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Poisson (co)homology and isolated singularities
Anne Pichereau
To cite this version:
Anne Pichereau. Poisson (co)homology and isolated singularities. Journal of Algebra, Elsevier, 2005,
299 (2), pp.747-777. �10.1016/j.jalgebra.2005.10.029�. �hal-00804880�
ANNEPICHEREAU
Abstrat. Toeahpolynomial'2F[x;y;z℄isassoiatedaPoissonstruture
onF 3
,asurfaeandaPoissonstrutureonthissurfae. When'isweightho-
mogeneouswithanisolatedsingularity,wedeterminethePoissonohomology
andhomologyofthetwoPoissonvarietiesobtained.
Contents
1. Introdution 1
2. ThePoissonohomologyomplexassoiatedtoapolynomial 5
3. IsolatedsingularitiesandtheKoszulomplex 8
4. Poisson ohomology assoiatedto a weighthomogeneous polynomial
withanisolatedsingularity 11
5. Poissonohomologyofthesingularsurfae 16
6. Poisson homologyassoiatedto aweighthomogeneouspolynomialwith
anisolatedsingularity 21
Referenes 26
1. Introdution
TherstPoissonstruturesappearedinlassialmehanis. In1809,D.Poisson
introduedabraketoffuntions,givenby:
ff;gg= r
X
i=1
f
q
i g
p
i f
p
i g
q
i
; (1)
for two smooth funtions f;g on R 2r
. It permits one to write the Hamilton's
equations as dierential equations, where positions (q
i
) and impulsions (p
i ) play
symmetri roles. Indeed, denoting by H the total energy of the system, these
equationsbeome:
_ q
i
= fq
i
;Hg;
_ p
i
= fp
i
;Hg;
1ir:
D. Poisson alsopointedoutthatiff andg areonstantsofmotion,then ff;ggis
alsoaonstantofmotionandthisphenomenonwasexplainedin1839byC.Jaobi,
whoprovedthat (1)satiseswhatisnowalled theJaobiidentity:
fff;gg;hg+ffg;hg;fg+ffh;fg;gg=0:
(2)
2000MathematisSubjetClassiation. 17B55,17B63.
Keywordsandphrases. Poissonohomology,Poissonhomology,isolatedsingularities.
This importantidentity leadsto thedenition of aPoissonalgebraasan algebra
Bequippedwithaskew-symmetribiderivation f;g,satisfying(2),forallf;g;h,
elementsofB. Saiddierently,aPoissonalgebraisaLiealgebra(B;f;g),where
f;gsatiestheLeibnizruleffg;hg=ffg;hg+ff;hgg,forallf;g;h2B. One
talks about a Poisson variety, when its algebra of funtions is equipped with a
Poissonstruture. Thisnotiongeneralizesthenotionof sympletimanifold.
For agivenPoissonalgebra(B;f;g),onedenesaohomology,alledPoisson
ohomology, introdued by A. Lihnerowiz in [12℄; see also [9℄ for an algebrai
approah. The ohains are the skew-symmetri multiderivations of A and the
oboundaryoperatoris [; ℄
S
,where:=f;gisthePoissonbraketand[;℄
S
istheShoutenbraket. TheresultingPoissonomplex,denedindetailinSetion
2.1,an beviewedas theontravariantversionofthedeRhamomplex. Itsoho-
mologygivesveryinterestinginformationaboutthePoissonstruture,asforsmall
k,thek-th PoissonohomologyspaeH k
(B;)hasthefollowinginterpretation:
H 0
(B;) = fCasimirfuntions g:=ff 2Bjff;g=0g;
H 1
(B;) =
fPoissonderivationsg
fHamiltonianderivationsg
;
H 2
(B;) =
fskew-symmetribiderivationsompatible withg
fLiederivativesofg
;
H 3
(B;) = fObstrutionsto deformationsofPoissonstruturesg:
Moreover,H 2
(B;)isfundamental inthestudy ofnormalformsofPoissonstru-
tures (see [4℄). We also denote by Cas(B;) the spae of all Casimir funtions
of (B;f;g) (that is to say H 0
(B;))and we point outthat eah H k
(B;) is a
Cas(B;)-modulein anaturalway.
TodeterminethePoissonohomologyofagivenPoissonalgebraexpliitlyis,in
general,diÆult. Oneof thereasonsseemsto bethat Poissonohomology isnot
afuntor: amorphism : A
1
!A
2
betweenPoissonalgebrasdoesnot leadto a
morphismbetweentheirohains(multiderivations),norbetweentheirorrespond-
ing Poisson ohomology groups. In afew spei ases, Poisson ohomologyhas
beendetermined. Fora sympleti manifold, there exists anatural isomorphism
between Poisson and de Rham ohomology (see [12℄). In [20℄ and [23℄, onends
somepartialresultsabouttheaseofregularPoissonmanifolds,while,forPoisson-
Liegroups,oneanreferto[7℄. Finally,thePoissonohomologyindimensiontwo
wasomputedinthegermiedandalgebraiasesin [14℄and[17℄.
Our purposeis to determine the Poisson ohomology of twolassesof Poisson
varieties, intimately linked. The rst lass is omposed of the singular surfaes
F
'
:f'=0gin F 3
(Fisaeldofharateristizero)thataredenedbythezeros
ofpolynomials'2F[x;y;z℄andtheseondoneisthelassofthePoissonvarieties
that are the ambient spae F 3
, equipped with Poisson strutures assoiated to
eah '. It means that we onsider Poisson strutures on the algebrasof regular
funtionsonF
' andF
3
,givenbyA
'
:=F[x;y;z℄=h'iandA:=F[x;y;z℄andthat
wedeterminethePoissonohomologyofthePoissonalgebrasobtained.
We pointout that thedimension three is therst onein whih there is areal
indeedtrivialindimensiontwoandeverypolynomial 2F[x;y℄leadstoaPoisson
struture on the aÆne spae F[x;y℄, given by
x
^
y
. One an onsider the
singular lous of suh a struture, given by : f = 0g. In [17℄, the authors
determinethedimensionsofthePoissonohomologyspaes,when is ahomoge-
neous polynomial. They observethat these dimensions are linked to the type of
thesingularityof . Conversely,in ourontext,weonsiderasurfaeF
'
,witha
singularity,andaPoisson braketthat donotbringothersingularities. That isto
say, thisPoisson struture issympletieverywhereexepton thesingularities of
F
'
. Infat,itwillbetherestritionofaPoisson struturef;g
' onF
3
,whihis
ompletelydened bythebrakets:
fx;yg
'
= '
z
; fy;zg
'
= '
x
; fz;xg
'
= '
y
; ('2A):
(3)
WesupposethatF
'
hasonlyoneweighthomogeneousisolatedsingularity(atthe
origin). Infat,thehypothesisisthat'isaweighthomogeneouspolynomialwith
anisolatedsingularity.
An other wayto approahourontext isto onsider thePoisson strutureson
A thatadmit aweighthomogeneousCasimir andasingular lousreduedto the
origin. ThatleadstostudythePoissonstruturesoftheformf;g
'
,with'weight
homogeneouswithanisolatedsingularity. As'isaCasimirforthisstruture,h'i
isaPoissonidealofthePoissonalgebra(A;f;g
'
). Thisimpliesthatf;g
' goes
downtothequotientalgebraA
'
=F[x;y;z℄=h'i. ThesingularsurfaeF
' isthen
theunionofasympletileaveoff;g
'
andtheorigin.
Foreah '2A weight homogeneouswithan isolated singularity,what wede-
termineisthePoissonohomologyofboththePoissonalgebrasintrodued. More-
over,weturnthese resultstogoodaounttogivethePoissonhomologyof these
algebras. The Poisson ohomology spaes are respetively denoted by H k
(A;')
for (A;f;g
'
) and H k
(A
'
) for the singularsurfae, while thePoisson homology
spaesaredenotedbyH
k
(A;') andH
k (A
' ).
Todeveloparstideaaboutourresults,onemaythinkof'asahomogeneous
polynomial,ofdegreedenotedby$('),suhthatitsthreepartialderivativeshave
onlyoneommonzerothatistheorigin. Thisimpliesthat
A
sing
:=A=h '
x
; '
y
; '
z i
is anitedimensional F-vetorspae. Itsdimensionis theso-alledMilnornum-
ber (see [13℄). This spae givesinformation aboutthe (isolated) singularity of
thesurfaeF
'
(likemultipliity, seealso[3℄)asitisexatlythealgebraofregular
funtionsonthissingularity. ItplaysanimportantroleinthePoissonohomology
of the algebra (A;f;g
'
), so that this Poisson ohomology is losely related to
the type of the singularity of F
'
. We onsider a family u
0
= 1;u
1
;:::;u
1 of
homogeneouselementsofA,whoseimagesinA
sing
giveaF-basisofthisF-vetor
spae.
ThealgebraofCasimirfuntionsofthealgebra(A;f;g
'
)isgiveninProposition
4.2andissimplythealgebrageneratedby',thatistosayCas(A;')=H 0
(A;')'
L
i2N F'
i
. InProposition4.5, wesee that therstPoisson ohomologyspae of
1
theCas(A;')-module givenby
H 1
(A;')'Cas(A;')~e ;
where~e:=(x;y;z)orresponds to theEuler derivationx
x +y
y +z
z
. Notie
that the ubi polynomials play a speial role here; in the weight homogeneous
ase, this role is played by the polynomials of degree the sum of the weights of
the three variables x;y;z. Moreover, with Proposition 4.8, we see that the ase
$(') = 3 is also the unique ase where the biderivation f;g
'
is not an exat
Poisson struture, i.e. f;g
'
, whih is a 2-oyle of the Poisson ohomology of
(A;f;g
'
),isnota2-oboundary(see[9℄). Proposition4.8aÆrmsindeedthatthe
seondPoissonohomologyspaeisexatly
H 2
(A;') '
M
j1
$(u
j
)6=$(') 3
Cas(A;')
~
r u
j
M
$(uj)=$(') 3
Cas(A;')u
j
~
r'
M
j1
$(uj)=$(') 3 F
~
ru
j :
ThiswritinghasbeenobtainedfromthethirdPoissonohomologyspae,whihis
determinedinProposition4.7,andisexatlythefreeCas(A;')-module
H 3
(A;')'Cas(A;')
F A
sing :
It may be remarked that H 2
(A;') is the unique Poisson ohomology spae of
(A;f;g
'
)whihisnotalwaysafreemoduleoverthealgebraofCasimirs.
InChapter5,wegivethePoissonohomologyspaesofthesingularsurfaeF
' ,
byonsidering thealgebraA
'
. ForthisPoisson algebra,theCasimirs aresimply
theelementsofFand,aordingtoPropositions5.5and5.6,wehave:
H 1
(A
' )'
M
$(uj)=$(') 3 Fu
j
~
e; H
2
(A
' )'
M
$(uj)=$(') 3 Fu
j
~
r ':
Finally,inChapter6,wedeterminethePoissonhomologyofthealgebra(F 3
;f;g
' )
andof thesingularsurfaeF
'
. Weexplainrst, in Proposition 6.1,that wehave
isomorphisms
H
k
(A;')'H 3 k
(A;'); forallk=0;1;2;3:
Then,using theresultsaboutPoissonohomologyof(A;f;g
'
),weomputethe
PoissonhomologyspaesofF
'
andweobtain,in Proposition6.5,
H
0 (A
' )'H
2 (A
' )'A
sing
; H
1 (A
' )'
1
M
j=1 F
~
ru
j :
Sine the oboundaryoperator is a weighthomogeneousoperator (see Setion
2.2),allourargumentsremaintrueifwereplaethealgebraA=F[x;y;z℄bythe
algebraofallformal powerseries
A:=F[[x;y;z℄℄, stillequippedwith thePoisson
struturef;g
'
,with'aweighthomogeneouselementofA. ItsuÆestoreplae
Cas(A;')=F['℄byCas(
A ;')=F[['℄℄,thealgebraofformalpowerseriesin'.
Iwouldliketotaketheopportunitytothankmythesisadvisor,PolVanhaeke,
forsuggestingtomethis interestingproblemandfor hisavailabilityallalongthis
waspreious forme,and CamilleLaurentforhisexplanationsaboutthe modular
lass.
I nallywould liketothankProf. M.vandenBergh. After writingthispaper,
hepointedouttomethat,inhisartile\Nonommutativehomologyofsomethree-
dimensionalquantumspaes"(see[21℄),heomputedthePoissonhomologyspaes
of the Poisson algebra (A;f;g
'
), for' = q1
3 (x
3
+y 3
+z 3
)+2p
1
xyz, where p
1
andq
1
areparameters. ThisaseisapartiularoneofthePoissonhomologythat
Idetermine,andthemethod isverysimilar.
2. The Poissonohomologyomplex assoiated to apolynomial
2.1. Poisson strutures on A =F[x;y;z℄ and their ohomology. Let A be
thepolynomialalgebraA=F[x;y;z℄, whereFisaeldofharateristizeroand
let'2A. APoissonstruture onAisdened bythebrakets:
fx;yg
'
= '
z
; fy;zg
'
= '
x
; fz;xg
'
= '
y : (4)
Reall that a Poisson braket on an assoiative and ommutative algebra B is a
skew-symmetribilinearmapf;g,fromB 2
toB(elementofHom(^
2
B;B)),whih
isaderivation ineahofitsargumentsandwhihsatisestheJaobiidentity:
fff;gg;hg+ffg;hg;fg+ffh;fg;gg=0;
(5)
foreahf;g;h2B. InthepartiularaseofA,thebraketsofthegeneratorsx;y;z
denetotallythePoissonbraket,inviewofthederivationproperty,andmoreover
theJaobiidentityissatisedforallf;g;h2Aifandonlyifitissatisedforx;y;z
(see[22℄). Here,aneasyomputationshowsthat thisonditionis satisedbythe
braketf;g
'
sothatitequips AwithaPoissonstruture,expliitlygivenby:
f;g
'
= '
z
x
^
y +
'
x
y
^
z +
'
y
z
^
x : (6)
Our rst purpose is to determine the Poisson ohomology of this Poisson al-
gebra (A;f;g
'
), when ' is a weight homogeneous polynomial with an isolated
singularityattheorigin.
Wereall thatthe Poisson omplexisonstrutedin the followingway(see[4℄
and [11℄ for details). First, the k-ohains of the Poisson omplex of (A;f;g
' )
aretheskew-symmetrik-derivations ofA(i.e.theskew-symmetrik-linearmaps
A k
! A that are derivations in eah of their arguments). We denote by X k
(A)
the A-module of all skew-symmetri k-derivations of A and the elements of the
A-module X
(A) = L
k 2N X
k
(A) are alled skew-symmetri multi-derivations of
A. Byonvention,theA-moduleofthe0-derivationsofAisX 0
(A)=A.
The Poisson oboundary operator Æ k
' : X
k
(A) ! X k +1
(A) is dened, for an
elementQ2X k
(A),by:
(7)
Æ k
' (Q)(f
0
;:::;f
k ):=
k
X
i=0 ( 1)
i n
f
i
;Q(f
0
;:::; b
f
i
;:::;f
k )
o
'
+
X
0i<jk ( 1)
i+j
Q(ff
i
;f
j g
'
;f
0
;:::; b
f
i
;:::; b
f
j
;:::;f
k );
wherethesymbol b
f
i
meansthat weomitthetermf
i
. It iseasytosee thatÆ k
' (Q)
isindeed askew-symmetri (k+1)-derivationwhile thefat thatÆ k +1
ÆÆ k
=0is
aneasyonsequeneoftheJaobiidentity(5). Theohomologyofthisomplexis
alledthePoissonohomologyof(A;f;g
'
). WedenotebyZ k
(A;'),respetively
B k
(A;'),thevetorspaeofallk-oyles,respetivelyofallk-oboundaries,and
wedenotebyH k
(A;'):=Z k
(A;')=B k
(A;'),thek-thohomologyspae. Asthe
spae H 0
(A;') is exatly the F-vetorspae of the Casimirs of f;g
'
(i.e. the
elementsthatbelong to theenter ofthis braket), wewillalso denotethis spae
by Cas(A;'). Notie that, if 2 Cas(A;') , the operator Æ
'
ommutes with
themultipliationby . This impliesthat eahof thePoissonohomologyspaes
H k
(A;') isaCas(A;')-module.
IntheaseofthepolynomialalgebraA=F[x;y;z℄,wehave:
X 0
(A)'X 3
(A)'A; X 1
(A)'X 2
(A)'A 3
; (8)
andX k
(A)' f0g,fork4. Wehoosethesenaturalisomorphismsasfollows:
X 1
(A) ! A
3
V 7 ! (V[x℄;V[y℄;V[z℄);
X 2
(A) ! A
3
V 7 ! (V[y;z℄;V[z;x℄;V[x;y℄);
andX 3
(A) !A:V 7 !(V[x;y;z℄).
The elements of A 3
are viewedas vetor-valued funtions on A, so we denote
them with anarrow,like
~
f 2A 3
. Sometimes, it will beimportantto distinguish
A 3
'X 1
(A)fromA 3
'X 2
(A);thenwewillratherwrite
~
f 2X 1
(A) or
~
f 2X 2
(A).
InA 3
,let,denoterespetivelytheusualinnerandrossproduts,while
~
r,
~
r,
Div denote respetivelythe gradient, the url and the divergene operators. For
example, with these notations and the above isomorphisms, the skew-symmetri
biderivationf;g
'
(denedin(6))isidentiedwiththeelement
~
r' ofA 3
.
Eahof thePoisson oboundaryoperatorsÆ k
'
, givenin(7), annowbewritten
inaompatform:
(9) Æ
0
' (f) =
~
rf
~
r '; forf 2A'X 0
(A);
Æ 1
' (
~
f) =
~
r(
~
f
~
r')+Div (
~
f)
~
r '; for
~
f 2A 3
'X 1
(A);
Æ 2
' (
~
f) =
~
r'(
~
r
~
f)= Div (
~
f
~
r'); for
~
f 2A 3
'X 2
(A);
andthePoissonohomologyspaesof(A;f;g
'
)takethefollowingforms
H 0
(A;') = Cas(A;')'ff2Aj
~
rf
~
r'=
~
0g;
H 1
(A;') ' f
~
f 2A 3
j
~
r(
~
f
~
r')+Div (
~
f)
~
r '=
~
0g
f
~
r f
~
r'jf 2Ag
;
H 2
(A;') ' f
~
f 2A 3
j
~
r'(
~
r
~
f)=0g
f
~
r(
~
f
~
r ')+Div (
~
f)
~
r'j
~
f 2A 3
g
;
H 3
(A;') '
A
~ ~ ~ ~
3 :
Inordertoomputetheseohomologyspaes,wewilloftenuse,for
~
f;~g;
~
h2A 3
andf 2A,thefollowingformulas,well-knownfromvetoralulusin R 3
:
~
r(f~g) =
~
rf~g+f(
~
r~g);
(10)
Div (f~g) =
~
rf~g+fDiv (~g);
(11)
Div(
~
f~g) = (
~
r
~
f)~g
~
f(
~
r~g):
(12)
2.2. Weight homogeneousmulti-derivations. Aswesaid,ourresultsonern
weight homogeneous Poisson strutures on A. A non-zero multi-derivation P 2
X
(A) issaidto be weighthomogeneous of (weighted) degreer2Z,ifthere exist
positive integers $
1
;$
2
;$
3 2 N
(the weights of the variables x;y;z), without
a ommondivisor, suh that L
~ e$
[P℄ =rP; where L
~ e$
is the Lie derivative with
respet to the (weighthomogeneous)Euler derivation~e
$
=$
1 x
x +$
2 y
y +
$
3 z
z
. Thedegreeof aweighthomogeneousmulti-derivationP 2X
(A) is also
denoted by$(P)2Z. Forf 2A,it amountsto theusual(weighted)degreeof a
polynomial. Notiethatthedegreeofanon-zerok-derivationmaybenegativefor
k>0. Byonvention,thezerok-derivationisweighthomogeneousofdegree 1.
TheEulerderivation~e
$
isidentied,withtheisomorphismsgiveninSetion2.1,
to theelement~e
$
= ($
1 x;$
2 y;$
3
z) 2 A 3
. We denote by j$j the sum of the
weights$
1 +$
2 +$
3
,sothatj$j=Div(~e
$
). Euler'sformulaforaweighthomo-
geneousf 2A,
(13)
~
rf~e
$
=$(f)f;
thenyields,using(11):
(14) Div (f~e
$
)=($(f)+j$j)f:
Fixingweights$
1
;$
2
;$
3 2 N
, itis learthat A= L
i2N A
i
, where A
0
=F
and for i 2 N
, A
i
is the F-vetor spae generated by all weight homogeneous
polynomialsofdegreei. DenotingbyX k
(A)
i
theF-vetorspaegivenbyX k
(A)
i :=
fP 2X k
(A)j$(P)=ig[f0g,wehavethefollowingisomorphisms:
(15)
X 0
(A)
i
' A
i
;
X 1
(A)
i
' A
i+$1 A
i+$2 A
i+$3
;
X 2
(A)
i
' A
i+$2+$3 A
i+$1+$3 A
i+$1+$2
;
X 3
(A)
i
' A
i+$1+$2+$3 :
Notie that even ifX 1
(A) ' X 2
(A) and X 0
(A) 'X 3
(A), these isomorphismsdo
notrespettheweightdeompositions(15).
OneofourpurposesistodeterminethePoissonohomologyof(A;f;g
' )when
'2Aisweighthomogeneouswithanisolatedsingularity. Theweighthomogeneity
of'willbeessentialfortheomputationofthesespaes. Itimpliesindeed,among
other things, thateah ofthe oboundaryoperatorsÆ k
'
is weighthomogeneousof
the samedegreeN
$
:=$(') j$j, asan be seenfrom (9). That is to say, we
have:
P2X k
(A)
i )Æ
k
'
(P)2X k +1
(A)
i+N
$ :
If P 2 X k
(A) is aoyle, then eah ofits weighthomogeneous omponents will
be a oyle. In the same way, if P 2 X k
(A) is a oboundary then eah of its
k
aweighthomogeneousoboundary,it istheoboundaryofaweighthomogeneous
elementin X k 1
(A).
3. Isolated singularities and the Koszulomplex
In the next hapters, we will study the Poisson ohomology assoiated to a
weighthomogeneouspolynomial'2A=F[x;y;z℄ (withhar(F)=0). As 'will
besupposedtohaveisolatedsingularities,wewill, inthispart,reallsomeresults
aboutthisnotion,see [19℄and[18℄forproofs.
Algebraially,wesaythat aweighthomogeneouselement'of F[x;y;z℄hasan
isolatedsingularity (attheorigin) if
A
sing
:=F[x;y;z℄=h '
x
; '
y
; '
z i (16)
isnite-dimensional,asaF-vetorspae. ThedimensionofA
sing
isthenalledthe
Milnornumberof the singularpoint. WhenF =C,this amounts,geometrially,
tosayingthatthesurfaeF
'
:f'=0ghasasingularpointonlyattheorigin.
Remark3.1. Bydenition,A
sing
isexatlytheF-algebraofregularfuntionsof
theaÆnevariety n
'
x
= '
y
= '
z
=0 o
whih isthesingularlousofthePoisson
struturef;g
'
(asanbeseenfrom(4)). ThisalgebraA
sing
willplayanimportant
rolein thePoissonohomologyofthealgebras(A;f;g
'
)and(A
'
;f;g
A
' ).
Now,withtheCohen-Maaulaytheorem,wewillseethat,if'2Aisaweightho-
mogeneouspolynomialwithanisolatedsingularity(whatwewilldenotebyw.h.i.s.),
then the sequene of its partial derivatives '
x
; '
y
; '
z
will be aregular sequene
of A. In order to explain that, we rst have to write down the denition of a
homogeneoussystemofparametersofanalgebra.
Denition 3.2. Let A be an assoiativeand ommutativegraded F-algebra. A
systemofhomogeneouselementsF
1
;:::;F
d
inA,wheredistheKrulldimensionof
A, isalled ahomogeneous systemof parameters of A(h.s.o.p.) ifA=hF
1
;:::;F
d i
isanite dimensionalF-vetorspae.
Forexample,ifweonsidertheF-algebraA=F[x;y;z℄,whihisgradedbythe
weighteddegree,wehaveanaturalh.s.o.p.givenbythesystemx;y;z. Moreover,
we have seen above that a weight homogeneous element ' 2 A has an isolated
singularity (that is to say is w.h.i.s.) if and only if the three partial derivatives
'
x
; '
y
; '
z
giveah.s.o.p.ofA.
Inordertounderstandthefollowingtheorem,thatwewillneed,westillhaveto
givethedenition ofaregularsequene.
Denition 3.3. A sequenea
1
;:::;a
n
in aommutativeassoiativealgebraAis
saidto beaA-regularsequene ifha
1
;:::;a
n
i6=A anda
i
is notazerodivisorof
A=ha
1
;:::;a
i 1
ifori=1;2;:::;n.
Forexample,itislearthatthesequenex;y;zisaregularsequeneinF[x;y;z℄.
But,whatabout '
x
; '
y
; '
z
,when'isw.h.i.s.?
Theorem 3.4 (Cohen-Maaulay). Let A be a Noetherian graded F-algebra. If
A has a h.s.o.p. whih is a regular sequene, then any h.s.o.p. in A is a regular
Thus,when'2F[x;y;z℄isw.h.i.s.,then '
x
; '
y
; '
z
isaregularsequene. This
isthekeyfatwhihleadstothefollowingproposition,thatwillplayafundamental
rolein ouromputationsofPoissonohomology,assoiatedtoapolynomial.
Proposition3.5. Forany '2Athe following diagram
F A
A 3
0 A
A 3
A 3
A
A
A 3
A 3
A
A
A 3
A 3
A
?
?
~
r
?
~
r
-
?
~
r -
~
r'
-
~
r'
?
~
r
-
~
r'
? Div
-
~
r'
?
~
r
-
~
r '
?
~
r
-
~
r'
? Div
-
~
r '
-
~
r '
-
~
r'
isommutativeandhasexatolumns. If'isw.h.i.s.thentherowsofthisdiagram
arealso exat.
Remark 3.6. If ' 2 A is weight homogeneous, then, as maps from X k
(A) to
X k 1
(A), eah of thevertialarrowsis weighthomogeneousofdegree zero,while
eahofthehorizontalarrowsisweighthomogeneousofdegree$('),the(weighted)
degreeof',leadingto:
X 3
(A)
r
X 2
(A)
r+$(')
X 3
(A)
r $(')
X 2
(A)
r
X 1
(A)
r+$(')
X 0
(A)
r+2$(')
X 2
(A)
r $(')
X 1
(A)
r
X 0
(A)
r+$(') -
~
r '
?
~
r
?
~
r
?
~
r
-
~
r '
-
~
r '
?
~
r
-
~
r'
? Div
-
~
r '
-
~
r'
Proof. Eaholumn ofthisdiagram iseasilyinterpretedasthedeRhamomplex
of A. The lassial argument of exatness of the de Rham omplexof C 1
(R n
)
is easily adapted to the algebrai ase: if
~
f = (f
1
;f
2
;f
3 ) 2 A
3
is omposed of
threehomogeneouspolynomialsofdegreedthenDiv(
~
f)=0impliesthat therst
omponentof
~
r(
~
f~e)isequalto
~
r(
~
f~e)
1
=2f
1 +
~
rf
1
~e xDiv(
~
f)=
(d+2)f
1
,in viewofEuler'sFormula(13)(~eistheEulerderivation (x;y;z)2A 3
,
thatistosay~e
$
,with$
1
=$
2
=$
3
=1),sothat
~
f = 1
d+2
~
r(
~
f~e). Similarly,
~
r
~
f =
~
0 implies that
~
r(
~
f~e)
1
= f
1 +
~
rf
1
~e = (d+1)f
1
, that lieds to
~
f = 1
d+1
~
r (
~
f~e),aordingagaintoEuler'sFormula.
Eahoftherowsofthediagramrepresents(partof)theso-alledKoszulomplex.
Let us provethat the Koszul omplex, assoiated to ' 2 A is exat, when ' is
w.h.i.s. If
~
f = (f
1
;f
2
;f
3 )2 A
3
satises the equation
~
f
~
r' =
~
0, then we have
three equalities like f
1 '
y f
2 '
x
= 0. Sine the partial derivatives of ' form a
regular sequene, '
is not azerodivisor in A=h '
i, so there exists 2A suh
thatf
1
= '
x
andthenf
2
= '
y
. Theotherequationsimplythatf
3
= '
z ,that
is tosay
~
f =
~
r '. Forthe seondpartof theexatitudeof theKoszulomplex,
thereasoningisexatlyofthesamekind.
Remark3.7. If'2Aisaweighthomogeneouspolynomialwithoutsquarefator
then the rst part of the Koszul omplex A
~
r'
! A 3
~
r'
! A 3
is exat, but the
seond partA 3
~
r'
! A 3
~
r'
!Aneednotbeexatif' isnotw.h.i.s. Forexample,
let'=xyz2A. Thepolynomial'issquarefreebut theoriginisnotanisolated
singularity for '. Then, the element
~
f = (x;y; 2z) 2 A satises the equation
~
f
~
r'=
~
0 but, by anargumentofdegree, there is noelement~g 2 A 3
suh that
~
f =~g
~
r'.
We will often apply Proposition 3.5 diretly but sometimes, we will use it in
termsofthefollowingorollary.
Corollary 3.8. Let '2A be w.h.i.s. and let
~
h2 A 3
. If (
~
r
~
h)
~
r ' =0then
thereexistf;g2Asuhthat
~
h=
~
rf+g
~
r'.
Proof. Aording tothe diagram in Remark 3.6, theoperator
~
h7!(
~
r
~
h)
~
r',
onsideredasamapbetweenX 2
(A)andX 0
(A),isaweighthomogeneousoperator
ofdegree$('). Therefore,itsuÆestoprovetheresultforanelement
~
h2X 2
(A)
r ,
withr2Z. If(
~
r
~
h)
~
r'=0then,byProposition3.5, thereexists
~
k2A 3
suh
that
~
r
~
h=
~
k
~
r'. Inview ofRemark 3.6,
~
k anbehosenin X 2
(A)
r $(') .
Summarizing,wehavetoprovethatanequationofthetype:
~
r
~
h=
~
k
~
r ';
~
h2X 2
(A)
r
;
~
k2X 2
(A)
r $(') (17)
impliesthat
~
h=
~
rf+g
~
r',withf;g2A.
We will do this by indution on r 2 Z, by proving the result diretly for all
r < $(') $ [2℄
, with $ [2℄
:= maxf$
1 +$
2
;$
1 +$
3
;$
2 +$
3
g, where the
integers$
1
;$
2
;$
3
aretheweightsofthevariables x;y;z.
Ifr<$(') $ [2℄
then,aordingtothedeompositionsin(15),X 2
(A)
r $(')
=
f0gsothattheequality(17)leadsto
~
r
~
h=
~
0. UsingProposition3.5, weobtain
~
h=
~
rf,withf 2Aasrequired.
Letr 0
$(') $ [2℄
andassumethat(17)implies,forallr<r 0
, theexistene
off;g2Asuhthat
~
h=
~
rf+g
~
r'. Letussupposethat anelement
~
l2X 2
(A)
r 0
satisesanequationlikein(17),namely,supposethatthereexists
~
h2X 2
(A)
r 0
$(')
suhthat
~
r
~
l=
~
h
~
r':
(18)
Then,
~
h satises (17), with r =r 0
$('). Indeed, omputing the divergene of
bothsummandsof(18)gives(
~
r
~
h)
~
r'=0andusingProposition3.5oneagain
leadstotheexisteneof
~
k2X 2
(A)
r 0
2$(')
suhthatwehave
~
r
~
h=
~
k
~
r'. By
indution hypothesis,there exist f;g 2Asuhthat
~
h=
~
rf+g
~
r'. Then, using
Formula(10),weobtain
~
r
~
l=
~
h
~
r'=
~
rf
~
r '=
~
r(f
~
r').
We annow onludewith Proposition 3.5 that there exists f 0
2 A suh that
~ ~ ~
0
Remark 3.9. AsZ 2
(A;') =f
~
h2A 3
j(
~
r
~
h)
~
r'=0g, Corollary3.8leadsto
theequality
Z 2
(A;')=f
~
rf+g
~
r'jf;g2Ag:
Thisidentitywill beusefulwhenwewilldetermineH 2
(A;') in Setion4.4.
4. Poissonohomologyassoiated to a weighthomogeneous
polynomial with an isolated singularity
Letus onsider the polynomialalgebraA =F[x;y;z℄ (har(F) =0), equipped
with the Poisson struture f;g
'
, where ' 2 A is w.h.i.s. (weight homogeneous
polynomialwith an isolated singularity). We determine the Poisson ohomology
spaesofthePoissonalgebra(A;f;g
' ).
Remark4.1. If'2Aisw.h.i.s.then$(') $
i
>0,fori=1;2;3(where$(')
is stillthe (weighted) degreeof 'and $
1
;$
2
;$
3
aretheweightsof thevariables
x;y;z),andin partiular,$(')>1.
4.1. The spae H 0
(A;'). A preise desriptionof the0-th Poisson ohomology
spae,whihisalsothealgebraoftheCasimirs,isgiveninthefollowingproposition.
Proposition4.2. If'2Aisw.h.i.s. thenthezerothPoisson ohomologyspaeof
(A;f;g
'
) isgivenby
H 0
(A;')=Cas(A;')' M
i2N F'
i
:
Proof. Letf 2A f0gbeaweighthomogeneous0-oyle,thussatisfyingÆ 0
' (f)=
~
rf
~
r' =
~
0. Write f as f =h' r
, where r 2 Nand where h 2 A f0gis a
polynomial that is not divisible by '. We have
~
r f = ' r
~
rh+rh' r 1
~
r', so
~
rh
~
r'=
~
0. Proposition3.5impliestheexisteneofg2Asuhthat
~
rh=g
~
r'.
Sinehand'areweighthomogeneousandin viewofEuler'sFormula(13),
$(h)h=
~
rh~e
$
=g
~
r'~e
$
=$(')g';
so$(h) =0,as his notdivisible by'. Thus h2 Fand f =h' r
2 L
i2N F'
i
.
Conversely,itislearthat Æ 0
' ('
r
)=
~
r(' r
)
~
r '=
~
0,foranyr2N.
Remark 4.3. Aording to Remark3.7, if'2Aisaweighthomogeneouspoly-
nomial without square fator but ' is not neessarly w.h.i.s., then the rst part
of theKoszul omplexis still exat, soProposition 4.2 isalso valid for this more
generallass of polynomials. However, if' has asquare fator, the resultis not
trueanymore. Forexample,if'= r
withr2and 2Aaweighthomogeneous
polynomialwithoutsquarefator,thenH 0
(A;')'H 0
(A; )' L
i2N F
i
sothat
H 0
(A;')6' L
i2N F'
i
.
4.2. ThespaeH 1
(A;'). Werstprovearesultwhihwillbeusefultodetermine
H 1
(A;').
Lemma 4.4. Let '2 A be w.h.i.s. and ~g 2 A 3
. Suppose that there exist r2 N
and2Fsuhthat
~g
~
r' = 0;
Div (~g) = ' r
: (19)
Proof. AordingtoRemark3.6,theoperator~g7!(~g
~
r';Div(~g))(fromA 3
toA 2
)
restritsforanyd2ZtoanoperatorbetweenX 1
(A)
d andX
0
(A)
d+$(') X
0
(A)
d .
Thereforeit suÆes to provethe lemma foran element~g 2 X 1
(A)
d
, with d2Z.
Suppose that suh anelement~g saties (19), then, aordingto Proposition 3.5,
therstequationimpliesthatthereexists
~
k2X 2
(A)
d $(')
,suhthat~g=
~
k
~
r'.
Wewillapplyindutiononr2N. First,ifr=0,then,aordingtoFormula(12),
=Div (~g)=Div (
~
k
~
r')=(
~
r
~
k)
~
r',sothat=0,fordegreereasons.
Assume now that for some xed r 0, any~g that satises (19) is divergene
free. Suppose that
~
h 2 A 3
satises
~
h
~
r' = 0and Div(
~
h) = 0
' r+1
, for some
0
2 F. Writing
~
h =
~
k
~
r', the Formulas (12), (13) and (14) show that ~g :=
~
r
~
k
0
$(') '
r
~e
$
satises (19), with = 0
($(')r+j$j)=$('), so that, by
indutionhypothesis,0== 0
($(')r+j$j)=$('). Itfollowsthat 0
=0.
Now,wean givethemain resultof thisSetion. Wereallthatj$jis thesum
oftheweightsofthethreevariablesx;y;z.
Proposition4.5. If '2Aisw.h.i.s., thenthe rstPoisson ohomology spae of
(A;f;g
'
) isafreemodule overCas(A;'), given by:
H 1
(A;')' (
f0g if $(')6=j$j;
Cas(A;')~e
$
= L
i2N F'
i
~e
$
if $(')=j$j:
Proof. Let
~
f 2 X 1
(A) be anon zero element of Z 1
(A;'), that is to say,
~
f 2A 3
satisestheequation:
~
r(
~
f
~
r')=Div (
~
f)
~
r':
(20)
AordingtoRemark3.6,wesupposethat
~
f isweighthomogeneous. Ourpurpose
is to write
~
f =
~
rk
~
r'+
$(') '
r
~e
$ 2 B
1
(A;')+ L
i2N F'
i
~ e
$
, where = 0
if$(') 6=j$j and need notbe 0otherwise. Ourproof will bedivided in three
parts.
1:First,usingoyleondition(20),wendanelement~g2A 3
whihsatises
theequations(19). ThisequalityimpliesindeedthatÆ 0
' (
~
f
~
r ')=
~
r(
~
f
~
r')
~
r '=
~
0, sothat the weighthomogeneouselement
~
f
~
r' of Ais a Casimir. Aording
to Proposition 4.2, there exist 2 F and r 2 N suh that
~
f
~
r' = ' r+1
.
UsingEquation(20)onemore,weobtainDiv(
~
f)=(r+1)' r
. Letting~g :=
~
f
$(') '
r
~ e
$
,Formulas(13)and(14)implythat~gsatises(19),where=(1 j$j
$(') ).
Lemma4.4leadsto
(
Div(~g)=0; ~g
~
r'=0;
0=
1 j$j
$(')
:
2:Now,wewillshowthat if~g 2A 3
satises Div (~g)=0and~g
~
r'=0,then
~g2B 1
(A;'). Let~g beasuh element. As~g
~
r'=0,Proposition3.5impliesthe
existeneofanelement
~
h2A 3
suhthat~g=
~
h
~
r'. Moreover,wehave
0=Div (~g)=Div(
~
h
~
r')=(
~
r
~
h)
~
r':
Corollary3.8leadsnowtotheexisteneofelementsk;l2Asuhthat
~
h=
~
rk+l
~
r',
sothat~g=
~
rk
~
r'=Æ 0
(k)2B 1
(A;').
3:Therst twoparts of thisproof leadto theexisteneof k2Aand 2F
suhthat
(
~
f =
~
rk
~
r'+
$(') '
r
~e
$
;
0=
1 j$j
$(')
: (21)
Now,wehavetoonsidertwoases: $(')6=j$jand$(')=j$j.
If$(')6=j$jthen=0and
~
f =
~
r k
~
r '=Æ 0
' (k)2B
1
(A;'). Thus,when
$(')6=j$j,thenH 1
(A;')'f0g.
Now, suppose that $(') = j$j, then (21) leads to Z 1
(A;') B 1
(A;')+
L
i2N F'
i
~ e
$
. Conversely,foranyi2N,Formulas(13)and(14)leadtoÆ 1
' ('
i
~e
$ )=
(j$j $('))' i
~
r'=0. Sothat
Z 1
(A;')=B 1
(A;')+ M
i2N F'
i
~e
$ :
Letusshowthatthis sumisadiretone. ItsuÆestoonsideraweighthomoge-
neouselement' i
~e
$ 2B
1
(A;'), 2F,i2N. Itmeans thatthereexists k2A
suhthat' i
~ e
$
=
~
rk
~
r'. Then(12)and(14)leadto
0=Div (
~
r k
~
r')=Div (' i
~e
$
)=j$j(i+1)' i
;
therefore = 0 and the sum B 1
(A;') L
i2N F'
i
~ e
$
is diret. Thus, when
$(')=j$j,thenH 1
(A;')' L
i2N F'
i
~e
$
.
Remark4.6. Weseethat thease$(')=j$jispartiular. When'ishomoge-
neous(i.e. weighthomogeneouswith$
1
=$
2
=$
3
=1),itistheasewherethe
degreeof'isthree, thatistosay,where'isaubipolynomial.
4.3. The spae H 3
(A;'). Now, we give thethird Poisson ohomology spae of
(A;f;g
'
),where '2A=F[x;y;z℄isw.h.i.s. Reallthat, inthisase,
A
sing
=F[x;y;z℄=h '
x
; '
y
; '
z i
is a nite dimensional F-vetor spae, whose dimension is the Milnor number,
denotedby. Letu
0
=1;u
1
;:::;u
1
beweighthomogeneouselementsofA,suh
thattheirimagesin A
sing
giveaF-basisofA
sing .
Proposition4.7. If'2A=F[x;y;z℄isw.h.i.s. thenthe thirdohomologyspae
H 3
(A;') isthefreeCas(A;')-module:
H 3
(A;')' 1
M
j=0
Cas(A;')u
j
'Cas(A;')
F A
sing :
Proof. Letf 2A'X 3
(A) beaweighthomogeneouspolynomialofdegreed2N.
1:Werstshowthatthereexist~g2A 3
,N 2Nandelements
i;j
2F,where
0iN and0j 1,suhthat:
f =
~
r'(
~
r~g)+ N
X
i=0 1
X
j=0
i;j '
i
u
j 2B
3
(A;')+ X
k 2N
0j 1
F' k
u
j : (22)
Let$ [1℄
:=max($
1
;$
2
;$
3
). Weapplyindutionond,provingdiretly theresult
[1℄
example, itontains theasef 2F). By denitionof the elements u
0
;:::;u
1 ,
wehave:
f =
~
r'
~
l+ 1
X
j=0
j u
j
; (23)
where
~
l2X 1
(A)
d $(') and
0
;:::;
1 2F.
If d $(') $ [1℄
then the orrespondenes (15) imply that
~
l is an element
(a;b;)ofF 3
sothatf isindeedoftheform (22),with~g=(bz;x;ay),N =0and
0;j
=
j .
Now,supposethatd>$(') $ [1℄
andthatanyweighthomogeneouspolynomial
ofdegreeatmostd 1isoftheform(22). Letusonsiderthedeomposition(23)
forf ofdegreed. Proposition3.5impliesthat thereexists~g2A 3
suhthat:
~
l
Div(
~
l )
d $(')+j$j
~e
$
=
~
r~g;
(24)
sineDiv
~
l
Div (
~
l)
d $(')+j$j
~e
$
=0,asfollowsfrom$(Div(
~
l))=d $(')and(14).
Using the indution hypothesis on Div (
~
l), weonlude that (23), with
~
l given
by (24), is indeed of the form (22) (one uses that, aording to Formula (10),
'(
~
r
~
k)
~
r'=(
~
r('
~
k))
~
r',for
~
k2A 3
).
2:So,wehavealreadyobtainedthat
(25)
A = f
~
r'(
~
r
~
l)j
~
l2A 3
g+ 1
X
j=0
Cas(A;')u
j
= B
3
(A;')+ 1
X
j=0
Cas(A;')u
j :
anditsuÆestoshowthatthis sumisdiret inA'X 3
(A).
Wesupposetheontrary. ThisallowsustoonsiderthesmallestintegerN
0 2N
suhthatwehaveanequationoftheform:
N
X
i=N
0 1
X
j=0
i;j '
i
u
j
=
~
r '(
~
r~g)= Æ 2
' (~g);
(26)
with ~g 2 A 3
, N N
0 and
i;j
2 F (for N
0
i N and 0 j 1) and
N
0
;j
0
6=0,forsome0j
0
1. Wewill showthat thishypothesisleadsto a
ontradition.
First,suppose that N
0
=0,thenthedenition of theu
j
, Euler'sFormula(13)
and(26)implythat
0;j
=0forall0j 1,whihontraditsthehypothesis
N0;j0 6=0.
Sowesuppose that N
0
>0,using Euler's Formula(13), theequation(26) an
bewrittenas
~
r' P
N
i=N0 P
1
j=0
i;j
$(') '
i 1
u
j
~e
$
!
=
~
r'(
~
r~g). Proposition3.5
impliesthatthereexists
~
h2A 3
suhthat:
N
X
i=N 1
X
j=0
i;j
$(') '
i 1
u
j
~e
$
=
~
r~g+
~
h
~
r':
ThedivergeneofbothsidesofthisequalityandFormula(14)give:
N
X
i=N1 1
X
j=0
0
i;j '
i
u
j
=(
~
r
~
h)
~
r'= Æ 2
' (
~
h);
where 0
i;j
= i+1;j
$(')
($(')i+$(u
j
)+j$j)andN
1
=N
0
1. So,wehaveobtained
anequationoftheform (26),withN
1
<N
0 and
0
N
1
;j
0
6=0. Thisfat ontradits
the hypothesis and we onlude that the sum (25) is diret. The desription of
H 3
(A;')follows.
4.4. The spae H 2
(A;'). Finally, using Proposition 4.7 (and in fat the writ-
ing of H 3
(A;')), we obtainthe seond Poisson ohomology spae of the algebra
(A;f;g
'
),when'2A=F[x;y;z℄isw.h.i.s.
Proposition 4.8. If '2 A=F[x;y;z℄ is w.h.i.s. then the seondPoisson oho-
mologyspaeof thealgebra(A;f;g
'
) istheCas(A;')-module:
H 2
(A;') '
1
M
j=1
$(uj)6=$(') j$j
Cas(A;')
~
r u
j
1
M
j=0
$(uj)=$(') j$j
Cas(A;')u
j
~
r'
1
M
j=1
$(u
j
)=$(') j$j F
~
ru
j
;
wherethe rstrow givesthe free part.
In partiular, we have: H 2
(A;') ' L
1
j=1
Cas(A;')
~
r u
j
, if $(') < j$j and
H 2
(A;')' L
1
j=1
Cas(A;')
~
r u
j
Cas(A;')
~
r ', when$(')=j$j.
Remark 4.9. We see that thePoisson struture f;g
'
will beexat (that is to
saya2-oboundary)ifandonlyif$(')6=j$j. This fatomes fromtheequality
Æ 1
' (~e
$
)= ($(') j$j)
~
r',aonsequeneofFormulas(13)and(14).
Remark 4.10. Contraryto the other ohomology spaes, H 2
(A;') is generally
notafreeCas(A;')-module. Infat,usingFormulas(13)and(14),weget:
Æ 1
' '
i
u
j
~ e
$
=( $(u
j
) $(')+j$j)' i
u
j
~
r ' $(')' i+1
~
r u
j : (27)
Thisequality,whihwillbealsousefullater,explainsthatwehavetodistinguish,
intheexpressionofH 2
(A;'),theu
j
satisfying$(u
j
)=$(') j$jfromtheother
ones. Ifjissuhthat$(u
j
)=$(') j$jthen(27)yieldsthat' k
~
ru
j 2B
2
(A;'),
forallk1,butthisisnottruewhen$(u
j
)6=$(') j$j. Thisisthereasonwhy
H 2
(A;')isnotalwaysafreemodule overCas(A;').
Moreover,for all j satisfying $(u
j
) 6= $(') j$j, (27) impliesthat ' i
u
j
~
r',
i0,anbewritten as' i+1
~
ru
j +Æ
1
0
' i
u
j
~e
$
,with; 0
2F f0g.
Proof. First,letusshowthat:
(28) Z
2
(A;') ' B 2
(A;')+
1
X
j=1
$(uj)6=$(') j$j
Cas(A;')
~
r u
j
+
1
X
j=0
$(u
j
)=$(') j$j
Cas(A;')u
j
~
r'+
1
X
j=1
$(u
j
)=$(') j$j F
~
r u
j :
Let
~
f 2Z 2
(A;'). AordingtoRemark3.9, thereexistsg;h2Asuhthat
~
f =
~
rg+h
~
r':
(29)
Moreover, Proposition 4.7 implies the existene of ~g
1
;
~
h
1 2 A
3
, N 2 N and of
elements
i;j
;Æ
i;j
2F,with 0iN and0j 1,suh that:
g=Æ 2
' (~g
1 )+
N
X
i=0 1
X
j=0
i;j '
i
u
j
; h=Æ 2
' (
~
h
1 )+
N
X
i=0 1
X
j=0 Æ
i;j '
i
u
j
; (30)
whilewehavethe2-oboundaries:
~
r(Æ 2
' (~g
1 )) =
~
r((
~
r~g
1 )
~
r')=Æ 1
' (
~
r~g
1 )2B
2
(A;');
Æ 2
' (
~
h
1 )
~
r ' =
(
~
r
~
h
1 )
~
r '
~
r'=Æ 1
' (
~
h
1
~
r')2B 2
(A;'):
Usingthisfat,(29)and(30),weobtain
~
f 2B 2
(A;')+ 1
X
j=1
Cas(A;')
~
ru
j +
1
X
j=0
Cas(A;')u
j
~
r':
Remark 4.10 then implies that
~
f an be deomposed as in the right hand side
of(28). Ontheotherhand,allelementsoftherighthandsideof(28)are2-oyles,
yielding equality in (28). (Indeed, using Formula (10), we have, forall f;g 2 A,
Æ 2
' ('
~
r f)=
~
r'(
~
r('
~
rf))=0andÆ 2
' (g
~
r')=
~
r'(
~
r(g
~
r'))=0).
For theproofthat thesumin(28)isadiret one,oneusesthedenitionofthe
u
j
and applies Propositions 3.5, 4.2 (expressionof H 0
(A;')) and 4.7 (writing of
H 3
(A;'))asintheproofsofPropositions4.5and4.7.
Remark 4.11. Using Euler's Formula (13) and the writings of the Poisson o-
homology spaes H 1
(A;') and H 2
(A;') given in Propositions 4.5 and 4.8, we
an make the ring struture on the spae H
(A;') :=
L
3
k =0 H
k
(A;'); indued
by the wedge produt, expliit. One obtains, for example, that ^ : H 1
(A;')
H 2
(A;') !H 3
(A;') issurjetivewhen$(')=j$j.
5. Poissonohomologyof the singular surfae
Inthis hapter,we still onsider anelement'2 A=F[x;y;z℄ (har(F) =0),
whihisw.h.i.s.(weighthomogeneouswithanisolatedsingularity)andwerestrit
thePoissonstruturef;g
'
tothesingularsurfaeF
'
:f'=0gandomputethe
5.1. ThePoissonomplexofthesingularsurfae F
'
. Thealgebraofregular
funtions onthesurfaeF
'
isthequotientalgebra:
A
' :=
F[x;y;z℄
h'i :
Beause ' is a Casimir, h'i is a Poisson ideal for (A;f;g
'
) and the Poisson
struturef;g
'
restritsnaturallytoF
'
,thatistosaygoesdownto thequotient
A
'
. That leadstoaPoissonstrutureon A
'
, denotedbyf;g
A
'
. Letus denote
by the natural projetion map A ! A
'
, then, for eah f;g 2 A, we have
f(f);(g)g
A
'
=
ff;gg
'
(that isto say, is aPoisson morphismbetweenA
andA
' ).
Denition 5.1. Wesaythat P 2X k
(A) and Q2 X k
(A
'
) are -related and we
writeQ=
(P)if
(P[f
1
;;f
k
℄)=Q[(f
1
);;(f
k )℄;
(31)
forallf
1
;;f
k 2A.
In thefollowingproposition, we givethe Poisson ohomologyspaes of theal-
gebra(A
'
;f;g
A
'
). That leadsto onsidertheskew-symmetrimulti-derivations
of the algebra A
'
and the Poisson oboundary operators, assoiated to f;g
A
' .
Theprevious denition will be usefulin this disussion. Byaslightabuseof no-
tationswewill, foran element
~
f =(f
1
;f
2
;f
3 )2A
3
, denote by(
~
f), theelement
((f
1 );(f
2 );(f
3 ))2A
3
' .
Proposition5.2. If'2Aisw.h.i.s.,thePoissonohomologyspaesofthealgebra
(A
'
;f;g
A'
), denotedbyH k
(A
'
),are given by:
Cas(A
'
) = H
0
(A
' )'
n
(f)2A
' j
~
rf
~
r'2h'i o
;
H 1
(A
' ) '
n
~
f
2A 3
' j
~
f
~
r'2h'i and
~
r(
~
f
~
r')+Div(
~
f)
~
r'2h'i o
n
~
rf
~
r'
jf 2A o
;
H 2
(A
' ) '
n
~
f
2A 3
' j
~
f
~
r'2h'i o
n
~
r(
~
f
~
r')+Div(
~
f)
~
r '
j
~
f 2A 3
;
~
f
~
r'2h'i o
;
andH 3
(A
'
)'f0g.
Subsequently, we denote by Z k
(A
'
) (respetively B k
(A
'
)) the spae of all k-
oyles(respetivelyk-oboundaries)ofA
' .
Proof. We rst have to determine the skew-symmetri multi-derivations of A
' .
Letus pointoutthat anyP 2X k
(A) is-relatedto aQ2X k
(A
'
)ifand onlyif
P[';f
2
;:::;f
k
℄2h'i, forallf
2
;:::;f
k
2A. Inthisase,theequality(31)denes
indeedanelementQof X k
(A
'
),in viewoftheskew-symmetryandthederivation
properties of P. Moreover, every Q 2 X k
(A
'
) is obtained in this way. Let us
Let Q 2X 1
(A
'
) and let us hoose
~
f =(f
1
;f
2
;f
3 ) 2 A
3
suh that Q[(x)℄ =
(f
1
), Q[(y)℄ = (f
2
) and Q[(z)℄ = (f
3
). Then, we get Q =
(P), with
P =f
1
x +f
2
y +f
3
z 2X
1
(A)andP['℄=f
1 '
x +f
2 '
y +f
3 '
z
=
~
f
~
r '2h'i.
Conversely, eah of (
~
f) 2 A 3
'
satisfying the equation
~
f
~
r' 2 h'i gives an
elementofX 1
(A
'
),denedby
f
1
x +f
2
y +f
3
z
. Thus,
X 1
(A
' )'f(
~
f)2A 3
' j
~
f
~
r'2h'ig:
With thesamereasoning,weobtain
X 2
(A
' )'f(
~
f)2A 3
' j
~
f
~
r'2h'ig:
AsitislearthatX 0
(A
' )'A
' andX
k
(A
'
)'f0g,fork4,letusnowonsider
the spae X 3
(A
'
). Inthe sameway that above, we getX 3
(A
'
) =f(f)2A
' j
f
~
r'2 h'ig. However,iff 2 Asatises f
~
r' ='~g, with ~g 2A 3
, then we have
~g
~
r'=
~
0andProposition3.5impliestheexisteneofanelementh2Asatifying
~g=h
~
r'sothatf =h'2h'i. ThatleadstoX 3
(A
'
)'f0g:
Now, letus onsider thePoisson oboundary operators of the Poisson algebra
(A
'
;f;g
A
'
), denoted by Æ k
A
'
. Using thedenition of Æ k
A
'
(similarly as(7)), we
obtain,forallP 2X k
(A),Æ k
A' (
(P))=
(Æ
k
'
(P)). Thatleadsto:
Æ 0
A'
((f)) =
~
rf
~
r'
; for(f)2A
' 'X
0
(A
' );
Æ 1
A
' ((
~
f)) =
~
r(
~
f
~
r')+Div(
~
f)
~
r'
;
for(
~
f)2f(~g)2A 3
' j~g
~
r'2h'ig'X 1
(A
' );
Æ 2
A' ((
~
f)) = 0; for(
~
f)2f(~g)2A 3
' j~g
~
r'2h'ig'X 2
(A
' );
whilethewritingofthePoissonohomologyspaesfollows.
5.2. The spae H 0
(A
'
). In this Setion, we onsider still ' 2 A w.h.i.s. and
thePoissonstrutureonA
'
, denotedbyf;g
A
'
. WedesribethezerothPoisson
ohomologyspae,thatis tosaythespaeoftheCasimirsof(A
'
;f;g
A
' )inthe
followingProposition.
Proposition5.3. If '2A=F[x;y;z℄isw.h.i.s., the zeroth Poisson ohomology
spaeof thesingular surfaedenedbythis polynomialisgiven by
H 0
(A
'
)=Cas(A
' )'F:
Proof. Letf 2Abeaweighthomogeneouspolynomialsuhthat(f)2H 0
(A
' ).
Then
~
rf
~
r'2h'ii.e.,there exists~g2A 3
satifying
~
rf
~
r'='~g. Itfollows
that~g
~
r'=0andProposition3.5impliestheexisteneofanelement
~
h2A 3
suh
that~g=
~
h
~
r'. Summingup,(
~
r f '
~
h)
~
r'=
~
0,andweanapplyProposition
3.5againtoobtainak2Asatifying
~
rf='
~
h+k
~
r'. Euler'sFormula(13)gives
$(f)f =
~
rf~e
$
='(
~
h~e
$
+$(')k):
So,f 2h'iunless$(f),the(weighted)degreeoff,iszero,thusH 0
(A )'F.
5.3. The spaeH 1
(A
'
). Thissetionisdevotedtothedeterminationoftherst
Poissonohomologyspaeof(A
'
;f;g
A
'
),where'2A=F[x;y;z℄isw.h.i.s.
Remark 5.4. UsingProposition5.3, weansimplify thewritingofZ 1
(A
' ). Let
indeed
~
f 2 A 3
be an element satisfying:
~
r(
~
f
~
r')+Div(
~
f)
~
r' 2 h'i. Then
~
r(
~
f
~
r')
~
r' 2 h'i, that is to say (
~
f
~
r') 2 H 0
(A
'
) ' F, aording to
Proposition 5.3. Fordegreereasons,this leadsto
~
f
~
r'2h'i. So,weansimply
write
Z 1
(A
' )=
n
(
~
f)2A 3
' j
~
r(
~
f
~
r')+Div (
~
f)
~
r '2h'i o
Now,letusgivethemainresultofthissetion(wereallthat j$jisthesumof
theweights$
1
;$
2
;$
3
ofthevariablesx;y;zandthatthefamilyfu
j
gisanF-basis
ofA
sing
andisdened inSetion4.3).
Proposition5.5. If'2A=F[x;y;z℄isw.h.i.s.thentherstPoissonohomology
spaeof thesingular surfaef'=0gisgiven by
H 1
(A
' )'
1
M
j=0
$(uj)=$(') j$j F(u
j
~e
$ ):
In partiular, if $(')<j$jthen H 1
(A
'
)'f0g.
Proof. Let
~
f 2A 3
satisfy
~
f
2Z 1
(A
'
),itmeansthatthereexists
~
k2A 3
satis-
fying Æ 1
' (
~
f)='
~
k. Then 0=Æ 2
' ('
~
k)='Æ 2
' (
~
k),beause,aswesaidinSetion 2.1,
the operator Æ 2
'
ommutes with the multipliation by '. So '
~
k 2 B 2
(A;') and
~
k2Z 2
(A;'). Aordingto Proposition4.8,
~
k 2 B
2
(A;')
1
M
j=1
$(uj)6=$(') j$j
Cas(A;')
~
r u
j
1
M
k =0
$(u
k
)=$(') j$j
Cas(A;')u
k
~
r'
1
M
l=1
$(u
l
)=$(') j$j F
~
ru
l :
Eah of therst three summands isstable by multipliation by', while Remark
4.10gives
1
M
l=1
$(ul)=$(') j$j 'F
~
ru
l B
2
(A;'):
Asaonsequene,sine'
~
k2B 2
(A;'),
~
k2B 2
(A;')
1
M
l=1
$(ul)=$(') j$j F
~
ru
l :
Sothereexist
~
h2A 3
andelements
l
2F,with lsatisfying$(u
l
)=$(') j$j,
suhthat
~
k=Æ 1
' (
~
h)+
1
X
l=1
$(ul)=$(') j$j
l
~
ru
l :
Forall1l 1suhthat$(u
l
)=$(') j$j,wehave'
~
ru
l
= Æ 1
'
1
$(') u
l
~ e
$
,
sothat
Æ 1
' (
~
f)='
~
k=Æ 1
' 0
B
B
'
~
h
1
X
l=1
$(u
l
)=$(') j$j
l
$(') u
l
~e
$ 1
C
C
A :
Thisimplies
~
f '
~
h+
1
X
l=1
$(ul)=$(') j$j
l
$(') u
l
~e
$ 2Z
1
(A;'):
(32)
If $(')6=j$j,then Proposition4.5impliesthat (32)belongsto B 1
(A;'),so
that
(
~
f)2
1
X
l=1
$(ul)=$(') j$j F(u
l
~ e
$ )+B
1
(A
' ):
If$(')=j$jthen(32)issimplytheequation
~
f '
~
h2Z 1
(A;')'B 1
(A;')+
Cas(A;')~e
$
,aordingtoProposition4.5. So,wehave(
~
f)2F(~e
$ )+B
1
(A
' ).
Aswehave$(u
l
)1,if1l 1,theresultofbothasesanbesummarized
asfollows:
Z 1
(A
' )B
1
(A
' )+
1
X
l=0
$(ul)=$(') j$j F(u
l
~e
$ ):
Euler's Formula (13) implies that (u
l
~e
$ ) 2 Z
1
(A
' ) (Æ
1
' (u
l
~e
$
) 2 h'i), when
$(u
l
)=$(') j$j,sothattheotherinlusionholdstoo. Italsoallowsustoshow
thattheabovesumisadiretone. Hene theresultaboutH 1
(A
'
).
5.4. The spae H 2
(A
'
). WenowomputetheseondPoissonohomologyspae
of(A
'
;f;g
A
'
),where'2A=F[x;y;z℄isw.h.i.s.
Proposition5.6. If '2A=F[x;y;z℄ isw.h.i.s. thenH 2
(A
'
)is given by
H 2
(A
' )'
1
M
j=0
$(uj)=$(') j$j F(u
j
~
r'):
Remark 5.7. It follows from Propositions 5.5 and 5.6 that there is a natural
isomorphism between H 1
(A
'
) and H 2
(A
'
), that maps the element u
j
~e
$ (with
$(u
j
)=$(') j$j)totheelementu
j
~
r'ofH 2
(A
' ).
Proof. First,weshowthatthefamily n
(u
j
~
r')j$(u
j
)=$(') j$j o
generates
theF-vetorspae H 2
(A
' ). Let
~
h2A 3
suh that (
~
h) 2Z 2
(A
'
),that is tosay,
suhthat there exists~g 2A 3
satisfying
~
h
~
r'='~g. Aordingto Remark 3.6,
we may suppose
~
h 2 X 2
(A)
d
and ~g 2 X 1
(A)
d
, with d 2 Z. Sine ~g
~
r' = 0,
Proposition3.5impliesthat~g=
~
k
~
r'and
~
h='
~
k+f
~
r',withf 2X 3
(A)
d $(')
and
~
k2X 2
(A)
d $(') .
Ifd<$(') j$jthenf=0and
~
h2h'i;otherwise(
~
h )=(f
~
r'),while,using
Formulas(13)and (14), weget Æ 1
(f~e
$
)=(d 2$(')+2j$j)f
~
r' $(')'
~
rf.
Thatleads,intheased6=2($(') j$j),to(
~
h)=(f
~
r')2B 2
(A
'
). Therefore,
let us suppose that d= 2($(') j$j), sothat $(f) = $(') j$j. For degree
reasons,theprojetionmap A!A
sing
=A=h '
x
; '
y
; '
z
irestritsto aninjetive
map A
$(') j$j
!A
sing
, sothat f is aF-linearombination ofthe u
j
satisfying
$(u
j
)=$(') j$j,thatleadsto
(
~
h )2
1
X
j=0
$(uj)=$(') j$j F(u
j
~
r');
andforallj,u
j
~
r '2Z 2
(A
' ).
ItsuÆesnowtoshowthatthisfamilyisF-free,moduloB 2
(A
'
). Itisemptyif
$(')<j$j,sowesuppose$(')j$j. Let
j
beelementsof Fwith j suh that
$(u
j
)=$(') j$jandlet
~
l ;~|2A 3
satisfying
(33)
1
X
j=0
$(u
j
)=$(') j$j
j u
j
~
r' =
~
r(
~
l
~
r')+Div(
~
l)
~
r'+'~|
= Æ 1
' (
~
l )+'~|;
where therighthand sideis anarbitraryrepresentativeof anelement ofB 2
(A
' ).
Asthe lefthand sidebelongs tothespae X 2
(A)
2$(') 2j$j
,wemaysuppose that
~
l2X 1
(A)
$(') j$j
and~|2X 2
(A)
$(') 2j$j .
Theequation(33)implies
~
r(
~
l
~
r ')
~
r '2h'i,sothat
~
l
~
r '
2Cas(A
' ). For
degreereasons,Proposition5.3leadstotheexisteneofg2Aofdegree$(') j$j
suh that
~
l
~
r ' = 'g = (g~e
$
~
r')=$('). Then Proposition 3.5 implies that
$(')
~
l=g~e
$ andÆ
1
' (
~
l )= '
~
rg,sothat
1
X
j=0
$(u
j
)=$(') j$j
j u
j
~
r'= '
~
rg+'~|='
~
F;
(34)
where
~
F=
~
rg+~|2X 2
(A)
$(') 2j$j
. Weget
~
F
~
r'=
~
0,butfordegreereasons,
Proposition 3.5 leadsto
~
F =
~
0so that, forall j,
j
=0, sinethefamily fu
j gif
F-freeinA.
6. Poissonhomology assoiated to aweight homogeneous polynomial
with an isolated singularity
Inthislast hapter,weonsider thealgebrasA=F[x;y;z℄(with har(F)=0)
andA
'
=A=h'i,where'2Aisweighthomogeneouswithanisolatedsingularity
(w.h.i.s.). ThesealgebrasarestillrespetivelyequippedwiththePoissonstrutures
f;g
'
and f;g
A'
. We use the Poisson ohomology of these Poisson algebras
(A;f;g
'
)and(A
'
;f;g
A
'
),givenintheprevioushapters4and5,todetermine
theirPoissonhomology.
