the questions when two posets are (universally) derived equivalent and how the derived
equivalenceisrelatedtothecombinatorialpropertiesoftheposetsinquestion.
As diagrams can be identified with sheaves and sometimes even with modules, both geo-
metrical tools from algebraic geometry and algebraic tools from representation theory can be
appliedwhenstudyingthequestionofderivedequivalence.
Thederivedequivalencerelation,eitheruniversalornot,isstrictlycoarserthanisomorphism.
AsimpleexampledemonstratingthisisgiveninFigure1. However,thereisnoknownalgorithm
whichdecides,giventwoposets,whethertheyarederivedequivalentornot.
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• • • • • •
Figure2: Hassediagramsoftwoposetsderivedequivalentbythebipartiteconstruction.
Therearetwodirectionstopursuehere. Thefirstistofindinvariantsofderivedequivalence,
that is, combinatorial properties of a poset which are shared among all other posets derived
equivalent to it. Examples of such invariants are the number of points, the Z-congruency class
oftheincidencematrix,andtheBettinumbers.
Theseconddirectionistosystematicallyconstruct,givenaposethavingcertaincombinato-
rial structure, new posets that are guaranteed to be derived equivalent to it. An example of an
analogousconstructionintherepresentationtheoryofquiversandfinitedimensionalalgebrasis
theBernstein-Gelfand-Ponomarevreflection[9].
The results
Constructionsofderivedequivalencesofposetsandotherobjects
Wehavefoundseveralkindsofcombinatorialconstructionsproducingderivedequivalencesand
they are described in Part I of this work. The common theme of these constructions is the
structuredreversaloforderrelations.
Thebipartiteconstruction
ThisconstructionisdescribedinChapter1,whereweshowthataposethavingabipartitestruc-
ture can be mirrored along that structure to obtain a derived equivalent poset. An example is
giveninFigure2.
Topresentthisconstructionmoreprecisely,weintroducethenotionofalexicographicsum
ofacollectionofposetsalongaposet,whichgeneralizestheknownnotionofanordinalsum. If
{X } isacollectionofposetsindexedbyaposetS,thelexicographicsumoftheX alongS
s s∈S s
istheposetwhoseunderlyingsetisthedisjointunionoftheX ,andtwoelementsarecompared
s
firstbasedontheindicesofthesetstheybelongto(usingthepartialorderonS),andwhenatie
occurs–i.e.theybelongtothesamesetX ,accordingtothepartialorderinsideX .
s s
AposetS iscalledbipartiteifitsHassediagramisabipartitedirectedgraph.
Theorem. LetS beabipartiteposet. Thenthelexicographicsumofacollectionofposetsalong
S isderivedequivalenttothelexicographicsumofthesamecollectionalongtheoppositeposet
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Sop.
Thisconstructionisinspiredbythegeometricalviewpointofdiagramsassheaves,building
on the notion of a strongly exceptional collection introduced in the study of derived categories
ofsheavesoveralgebraicvarieties.
Asacorollary,weseethatthederivedequivalenceclassofanordinalsumofanytwoposets
doesnotdependontheorderofsummands. However,wegaveanexampleshowingthatthisis
nottrueforthreesummands.
Byusingtheother,algebraic,viewpointofdiagramsasmodules,wehaveextendedtheabove
result to general triangular matrix rings, as described in Chapter 2. Instead of formulating the
resultinitsmostgenerality(seeTheorem2.4.5),weshalldemonstrateitinthecaseofalgebras
overafield,asexpressedinthefollowingtheorem.
Theorem. Let k be a field and let R, S be k-algebras. Assume that (at least) one of R, S is
finitedimensionalandoffiniteglobaldimension. ThenforanyfinitedimensionalS-R-bimodule
M ,thetriangularmatrixalgebras
R S
(cid:18) (cid:19) (cid:18) (cid:19)
R M S DM
and
0 S 0 R
arederivedequivalent,wherethedualDM = Hom (M,k)isviewedasanR-S-bimodule.
k
Generalizedreflections
This construction, described in Chapter 3, produces universal derived equivalences that can be
considered as generalized reflections, described in very simple, explicit combinatorial terms.
This vastly generalizes the well-known Bernstein-Gelfand-Ponomarev reflections for quivers,
and has found applications in the representation theory of algebras for the study of the partial
orders of tilting modules and cluster-tilting objects over path algebras of quivers, as described
laterinPartIIofthiswork.
Weshalldemonstratethisconstructioninthefollowingsituation,calledflip-flop. Foramore
general setup, see Section 3.1.1. If X and Y are two posets and f : X → Y is an order-
preserving map, one can consider two partial orders ≤f and ≤f on the disjoint union X tY,
+ −
definedasfollows. InsideeachofthesetsX andY,bothordersagreewiththeoriginalorders,
butbetweenthem,x ≤f y iff(x) ≤ y andy ≤f xify ≤ f(x),wherex ∈ X andy ∈ Y.
+ −
Theorem. Theposets(X tY,≤f )and(X tY,≤f )areuniversallyderivedequivalent.
+ −
NotethatthistheoremistruealsowhentheposetsX andY areinfinite. Themaintoolused
intheconstructionisthenotionofaformula,whichconsistsofcombinatorialdatathatproduces,
simultaneously for any abelian category A, a functor between the categories of complexes of
diagrams over two posets Z and Z0 with values in A, inducing a triangulated functor between
the corresponding derived categories. When Z and Z0 have certain combinatorial structure, as
forexampleintheaboveconstructions,webuildsuchfunctorsthatareequivalences.
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•
•
Figure3: Hassediagramsoftwouniversallyderivedequivalentposetsoftiltingmodulesofpath
algebrasofquiverswhoseunderlyinggraphistheDynkindiagramA .
4
Combinatorialapplicationsfortiltingobjects
In Part II of this work we demonstrate that the flip-flop constructions appear naturally in com-
binatorialcontextsconcerningpartialordersoftiltingmodulesandclustertiltingobjectsarising
frompathalgebrasofquivers.
First we show that the posets of tilting modules, in the sense of Riedtmann-Schofield [75],
of any two derived equivalent path algebras of quivers without oriented cycles, are always uni-
versallyderivedequivalent. ThisisdescribedinChapter4. AnexampleisgiveninFigure3.
Thenweshowasimilarresultfortheposetsofclustertiltingobjects,describedinChapter5.
Cluster categories corresponding to quivers without oriented cycles were introduced in [16] as
arepresentationtheoreticapproachtotheclusteralgebrasintroducedandstudiedbyFominand
Zelevinsky [28]. In these categories one can define, similarly to tilting modules, cluster tilting
objects, which correspond to the clusters of the cluster algebra. The set T of cluster tilting
CQ
objects in the cluster category of a quiver Q without oriented cycles admits a partial order as
describedin[48].
Theorem. LetQandQ0betwoquiverswithoutorientedcycleswhosepathalgebrasarederived
equivalent. ThentheposetsT andT areuniversallyderivedequivalent.
CQ CQ0
When the quiver Q is of finite type, the poset T is a Cambrian lattice as introduced by
CQ
Reading [72], defined as a certain quotient of the weak order on the corresponding Coxeter
group. ItsHassediagramisthe1-skeletonofapolytopeknownasthecorrespondinggeneralized
Associhedron.
Piecewisehereditarycategoriesandposets
InPartIIIoftheworkwestudyposetswhosecategoriesofdiagrams(overafield)arepiecewise
hereditary, that is, derived equivalent to an abelian category of global dimension one. One
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should think of such categories as the simplest ones after the semi-simple ones. We present
threeresultsonsuchcategoriesandposets.
Thefirstresultconcernstheglobaldimensionofapiecewisehereditaryabeliancategory. In
general,thisquantitycanbearbitrarilylarge. However,weshowthatforapiecewisehereditary
categoryofdiagramsoveraposet,theglobaldimensioncannotexceed3. Moreover,weextend
this result to categories of modules over sincere algebras and more generally to a wide class of
finitelengthpiecewisehereditarycategoriessatisfyingcertainconnectivityconditionsexpressed
via their graphs of indecomposable objects. Note that the bound of 3 is sharp. This result is
describedinChapter6.
Second,weexploretherelationshipsbetweenspectralpropertiesoftheCoxetertransforma-
tionandpositivitypropertiestheEulerform,forfinitedimensionalalgebraswhicharepiecewise
hereditary. Weshowthatforsuchalgebras,iftheCoxetertransformationisoffiniteorder,then
the Euler bilinear form is non-negative. We also demonstrate, through incidence algebras of
posets, that the assumption of being piecewise hereditary cannot be omitted. This is done in
Chapter7.
Finally, we give a complete description of all the canonical algebras (which form a special
classofpiecewisehereditaryalgebras,introducedbyRingel[76])thatarederivedequivalentto
incidence algebras of posets. This is expressed in the following theorem, whose proof can be
foundinChapter8.
Theorem. Acanonicalalgebraoftype(p,λλλ)overanalgebraicallyclosedfieldisderivedequiv-
alent to an incidence algebra of a poset if and only if the number of weights of p is either 2 or
3.
Somepartsofthisworkhaveappearedinjournalpapers;Chapter1isbasedonthepaper[59]
andChapters6,7and8arebasedonthepapers[57],[60]and[58],respectively.
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