The homology of digraphs as a generalisation of Hochschild homology

0.31 MB·English

by  Paul Turner

ThehomologyofdigraphsasageneralisationofHochschild
homology
PaulTurner(cid:63) andEmmanuelWagner
Abstract. J.PrzytyckihasestablishedaconnectionbetweentheHochschildhomologyofanalgebra
A and the chromatic graph homology of a polygon graph with coefficients in A. In general the
chromaticgraphhomologyisnotdefinedinthecasewherethecoefficientringisanon-commutative
algebra.Inthispaperwedefineanewhomologytheoryfordirectedgraphswhichtakescoefficients
0
in an arbitrary A−A bimodule, for A possibly non-commutative, which on polygons agrees with
1
Hochschildhomologythrougharangeofdimensions.
0
2

n
a
J Introduction

9
2 ThispaperaddressesaquestionofJ.Przytycki.WhendefiningtheHochschildho-
mology of an algebra with coefficients in a bimodule, the differential displays a
]
T certain cyclical feature which makes it sometimes convenient to write the tensor
G factorsofn-chaingenerators,notlinearly,butinsteadastheverticesofan(n+1)-
. sidedpolygon.Przytyckihasabeautifulinterpretationofthisappearanceofn-gons
h
intermsofthechromatichomologyofgraphsdevelopedbyofHelme-Guizonand
t
a
Rong ([3]): a variant of this theory constructed using an algebra A and an A−A-
m
bimodule M applied to the n-gon is the Hochschild homology of A with coeffi-
[
cientsinM througharangeofdimensionsincreasingwithn.Ingeneralchromatic

1 homology of a graph is only defined when the algebra is commutative, but in the
v
case of the n-gon or a line graph, the non-commutative case also makes sense. In
9
7 speculatingaboutpossiblegeneralisationsofHochschildhomologyin[4]Przytycki
3 writesthathebelieves“graphhomologyisthepropergeneralizationofHochschild
5
homology:fromapolygontoanygraph”.Theremainingproblembeing,however,
.
1 thatonedoesnotknowingeneralhowtodefinechromaticgraphhomologyinvolv-
0
ingnon-commutativealgebras.Toparaphrase,Przytycki’squestionis:
0
1
Given a (possibly non-commutative) algebra A and an A−A bimodule
:
v M, can one construct a functor from some category of graphs to graded
i
X modulessuchthatonn-gonsthisfunctoragreeswithHochschildhomology
througharangeofdimensions?
r
a
In this paper we consider finite based directed graphs, that is digraphs whose
vertex set is finite and for which there is a distinguished vertex, the base vertex.
Given a triple (Γ,A,M) consisting of a based digraph Γ, an R-algebra A and an
A−A bimodule M our main purpose is to define homology groups H (Γ,A,M)
∗
PAUL TURNER: De´partement de mathe´matiques, Universite´ de Fribourg, CH-1700 Fri-
bourg, Switzerland. e-mail: [email protected]. EMMANUEL WAGNER: Institut de
Mathe´matiques de Bourgogne, Universite´ de Bourgogne, UMR 5584 du CNRS, BP47870, 21078
DijonCedex,France.e-mail:[email protected].
(cid:63) ThefirstauthorwaspartiallysupportedbySNFprojectno.200020-121506/1
2 PaulTurnerandEmmanuelWagner
withnicefunctorialproperties.Therearetwokeyideasnecessarytoembracenon-
commutative algebras: firstly one must give up on having “cube” (as seen in the
constructionofchromaticgraphhomology)andsecondlyoneneedsdirectededges
inordertobeabletomultiplynon-commutingelements(theheadandtailproviding
information on which element comes first). The base vertex is necessitated by the
appearance of the bimodule M. In order to obtain a functor it is essential that we
usetheapproachinitiatedin[1]relatingKhovanov-typehomologytothehomology
ofposetswithcoefficientsinapresheaf.
The construction goes roughly as follows. We replace a digraph Γ by its poset
of directed multipaths P(Γ) and then define the homology groups H (Γ,F) as
∗
the homology of the poset P(Γ) with coefficients in an arbitrary coefficient sys-
tem(pre-sheaf)F.Armedwiththesegeneralitiesweconstructfromapair(A,M)
where A is an algebra and M an A−A bimodule, a particular coefficient system
F . This coefficient system generalises the construction of the chromatic ho-
A,M
mologyofgraphs.Thehomologygroupsweareafterarethendefinedby
H (Γ,A,M) = H (Γ,F ).
∗ ∗ A,M
Thesehomologygroupssatisfysomenicepropertiesandgiveonepossiblean-
swer to Przytycki’s question. Let DirGr denote the category of finite based di-
b
graphs with base vertex preserving inclusions of digraphs as morphisms and let
GrModdenotethecategoryofZ-gradedR-modules.Wethenhave:
Theorem1LetAbeanR-algebraandM anA−Abimodule.Then
H (−,A,M): DirGr → GrMod
∗ b
is a functor with the property that if Γ is a consistently directed n-gon then for
0 ≤ i ≤ n−2
H (Γ,A,M) ∼= HH (A;M)
i i
where on the right hand side we have the Hochschild homology of the algebra A
withcoefficientsinthebimoduleM.
An interestingspecial case ariseswhenM = A. Inthis case unbaseddigraphs
sufficeandwewriteH (Γ,A) = H (Γ,A,A).Thisisfunctorialinbothvariables:
∗ ∗
Theorem2 H (−,−): DirGr×Alg → GrModisabifunctor.
∗
By the previous theorem this bifunctor has the property if Γ is a consistently
directedn-gonthenfor0 ≤ i ≤ n−2,
H (Γ,A) ∼= HH (A).
i i
1. Thehomologyofdirectedgraphs
InthissectionwewillbedealingwiththecategoryDirGrwhoseobjectsarefinite
directedgraphs(unbased)andwhosemorphismsareinclusionsofdirectedgraphs.
Ourinterestwillbetoconstructhomologyofdirectedgraphsforanarbitrarycoef-
ficientsystem.NotethatadirectedgraphΓ comeswithtailandhead functions
t,h: Edge(Γ) → Vert(Γ)
Homologyofdigraphs 3
Fig.1.Adigraphanditspathposet
takingadirectededgeetoitstailanditsheadrespectively.
We will assume familiarity with the basics of posets. As usual we will denote
a partial ordering by ≤ with x < y meaning x ≤ y and x (cid:54)= y. We recall that an
element y is said to cover another element x if x < y and there is no z such that
x < z < y.Insuchacircumstancewewritex ≺ y.TheHassediagramofaposet
P is the directed graph with one vertex for each element of P and an oriented arc
fromxtoyifandonlyifx ≺ y.TheBooleanlatticeonasetistheposetofsubsets
partially ordered by inclusion and its Hasse diagram is a hypercube. A poset may
beregardedasacategorywithoneobjectforeachelementandauniquemorphism
fromxtoy wheneverx ≤ y.Suchmorphismscomposeintheobviousway.
Let Γ be a finite digraph and let B(Γ) be the Boolean lattice on its edge set
Edge(Γ).
Definition1.AsimplepathinΓ isasequenceofedgese ,...,e suchthath(e ) =
1 n i
t(e )andnovertexisencounteredtwice.AmultipathinΓ isacollectionofdis-
i+1
jointsimplepaths.
LetP(Γ)bethesubposetofB(Γ)consistingofmultipathsinΓ.Wewillrefer
to it as the path poset of Γ. By convention there is one empty path ∅ and this is
a (global) minimal element for P(Γ) which we will denote by 0. There may be
several (local) maxima. As mentioned in the previous paragraph we may choose
toviewP(Γ)asacategorywithauniquemorphismbetweenanytworelatedele-
ments.
Example1.In Figure 1 we see a digraph Γ along with an illustration of its path
poset.
Theassignmentofadigraphtoitspathposetgivesacovariantfunctor
P(−): DirGr → Posets.
Given an inclusion f: Γ(cid:48) → Γ we write f˜: P(Γ(cid:48)) → P(Γ) for P(f). Note that
this is an injective map of posets with the property that if x ≺ y in P(Γ(cid:48)) then
f˜(x) ≺ f˜(y)inP(Γ).
Wenoteforlaterusethefollowinglemmaconcerningsuchpathposets.
4 PaulTurnerandEmmanuelWagner
Lemma1.
(i)For x ∈ P(Γ), the interval [0,x] = {y ∈ P(Γ) | y ≤ x} is a Boolean lattice
ofrank|x|.
(ii)Ifx ≺ y ≺ z inP(Γ)thenthereexistsauniquey(cid:48) (cid:54)= y suchthatx ≺ y(cid:48) ≺ z.
Proof. For (i) we simply note that given a multipath then each subset of its edges
isagainamultipath.For(ii),thestatedpropertyistrueforBooleanlatticesandso
theresultfollowsbyconsideringtheinterval[0,z]. (cid:116)(cid:117)
In general the homology of posets only becomes interesting (especially in the
presence of a global minimum) if one allows local systems of coefficients. Let R
beacommutativeringandletModbethecategoryofR-modules.
Definition2.A coefficient system for a digraph Γ consists of a covariant functor
P(Γ) → Mod. These form the objects of a category Coeff(Γ) whose morphisms
arenaturaltransformationsoffunctors.
Onecantakethehomologyofanysmallcategorywithcoefficientsinafunctor
andwenowrecallthisconstructionrestrictedtoourparticularsettingi.e.wherethe
category is a path category of a digraph Γ and the functor F: P(Γ) → Mod is a
coefficient system. We will define a chain complex S (Γ,F) whose homology, by
∗
definition,isthehomologyofΓ withcoefficientsinF.
Weset
(cid:77)
S (Γ;F) = F(x )
k 0
x0x1...xk
wherethesumisoverallsequencesx ≤ x ≤ ··· ≤ x inP(Γ)oflengthk+1.
0 1 k
A typical element is therefore a sum of elements of the form λx x ...x where
0 1 k
λ ∈ F(x ).Toturnthisintoacomplexwedefined: S (Γ;F) → S (Γ;F)by
0 k k−1
k
(cid:88)
d(λx x ...x ) = F(x ≤ x )(λ)x ...x + (−1)iλx ...xˆ ...x .
0 1 k 0 1 1 k 0 i k
i=1
It is a standard fact (easily checked) that d2 = 0 and so (S (Γ;F),d) is a chain
∗
complex. We are now free to take homology and we define the homology of the
directedgraphΓ withcoefficientsin F tobethegradedR-module
H (Γ;F) = H(S (Γ;F),d).
∗ ∗
Homologyhasnicefunctorialpropertiesasweseeinthenextproposition.
Proposition1.
(i)LetΓ beafinitedirectedgraph.Then
H (Γ;−): Coeff(Γ) → Mod
∗
isacovariantfunctor.
(ii)Let f: Γ(cid:48) → Γ be an inclusion and let F: P(Γ) → Mod be a coefficient
systemforΓ.Thenthereisaninducedhomomorphism
f : H (Γ(cid:48);F◦f˜) → H (Γ;F).
∗ ∗ ∗
Such induced homomorphisms are well behaved under composition of inclu-
sions.
Homologyofdigraphs 5
Proof. (i) Let F and G be coefficient systems and let τ be a morphism from F to
G. That is, F and G are functors P(Γ) → Mod and τ is a natural transformation
F −→• G consisting of a map τ : F(x) → G(x) for each x ∈ P(Γ) satisfying the
x
usualnaturalityrequirements.Wenowdefineahomomorphism
τ(cid:48): S (Γ;F) → S (Γ;G)
k k
bysetting
τ(cid:48)(λx ...x ) = τ (λ)x ...x .
0 k x0 0 k
Thenaturalityofτ guaranteesthatthisisachainmapandthusinduces
τ : H (Γ;F) → H (Γ;G)
∗ ∗ ∗
asrequired.
Furthermore,givenanothernaturaltransformationσ: G −→• Konehas(στ)(cid:48) =
σ(cid:48)◦τ(cid:48) fromwhichitfollowsthat(στ) = σ ◦τ .
∗ ∗ ∗
(ii)Recallingthatf˜istheinducedmaponpathposets,thereisahomomorphism
f(cid:48): S (Γ(cid:48);F◦f˜) → S (Γ;F)
k k
definedby
f(cid:48)(λx ...x ) = λf˜(x )...f˜(x ).
0 k 0 k
This is a chain map since, by definition, F ◦ f˜(x ≤ y) = F(f˜(x) ≤ f˜(y)). In
homologythisdefinesf .
∗
Giveng: Γ(cid:48)(cid:48) → Γ(cid:48) wehavef(cid:102)g = f˜◦g˜fromwhichitfollowsimmediatelythat
(fg) = f ◦g . (cid:116)(cid:117)
∗ ∗ ∗
Similar calculations to those in the above proof show that the maps f are
∗
natural with respect to morphisms of coefficient systems. Spelt out more clearly
this means the following. Let f: Γ(cid:48) → Γ be an inclusion of finite digraphs and
let F ,F : P(Γ) → Mod be coefficient systems. Given a natural transformation
1 2
τ: F −→• F ,defineanaturaltransformationτ˜fromF ◦f˜toF ◦f˜byτ˜ = τ .
1 2 1 2 x f˜(x)
Undersuchcircumstancesthefollowingdiagramcommutes.
H (Γ(cid:48);F ◦f˜) f1∗ (cid:47)(cid:47)H (Γ;F )
∗ 1 ∗ 1
τ˜∗ τ∗
(cid:15)(cid:15) (cid:15)(cid:15)
H (Γ(cid:48);F ◦f˜) f2∗ (cid:47)(cid:47)H (Γ;F )
∗ 2 ∗ 2
2. ThehomologygroupsH (Γ,A,M)
∗
The category DirGr has as objects finite digraphs that are equipped with a pre-
b
ferred base vertex. Morphisms are inclusions that take base vertex to base vertex.
There is a forgetful functor DirGr → DirGr and we can take homology by first
b
applyingthisfunctorandthenproceedingasintheprevioussection.
Our task in this section is to construct a coefficient system F : P(Γ) →
A,M
Mod, given a based digraph Γ, a (possibly non-commutative) unital R-algebra A
andanA−AbimoduleM.Onceachievedthemaindefinitionofthepaperwillbe
H (Γ,A,M) = H (Γ,F ).
∗ ∗ A,M
6 PaulTurnerandEmmanuelWagner
Wewilltakethetensorproductofmodulesoverunorderedsetssowerecallhere
whatthismeans.LetS beafinitesetandsupposewehaveafamilyofR-modules
indexed by S, that is for each α ∈ S we have an R-module M . The unordered
α
tensorproductofthisfamily,denoted
(cid:79)
M
α
α∈S
is formed by considering all possible orderings of the set S, taking the direct sum
of the ordered tensor product for each and then identifying these via the obvious
canonicalisomorphismsinducedfrompermutations.
WenowproceedwiththeconstructionofF : P(Γ) → Mod.Forx ∈ P(Γ)
A,M
let Γ be the directed graph with the same vertex set as Γ and with edge set con-
x
sisting of the edges in the multipath x (along with their directions). We will write
π (Γ ) for the set of connected components of Γ . We note that if x ≺ y then
0 x x
Γ contains all the edges of Γ with one addition. Since all paths are simple this
y x
additional edge clearly joins two separate components of Γ . There is evidently a
x
canonicalidentificationofthecomponentsofΓ andΓ notinvolvedinthisfusion.
x y
For x ∈ P(Γ) consider the following family of R-modules {M } indexed
α
by the set π (Γ ). If the component indexed by α contains the base vertex then
0 x
M = M otherwiseM = A.NowwedefineF (x)tobetheunorderedtensor
α α A,M
product
(cid:79)
F (x) = M .
A,M α
α∈π0(Γx)
To define the homomorphisms F (x ≤ y) we first consider what happens
A,M
in the case x ≺ y. Here Γ consists of Γ with an additional edge e and as noted
y x
above two distinct components in Γ become one in Γ . The idea is to define a
x y
homomorphismsusingthecanonicalidentificationbetweencomponentsawayfrom
thosethatfuse,andmultiplicationortheactionsofAonM forthosethatfuse.The
keypointisthattheorderofmultiplicationisdeterminedbytheheadandtailofe.
For temporary purposes let I be an ordering of π (Γ ) and J be an ordering
0 x
ofπ (Γ ).WithrespecttotheseorderingssupposethetwocomponentsofΓ that
0 y x
fuse are indexed by i and i(cid:48) and the new fused component in Γ is indexed by j.
y
Wenowdefineahomomorphism(hereweusetheorderedtensorproduct)
µ: M ⊗M −→ M
i i(cid:48) j
by
(cid:40)
ab ifiindexesthecomponentcontainingt(e)
µ(a⊗b) =
ba ifiindexesthecomponentcontainingh(e)
Here the expression ab has several possible meanings: if the base vertex is not
involvedinthefusionofcomponentsthenitmeansthemultiplicationinthealgebra
A; if M = M (i.e. i indexes the component containing the base vertex) then
i
M = AandM = M andabmeanstherightactionofAonM;ifM = M (i.e.j
j k j
indexesthecomponentcontainingthebasevertex)thenM = AandM = M and
i k
abmeanstheleftactionofAonM.Onesimilarlyinterpretsba.Bycombiningthis
map with the canonical permutation identification on the remaining tensor factors
(cid:78) (cid:78)
thisgivesahomomorphismoforderedtensorproducts M → M .
I i J j
Homologyofdigraphs 7
3
e’
2
e
1
Fig.2.Pathordering
Lemma2.Theabovedefinesahomomorphism
F (x ≺ y): F (x) → F (y).
A,M A,M A,M
Proof. Let σ be a permutation taking an ordering I to another I(cid:48). In the above
constructionthemapsµdependonthetensorfactorscorrespondingtot(e)andh(e)
not on the factors position in any ordering. It follows that there is a commutative
diagram
(cid:78) µ (cid:47)(cid:47)(cid:78)
M M
σI(cid:15)(cid:15) i(cid:115)(cid:115)(cid:115)(cid:115)µ(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:115)(cid:57)(cid:57) J j
(cid:78)
M
I(cid:48) i(cid:48)
andsothemapsµarecompatiblewiththesymmetricgroupaction. (cid:116)(cid:117)
Lemma3.Ifx ≺ y ≺ z andx ≺ y(cid:48) ≺ z then
F (y ≺ z)◦F (x ≺ y) = F (y(cid:48) ≺ z)◦F (x ≺ y(cid:48)).
A,M A,M A,M A,M
Proof. By Lemma 1 (ii) we know that y and y(cid:48) are the only two elements lying
betweenxandz inthisway.SupposeΓ = Γ ∪{e}andΓ = Γ ∪{e(cid:48)}.Ifeand
y x y(cid:48) x
e(cid:48)arenotbothcontainedinasinglesimplepathofΓ thentheresultisclear.Ifthey
z
arebothcontainedinthesamesimplepathofΓ thenwithoutlossofgeneralitywe
z
may suppose that e comes before e(cid:48). Now choose an ordering on the components
ofΓ sothatthesimplepathbeforeeislabelled1,thesimplepathbetweeneand
x
e(cid:48) islabelled2andthesimplepathaftere(cid:48) islabelled3(seeFigure2).
Now suppose we have ordering of the components of Γ such that 1 indexes
y
the component of Γ containing e and 2 indexes the component at the head of e(cid:48).
y
Similarly,suppose wehave orderingofthe componentsofΓ suchthat1indexes
y(cid:48)
thecomponentatthetailofeand2indexesthecomponentcontaininge(cid:48).InΓ the
z
simplepathcontainingeande(cid:48) isindexedby1.Thenbytheassociativityofµthe
followingdiagramcommutes
µ⊗1 (cid:47)(cid:47)
M ⊗M ⊗M M ⊗M
1 2 3 1 2
1⊗µ µ
(cid:15)(cid:15) (cid:15)(cid:15)
µ (cid:47)(cid:47)
M ⊗M M
1 2 1
fromwhichtheresulteasilyfollows. (cid:116)(cid:117)
8 PaulTurnerandEmmanuelWagner
We now extend this to define a map F (x ≤ y): F (x) → F (y) for
A,M A,M A,M
anyx ≤ y.Pickasequencex ≺ x ≺ ··· ≺ x ≺ y andset
1 l
F (x ≤ y) = F (x ≺ y)◦···◦F (x ≺ x ).
A,M A,M l A,M 1
CourtesyofLemma3weimmediatelyseethatthisdoesnotdependonthepartic-
ularchoiceofsequence.Wehavethusshownthat
Proposition2.F : P(Γ) → Mod as defined above is a covariant functor, i.e.
A,M
F isacoefficientsystemforΓ.
A,M
Wefinallyarriveattheprincipaldefinitionofthissection.
Definition3.Let A be a unital R-algebra, M an A−A bimodule and Γ a finite
baseddigraph.UsingthecoefficientsystemF abovewedefine
A,M
H (Γ,A,M) = H (Γ,F ).
∗ ∗ A,M
ThefollowingtheoremprovidesonepossibleanswertoPrzytycki’squestion.
Theorem1.LetAbeanR-algebraandM anA−Abimodule.Then
H (−,A,M): DirGr → GrMod
∗ b
is a functor with the property that if Γ is a consistently directed n-gon then for
0 ≤ i ≤ n−2
H (Γ,A,M) ∼= HH (A;M)
i i
where on the right hand side we have the Hochschild homology of the algebra A
withcoefficientsinthebimoduleM.
Before proving this Theorem let us recall Przytycki’s result relating the chro-
matic homology of graphs to Hochschild homology [4]. We will state the results
using homological grading conventions. Firstly recall that (homologically graded)
chromatichomologyofagraphGisdefinedasfollows.LetAbeacommutativeR-
algebra.LetBbetheBooleanlattice(the“cube”asitisusuallyreferredto)onthe
edgesofG.AnelementofBisasubgraphofGwiththesamevertexsetasV and
willtypicallycontainsomeisolatedvertices.Toeachsuchsubgraphxassociatethe
moduleF(x)beingatensorproductofcopiesofA,oneforeachconnectedcompo-
nent.Toeachedgeζ ofthecube(coveringrelationintheBooleanlattice)associate
a map d being given by the algebra multiplication if two connected components
ζ
fuse, or the identity map otherwise. Letting N be the number of edges in G, one
defines,fori = 0,1,...,N
(cid:77)
C (Γ) = F(x).
i
x∈P(Γ),|x|=N−i
Adifferentiald: C (Γ) → C (Γ)canbedefinedfora ∈ F(x)by
i i−1
(cid:77)
d(a) = (cid:15)(ζ)d (a),
ζ
ζ
where (cid:15)(ζ) = ±1. This gives a complex, whose homology is the chromatic graph
homology of G using the algebra A. All this is well documented elsewhere (see
[2,3]fordetails).
Homologyofdigraphs 9
Przytyckiextendstheaboveinthefollowingway(see[4]fordetails).Suppose
M is an A−A-bimodule (where as above A is commutative) and suppose v is
1
a chosen base vertex of G. Modify the above construction by replacing A by M
whenever associating a module to a component containing v . Moreover, in the
1
definitionofthedifferential,partialderivativesbetweensubgraphshavingthesame
numberofcomponentsaresettozero.DenotetheresultHˆA,M(G).
∗
IfwetakeG = P ,ann-sidedpolygon,thentakingcoherentdirectionsoneach
n
edge(sothatthewholepolygonisorientedclockwiseoranti-clockwise)thenPrzy-
tycki argues the above may be extended to the case where A is non-commutative.
(From our point of view, the existence of this global orientation, means that each
subgraphisagainconsistentlydirectedandisthusamultipathinoursetup).
Przytycki’smainresult(withhomologicalgradingconventions)is:
Theorem(Przytycki)
HˆA,M(P ) ∼= HH (A;M) for1 ≤ i ≤ n−1.
i n i−1
Thus, modified chromatic homology of polygons agrees with Hochschild ho-
mology through a range of dimensions. We are now ready to prove Theorem 1
above.
Proof. (of Theorem 1) Let f: Γ(cid:48) → Γ be a (basepoint preserving) inclusion. We
wishtodefineachainmap
f(cid:48): S (Γ(cid:48),F(cid:48) ) → S (Γ,F )
∗ A,M ∗ A,M
Here we are writing F(cid:48) for the coefficient system constructed above for the
A,M
graphΓ(cid:48).
Firstly,weconstructamorphismofcoefficientsystemsτ: F(cid:48) −→• F ◦f˜.
A,M A,M
Forx ∈ P(Γ)thegraphΓ isisomorphictothegraphΓ(cid:48)∪W whereW consists
f˜(x) x
of the vertices in Γ which are not in (the image of) Γ(cid:48). We can thus make the
identification
F (f˜(x)) ∼= F(cid:48) (x)⊗(cid:79)A.
A,M A,M
W
Moreover, if x ≺ y in Γ(cid:48) then these identifications make the following diagram
commute.
F (f˜(x)) ∼= F(cid:48) (x)⊗(cid:78)A
A,M A,M
FA,M◦f˜(x≺y)(cid:15)(cid:15) (cid:15)(cid:15)FA(cid:48) ,M(x≺y)
F (f˜(y)) ∼= F(cid:48) (y)⊗(cid:78)A
A,M A,M
Wenowdefineτ tobethecomposition
x
F(cid:48) (x) ∼= F(cid:48) (x)⊗ (cid:79)R → F(cid:48) (x)⊗ (cid:79)A ∼= F (f˜(x))
A,M A,M R A,M R A,M
where the map shown is just the identity on F(cid:48) (x) and the unit map of the
A,M
algebraR → Aontheremainingtensorfactors.
Forx ≺ y inΓ(cid:48),thediagramaboveshowsthat
F(cid:48) (x) τx (cid:47)(cid:47)F(cid:48) (x)⊗(cid:78)A
A,M A,M
F(cid:48) (x≺y) F(cid:48) (x≺y)⊗Id
A,M (cid:15)(cid:15) (cid:15)(cid:15) A,M
F(cid:48) (y) (cid:47)(cid:47)F(cid:48) (y)⊗(cid:78)A
A,M τy A,M
10 PaulTurnerandEmmanuelWagner
Nowsupposex ≤ yinΓ(cid:48).Wemaychooseasequencex ≺ x ≺ ··· ≺ x ≺ yand
1 m
notingthatf˜(x) ≺ f˜(x ) ≺ ··· ≺ f˜(x ) ≺ f˜(y)weseeτ isnaturalbyrepeated
1 m
use of the above diagram. Thus it is thus a morphism of coefficient systems as
desired.
FromProposition1(i)wenowgetahomomorphism
τ : H (Γ(cid:48),F(cid:48) ) → H (Γ(cid:48),F ◦f˜).
∗ ∗ A,M ∗ A,M
Invokingpart(ii)ofProposition1wealsohaveahomomorphism
f : H (Γ(cid:48),F ◦f˜) → H (Γ,F ).
∗ ∗ A,M ∗ A,M
Thecompositionf ◦τ givesahomomorphism
∗ ∗
f : H (Γ(cid:48),A,M) → H (Γ,A,M)
• ∗ ∗
whichisthemapinhomologyinducedbyf.
Itremainstoshowthatifg: Γ(cid:48)(cid:48) → Γ(cid:48)isaninclusionthen(fg) = f ◦g .This
• • •
amounts to showing that the top and bottom routes around the following diagram
are the same (where we have omitted the subscripts A,M and are being a little
liberalinourmultipleusesoftheletterτ).
H∗(Γ(cid:48)(cid:48),F(cid:48)(cid:111)(cid:48))(cid:111)(cid:111)(cid:87)(cid:111)(cid:87)(cid:111)τ(cid:87)(cid:111)∗(cid:87)(cid:111)(cid:111)(cid:87)(cid:111)(cid:87)H(cid:111)(cid:87)(cid:111)(cid:87)∗(cid:111)(cid:87)((cid:55)(cid:55)Γ(cid:87)(cid:87)(cid:48)(cid:48)(cid:87)τ,∗(cid:87)F(cid:87)(cid:66)(cid:48)(cid:66)(cid:87)◦(cid:66)(cid:87)(cid:66)g˜(cid:87)(cid:66))(cid:66)(cid:87)(cid:66)(cid:87)(cid:66)τ˜(cid:87)(cid:66)∗(cid:66)g(cid:87)(cid:66)∗(cid:87)(cid:66)(cid:66)(cid:87)(cid:66)(cid:43)(cid:43)(cid:66)(cid:66)(cid:66)(cid:47)(cid:47)H(cid:66)(cid:66)(cid:66)∗(cid:33)(cid:33)(Γ(cid:48),F(cid:125)(cid:125)(cid:48)(cid:125))(cid:125)(cid:125)(cid:125)(cid:125)(cid:103)(cid:125)(cid:103)(cid:125)τ(cid:125)(cid:103)∗(cid:125)g(cid:103)(cid:125)∗(cid:103)(cid:125)(cid:125)(cid:103)(cid:47)(cid:47)(cid:125)H(cid:103)(cid:125)(cid:103)(cid:125)∗(cid:103)(cid:125)((cid:125)((cid:103)Γf(cid:62)(cid:62)(cid:103)g(cid:48)(cid:103),)∗(cid:103)F(cid:103)◦(cid:103)(cid:78)(cid:103)f(cid:78)˜(cid:103))(cid:78)(cid:103)(cid:78)(cid:103)(cid:78)(cid:103)(cid:78)f(cid:103)(cid:78)∗(cid:78)(cid:103)(cid:78)(cid:103)(cid:78)(cid:51)(cid:51)(cid:78)H(cid:38)(cid:38) ∗(Γ,F)
H∗(Γ(cid:48)(cid:48),F◦f(cid:102)g)
Theleft-handtrianglecommutesdirectlyfromthedefinitionofthemapsτ,and
the right-hand triangle commutes from the statement in Proposition 1 (ii) that the
inducedmapsbehavewellundercomposition.Themiddlesquarecommutesbythe
commentsimmediatelyaftertheproofofProposition1.
InordertomaketheconnectionwithHochschildhomologywecombinePrzy-
tycki’s result with the work of Everitt and the first author [1]. It is clear that if
Γ = P with consistent directions, then P(Γ) is the Boolean lattice on the edges
n
ofP minusitsmaximumelement.
n
Moreover the functor F constructed on P(Γ) agrees with the (implicit)
A,M
functorusedintheconstructionofHˆA,M inthiscase.Sincethegraphisapolygon,
i+1
the only partial derivative in Przytycki’s set-up that are set to zero are those from
corank 1 elements to the maximal element and this corresponds to the absence of
themaximumelementoftheBooleanlatticeinP(Γ).
It now follows from the main result of [1] that the homology of the path cate-
gory of category P with coefficients in F is isomorphic to HˆA,M(P ) with a
n A,M i+1 n
gradingshift:
H (P ,A,M) ∼= H (P ,F ) ∼= HˆA,M(P ).
i n i n A,m i+1 n
CombiningthiswithPrzytycki’sresultgivesthedesiredisomorphism. (cid:116)(cid:117)