a Xi :1201.3594 4 [ma h.AG] 25 Jun 2015
A duali y app oach o he symme y o
Be ns ein-Sa o polynomials o ee di iso s
Luis Na áez Maca o∗
Depa amen o de Álgeb a & Ins i u o de Ma emá icas (IMUS)
Uni e si y o Se illa
June 2015
Abs ac
In his pape we p o e ha he Be ns ein-Sa o polynomial o any ee
di iso o which he D[s]-module D[s]hsadmi s a Spence loga i hmic
esolu ion sa is ies he symme y p ope y b(−s−2) = ±b(s). This applies
in pa icula o locally quasi-homogeneous ee di iso s ( o ins ance, o
ee hype plane a angemen s), o mo e gene ally, o ee di iso s o linea
Jacobian ype. We also p o e ha he Be ns ein-Sa o polynomial o an
in eg able loga i hmic connec ion Eand o i s dual E∗wi h espec o a
ee di iso o linea Jacobian ype a e ela ed by he equali y bE(s) =
±bE∗(−s−2). Ou esul s a e based on he beha iou o he modules
D[s]hsand D[s]E[s]hsunde duali y.
Keywo ds: Be ns ein-Sa o polynomials, ee di iso s, loga i hmic di e -
en ial ope a o s, Spence esolu ions, Lie-Rineha algeb as, loga i hmic
connec ions.
MSC: 14F10, 32C38
In oduc ion
In [16] G ange and Schulze p o ed ha he Be ns ein-Sa o polynomial o any
educ i e p ehomogeneous de e minan o o any egula special linea ee di-
iso sa is ies he equali y b(−s−2) = ±b(s). Thei p oo is based on Sa o’s
undamen al heo em o i educible educ i e p ehomogeneous spaces. This
symme y p ope y has been also checked o many o he examples o linea (see
o ins ance [16] and [32]) and non-linea ee di iso s (e.g. quasi-homogeneous
plane cu es and he examples in [27]). In his pape we p o e he abo e sym-
me y p ope y o ee di iso s o which he D[s]-module D[s]hsadmi s a log-
a i hmic Spence esolu ion (see Theo em (4.1) o a p ecise s a emen ). This
hypo hesis holds o any ee di iso o linea Jacobian ype, and so o any
locally quasi-homogeneous ee di iso ( o ins ance, ee hype plane a ange-
men s o disc iminan s o s able maps [22, Co olla y 6.13] in Ma he ’s “nice
dimensions” [24]; “nice dimensions” a e hose dimensions o sou ce and a ge
mani olds o which s able p ope mappings a e dense in he p ope mappings).
∗Pa ially suppo ed by MTM2010-19298, P12-FQM-2696, MTM2013-46231-P and
FEDER.
1
The main ing edien o he p oo is he explici desc ip ion o he D[s]-dual
o D[s]hsby means o he loga i hmic duali y o mula in [7, 8].
Le us men ion ha o any quasi-homogeneous ge m h: (Cd,0) →(C,0)
wi h isola ed singula i y, i s educed Be ns ein-Sa o polynomial eb(s) = b(s)
s+1
sa is ies he equali y eb(s) = ±eb(−s−d). This esul and ou s sugges ha bo h
a e ex emal cases o a whole amily o “pu e” cases whe e symme y p ope ies
occu wi h o he in e media e shi ings (see Ques ion (6.2)). One can expec
e en ha in he “non-pu e” cases, he ac o s o he Be ns ein-Sa o polynomial
which b eak he symme y appea as minimal polynomials o he ac ion o son
o he D[s]-modules a ached o ou singula i y (see o ins ance he examples
in [28, §3]), possibly ela ed wi h he mic olocal s uc u e.
Le us now commen on he con en o he pape .
In sec ion §1 we ecall he di e en condi ions and hypo heses on ee di i-
so s we will use h oughou he pape . In sec ion §2 we ecall he loga i hmic
Be ns ein cons uc ion and we s udy he hypo heses we will need la e o p o e
ou main esul s. In sec ion §3 we apply he duali y o mula in [8] o desc ibe
he D[s]-dual o D[s]hϕ(s), whe e ϕis a C-algeb a au omo phism o C[s], unde
he hypo heses s udied in sec ion §2. In sec ion §4 we p o e he symme y p op-
e y b(−s−2) = ±b(s)unde he abo e hypo heses. The idea o he p oo is
he ollowing: once we know ha he D[s]-dual o D[s]hs( esp. o D[s]hs+1) is
concen a ed in deg ee 0and is isomo phic o D[s]h−s−1( esp. o D[s]h−s−2),
we can compu e he D[s]-dual o he exac sequence
0→D[s]hs+1 →D[s]hs→Q:= (D[s]hs)/D[s]hs+1→0
and deduce ha he D[s]-dual o Qis concen a ed in deg ee 1and is isomo -
phic o D[s]h−s−2/D[s]h−s−1. F om he e he symme y p ope y comes up. A
he end o he sec ion we gi e some applica ions o he loga i hmic compa i-
son p oblem and a cha ac e iza ion o he loga i hmic compa ison heo em o
Koszul ee di iso s. In sec ion §5 we gene alize he abo e esul s o he case
o in eg able loga i hmic connec ions wi h espec o ee di iso s o linea Ja-
cobian ype. In sec ion §6 we ha e included some open ques ions dealing wi h
he ela ionship be ween he esul s o [16] and ou s, and wi h he symme y
p ope ies o ( educed) Be ns ein-Sa o polynomials in he non- ee case. Finally,
and o he ease o he eade , we ha e included an Appendix A wi h a de ailed
p oo o he duali y o mula (A.32), and he needed no ions and esul s abou
Lie-Rineha algeb as. This ma e ial includes a simple p oo o he associa i i y
law (see Theo em (A.11) and Co olla y (A.14) ), he o iginal p oo in [8] being
unpleasan .
I would like o hank F ancisco Cas o, Michel G ange and Ma hias Schulze
o use ul discussions and commen s. I would also like o hank he e e ees o
hei commen s.
1 No a ions and linea i y condi ions
In his pape Xwill deno e a complex mani old o pu e dimension d,D⊂Xa
hype su ace (= di iso ), OX[⋆D] he shea o me omo phic unc ions along D,
2
OX(D) he shea o me omo phic unc ions along Dwi h poles o o de ≤1and
DX he shea o linea di e en ial ope a o s wi h coe icien s in OX. On DX[s]
we will conside wo il a ions: he one induced by he usual o de il a ion
DX[s]i=Di
X[s], and he o al o de il a ion Fi
TDX[s] = ⊕p+q=iDp
Xsq,i≥0.
The associa ed g aded ings o bo h il a ions a e isomo phic o (g DX)[s], bu
he co esponding g adings on he las shea a e di e en .
We will also deno e by J
D⊂OX he Jacobian ideal o D⊂X, i.e. he cohe -
en ideal o OXwhose s alk a any p∈Xis he ideal gene a ed by h, h′
x1,...,h′
xd,
whe e h∈OX,p is any educed local equa ion o Da pand x1,...,xd∈OX,p is
a sys em o local coo dina es cen e ed a p.
We ecall ha Dis a ee di iso , in he sense o K. Sai o [31], i he
cohe en OX-module De C(−log D)o loga i hmic ec o ields wi h espec
o Dis locally ee (o ank d). In such a case we will deno e by V
X=
OX[De C(−log D)] ⊂DX he shea o loga i hmic di e en ial ope a o s wi h
espec o D[4].
(1.1) De ini ion. (C . [37, §7.2]) Le Abe a commu a i e ing and I⊂Aan
ideal. We say ha Iis o linea ype i he canonical (su jec i e) map o g aded
A-algeb as SymA(I)→Rees(I)is an isomo phism.
In he abo e de ini ion, i I= (a1,...,a )and (si1,...,si ),i∈L, is
a sys em o gene a o s o he syzygies o a1,...,a , o say ha he ideal I
is o linea ype is equi alen o saying ha any homogeneous polynomial
F(ξ1,...,ξ )∈A[ξ]such ha F(a1,...,a ) = 0 is a linea combina ion wi h
coe icien s in A[ξ]o he linea o ms si1ξ1+···+si ξ ,i∈L.
(1.2) Example. ([26, Théo ème 1]; see also [6, P oposi ion 2.4]) An ideal
gene a ed by a egula sequence is o linea ype.
(1.3) De ini ion. (See [9, De ini ions 1.11, 1.14].) (a) We say ha he di iso
Dis o linea Jacobian ype a p∈Di J
D,p ⊂OX,p is o linea ype. We say
ha Dis o linea Jacobian ype i i is so a any p∈D.
(b) We say ha he di iso Dis o di e en ial linea ype a p∈Di o
some (and hence o any) educed local equa ion h∈OX,p o Da p, he ideal
annDX,p[s]hsis gene a ed by o de 1 ope a o s (wi h espec o he usual o o
he o al o de il a ion). We say ha Dis o di e en ial linea ype i i is so
a any p∈D.
The ollowing P oposi ion is p o en in [9, P oposi ion 1.15].
(1.4) P oposi ion. Any di iso o linea Jacobian ype is o di e en ial linea
ype.
(1.5) De ini ion. (a) We say ha he di iso Dis s ongly Eule homoge-
neous i o any p∈Dand o some (and hence o any) educed local equa ion
h∈OX,p o Da p he e is a ge m o ec o ield χa p anishing a psuch
ha χ(h) = h.
(b) We say ha he di iso Dis locally quasi-homogeneous i o any p∈D
he e is a sys em o local coo dina es x= (x1,...,xd)cen e ed a psuch ha he
ge m (D, p)has a educed weigh ed homogeneous de ining equa ion (wi h s ic ly
posi i e weigh s) wi h espec o x.
3
The ollowing heo em has been p o en in [6, Theo em 5.6].
(1.6) Theo em. Any locally quasi-homogeneous ee di iso is o linea Jaco-
bian ype.
We do no know any example o a ee di iso o linea Jacobian ype which
is no locally quasi-homogeneous.
(1.7) Rema k. The e is also he no ion o Eule homogenei y. Namely, we
say ha he di iso Dis Eule homogeneous a a poin p∈Di he e is a
educed local equa ion h∈OX,p o Da pand a ge m o ec o ield χa p
(no necessa ily anishing a p) such ha χ(h) = h. In ha case we also say
ha his Eule homogeneous. I is clea ha i Dis Eule homogeneous a p,
hen i is also Eule homogeneous a any poin q∈Dclose enough o p. No ice
ha no any local educed equa ion o an Eule homogeneous di iso is Eule
homogeneous. No ice also ha o any di iso D⊂X, which migh no be
Eule homogeneous, he di iso D′=D×C⊂X′=X×Cis always Eule
homogeneous. Ne e heless, a di iso D⊂Xis s ongly Eule homogeneous i
and only i D′=D×C⊂X′=X×Cis s ongly Eule homogeneous. Le us
also no ice ha a di iso D⊂Xis o linea Jacobian ype a p∈Di and only
D′=D×C⊂X′=X×Cis o linea Jacobian ype a (p, 0) ∈D′.
(1.8) Example. Any smoo h hype su ace is o linea Jacobian ype. Mo e
gene ally, any quasi-homogeneous (wi h s ic ly posi i e weigh s) isola ed sin-
gula i y is o linea Jacobian ype, since any educed equa ion hbelongs o he
ideal gene a ed by he pa ial de i a i es and so he Jacobian ideal is gene a ed
by he egula sequence h′
x1,...,h′
xd(see Example (1.2)).
(1.9) P oposi ion. I he di iso Dis o linea Jacobian ype, hen i is
s ongly Eule homogeneous.
P oo . Le h∈OX,p be a educed local equa ion o (D, p)and x= (x1,...,xd)
a sys em o local coo dina es cen e ed a p. We ecall he a gumen in [9, Rema k
1.26 (a)] o he ease o he eade . Since hbelongs o he in eg al closu e o he
ideal I= (h′
x1,...,h′
xd)(c . [33, §0.5, 1]), he e is a homogeneous polynomial
F∈OX,p[s, ξ1, . . ., ξd]o deg ee m > 0such ha F(h, h′
x1,...,h′
xd) = 0 and
F(s, 0,...,0) = sm. Le
δi=
d
X
j=1
aij
∂
∂xj
,1≤i≤n
be a sys em o gene a o s o De (−log D)pand le us w i e δi(h) = αih. In
o he wo ds, (−αi, ai1,...,aid),1≤i≤n, is a sys em o gene a o s o he
syzygies o h, h′
x1,...,h′
xd. F om ou hypo hesis, he polynomial Fmus be a
linea combina ion o he polynomials
−αis+ai1ξ1+···+aidξd,1≤i≤m
wi h coe icien s in OX,p[s, ξ]. Pu ing ξ1=···=ξd= 0 we deduce ha a leas
one o he αimus be a uni , i.e. h∈(h′
x1,...,h′
xd)and his Eule homogeneous.
Once we know ha his Eule homogeneous, le us p o e ha his s ongly
Eule homogeneous by induc ion on he ambien dimension d. The case d= 1 is
4
ob ious. Fo d > 1, i he Eule ec o ield χ(χ(h) = h) does no anish a p,
we can in eg a e i and p o e ha (D, X, p)≃(D′×C,Cd−1×C,(0,0)), whe e
(D′,0) ⊂(Cd−1,0) is a ge m o a di iso . Fi s , we deduce ha D′is o linea
Jacobian ype, and second, om he induc ion hypo hesis, ha D′is s ongly
Eule homogeneous and so Dis s ongly Eule homogeneous oo (see Rema k
(1.7)). Q.E.D.
Le us ecall ha a ee di iso Dis said o be Koszul ([4, De ini ion 4.1.1])
a p∈Di o some (and hence any) basis δ1,...,δdo De C(−log D)p, he
sequence σ(δ1), . . . , σ(δd)is egula in g DX,p. I u ns ou ha his p ope y
is equi alen o being holonomic in he sense o Sai o [15, Theo em 7.4].
The ollowing de ini ion is inspi ed by [16, De ini ion 7.1], which only applies
o he case o linea ee di iso s (see also P oposi ion 7.2 and he subsequen
ema k in [16].
(1.10) De ini ion. Assume ha Dis a ee di iso . We say ha Dis s ongly
Koszul a p∈Di o some (and hence any) basis δ1,...,δdo De C(−log D)p
and o some (and hence any) educed equa ion h∈OX,p o (D, p), he sequence
h, σ(δ1)−α1s,...,σ(δd)−αds, wi h δi(h) = αih,
is egula in g DX,p[s](since he sequence is o med by homogeneous elemen s,
i s egula i y does no depend on he o de ).
(1.11) P oposi ion. Assume ha Dis a ee di iso and p∈D. The
ollowing p ope ies a e equi alen :
(a) Dis o linea Jacobian ype a p.
(b) Dis s ongly Koszul a p.
P oo . Le x1, . . . , xd∈O:= OX,p be a sys em o local coo dina es cen e ed
a p,h∈Oa educed local equa ion o Da pand J=J
D,p = (h, h′
x1,...,h′
xd).
Le {δi=Pd
j=1 aij ∂
∂xj}1≤i≤dbe a basis o De (−log D)p, and le us w i e
δi(h) = αihand σi:= σ(δi) = Pd
j=1 aijξj∈g DX,p =O[ξ]. The amily
{(−αi, ai1,...,aid)}1≤i≤dis a basis o he syzygies o h, h′
x1,...,h′
xd.
(a) ⇒(b): F om P oposi ion (1.9) we know ha his Eule homogeneous, i.e.
h∈(h′
x1,...,h′
xd), and we can ake α1=···=αd−1= 0 and αd= 1. In o he
wo ds, {(ai1,...,aid)}1≤i≤d−1)is a basis o he syzygies o h′
x1,...,h′
xd.
Le ϕ:O[ξ]−→ Rees(J) = O[h′
x1 ,...,h′
xd ]be he su jec i e map o O-
algeb as de ined by ϕ(ξi) = h′
xi . Since Jis an ideal o linea ype, he ke nel
o ϕis gene a ed by he σi,1≤i≤d−1. So
dim O[ξ]
(σ1,...,σd−1)= dim Rees(J) = d+ 1
and σ1,...,σd−1is a egula sequence in O[ξ].
On he o he hand, since ke ϕ= (σ1,...,σd−1)is a p ime ideal and h /∈
ke ϕ, we deduce ha h, σ1,...,σd−1is also a egula sequence in O[ξ]and,
indeed h, σ1,...,σd−1, σd−sis a egula sequence in O[ξ, s] = g DX,p[s]and D
is s ongly Koszul a p.
5

(b) ⇒(a): Assume ha Xis a small enough open neighbo hood o p,K(1) =
(σ1−α1s,...,σd−αds)⊂OX[s, ξ1,...,ξd]and le Kbe he ke nel o he
canonical g aded su jec i e map
Φ : OX[s, ξ1,...,ξd]−→ Rees(J
D), s 7→ h , ξi7→ h′
xi . (1)
The homogeneous componen s o K(1) and Ka e cohe en OX-modules. Since
Dis o linea Jacobian ype a any smoo h poin , we deduce ha K/K(1)
is suppo ed by he singula locus o D. In pa icula , o any homogeneous
polynomial F∈Kp he e is an N > 0such ha hNF∈K(1)
p, bu h, σ1−
α1s,...,σd−αdsis a egula sequence and so F∈K(1)
p. We deduce ha
Kp=K(1)
pand Dis o linea Jacobian ype a p. Q.E.D.
(1.12) Co olla y. Assume ha Dis a ee di iso and p∈D. The ollowing
p ope ies a e equi alen :
(a) Dis s ongly Koszul a p.
(b) Dis Eule homogeneous a pand o any educed Eule homogeneous equa-
ion h∈OX,p o (D, p)and any basis δ1,...,δdo De C(−log D)pwi h
δ1(h) = ···=δd−1(h) = 0 and δd(h) = h, he sequence h, σ(δ1),...,σ(δd−1)
is egula in g DX,p.
(c) The e is a educed equa ion h∈OX,p o (D, p)and a basis δ1,...,δdo
De C(−log D)pwi h δ1(h) = ··· =δd−1(h) = 0 and δd(h) = hsuch ha
he sequence h, σ(δ1),...,σ(δd−1)is egula in g DX,p.
P oo . I is a s aigh o wa d consequence o P oposi ions (1.11) and (1.9).
Q.E.D.
Le us no ice ha p ope y (c) in he abo e co olla y appea ed as condi ion
(c’) in [35, Co olla y 1.8].
The ollowing no ion was in oduced in [28, page 257] and was called “(GK)”.
(1.13) De ini ion. Assume ha Dis a ee di iso . We say ha Dis weakly
Koszul a p∈Di o some (and hence any) basis δ1,...,δdo De C(−log D)p
and some (and hence any) educed local equa ion h∈OX,p o (D, p), he sequence
σ(δ1)−α1s,...,σ(δd)−αds, wi h δi(h) = αih,
is egula in g DX,p[s]. We say ha Dis weakly Koszul i i is so a any p∈D.
(1.14) P oposi ion. Fo a ee di iso , he ollowing implica ions hold:
(a) s ongly Koszul ⇒Koszul.
(b) Koszul ⇒weakly Koszul.
P oo . The i s implica ion is a consequence o P oposi ion (1.11) and [9,
P oposi ion 1.27]. The second one comes om [28, P oposi ion 2.2.14], [9,
P oposi ion 1.22]. Q.E.D.
6
The example x1x2(x1+x2)(x1+x3x2) = 0 is a weakly Koszul ee di iso
which is no Koszul [28, Example 3.1] and any non-quasihomogeneous plane
cu e is a Koszul ee di iso non-s ongly Koszul (P oposi ion 2.3.1 in [28]).
(1.15) Rema k. Le Dbe a ee di iso and h∈OX,p a educed local equa ion
o (D, p). I he e is a ge m o ec o ield χa psuch ha χ(h) = h(i.e. h
is Eule homogeneous), hen Dis weakly Koszul a pi and only i o some
(and hence any) basis δ1,...,δd−1o ge ms o ec o ields anishing on h, he
sequence σ(δ1), . . . , σ(δd−1)is egula in g DX,p.
2 Loga i hmic–me omo phic compa ison o Be n-
s ein modules
F om now on we assume ha h: (Cd,0) →(C,0) is a educed local equa ion o
a ge m o a ee di iso (D, 0) ⊂(Cd,0). Le us w i e o sho O=OCd,0,D=
DCd,0and V=V
Cd,0=O[De C(−log D)0]⊂D. We conside he loga i hmic
Be ns ein module O[s]hs[9, §1.6], which is a V[s]-submodule o he Be ns ein
D[s]-module O[s, h−1]hs[2]. Ob iously O[s]hsis gene a ed by hso e V[s]and
annV[s]hsis he le V[s]-ideal gene a ed by he Lie-Rineha algeb a o e (C,O)
(see Appendix A)
Θh:= {δ−αs |δ∈De (−log D)0, δ(h) = αh} ⊂ V[s].
The ollowing esul gene alizes [35, P oposi ion 4.4] o he non-Eule homo-
geneous case and comple es P oposi ion (1.11).
(2.1) P oposi ion. Wi h he abo e hypo heses, he ollowing p ope ies a e
equi alen :
(a) (D, 0) is o di e en ial linea ype and weakly Koszul.
(b) (D, 0) is s ongly Koszul (o equi alen ly, o linea Jacobian ype).
P oo . (b) ⇒(a): I is a consequence o P oposi ions (1.11), (1.4) and (1.14).
(a) ⇒(b): We ollow To elli’s a gumen in 3 ⇒4 o [35, P oposi ion 4.4].
Le δ1,...,δdbe a basis o De C(−log D)0wi h δi(h) = αihand le us w i e
K= annD[s]hs. I is clea ha Θhis eely gene a ed as O-module by δ1−
α1s,...,δd−αds. Since (D, 0) is o di e en ial linea ype, we ha e K=
D[s]Θh. Since σ(δ1)−α1s,...,σ(δd)−αdsis a egula sequence in g D[s] =
g FT(D[s]), we deduce ha σFT(K)is he ideal o g FT(D[s]) gene a ed by
σ(δ1)−α1s,...,σ(δd)−αds. We know ha he cha ac e is ic a ie y
W=
V(σFT(K)) ⊂C×T∗Cdo D[s]hsis i educible o dimension d+ 1 ([20, §5],
[38, P oposi ion 2.3]). In ac I(
W) = ke Φ, whe e Φhas been de ined in
(1). Since Φ(h)6= 0 we deduce ha dim V(h, σ(δ1)−α1s,...,σ(δd)−αds) =
dim(W∩V(h)) = dand so h, σ(δ1)−α1s,...,σ(δd)−αdsis a egula sequence.
Q.E.D.
Le us deno e by Sp•
Θh,V[s]=V[s]⊗U(Θh)Sp•
Θh, whe e he complex Sp•
Θhis
de ined in (A.18). F om P oposi ion 1.21 in [9], we know ha Sp•
Θh,V[s]becomes
7
aV[s]- ee esolu ion o O[s]hswi h he augmen a ion ε0: Sp0
Θh,V[s]=V[s]→
O[s]hs,ε0(P) = Phs.
The p oo o he ollowing p oposi ion is clea .
(2.2) P oposi ion. Unde he abo e hypo heses, he ollowing p ope ies a e
equi alen :
(a) The canonical map D[s]
L
⊗V[s](O[s]hs)−→ D[s]hsis an isomo phism in
he de i ed ca ego y o le D[s]-modules.
(b) The di iso Dis o di e en ial linea ype a 0and he complex D[s]⊗V[s]
Sp•
Θh,V[s]is exac in deg ees 6= 0.
(2.3) P oposi ion. Any ge m o ee di iso (D, 0) ⊂(Cd,0) o di e en ial
linea ype and weakly Koszul a 0sa is ies he equi alen p ope ies o P oposi-
ion (2.2).
P oo . To p o e ha he complex D[s]⊗V[s]Sp•
Θh,V[s]is exac in deg ees
6= 0, we il e i in such a way ha i s g aded complex is he Koszul complex
associa ed wi h he sequence σ(δ1)−α1s,...,σ(δd)−αdswi h δ1,...,δda basis
o De C(−log D)0and δi(h) = αih(see [9, P oposi ion 1.18]). Q.E.D.
The ollowing co olla y is a pa icula case o [9, Theo em 3.1].
(2.4) Co olla y. Unde he abo e hypo heses, i (D, 0) ⊂(Cd,0) is a ge m
o a ee di iso o linea Jacobian ype, hen he equi alen p ope ies o P opo-
si ion (2.2) hold.
P oo . I is clea om P oposi ion (2.1). Q.E.D.
(2.5) De ini ion. Fo any polynomial q(s)∈C[s]we de ine:
(1) The q(s)-Be ns ein module as he ee O[s, h−1]-module O[s, h−1]hq(s)wi h
basis hq(s)endowed wi h he le D[s]-module s uc u e gi en by
δ·(ahq(s)) = δ(a) + q(s)δ(h)h−1ahq(s)
o any δ∈De C(O).
(2) The loga i hmic q(s)-Be ns ein module as he le V[s]-submodule O[s]hq(s)
o he q(s)-Be ns ein module O[s, h−1]hq(s).
I is clea ha O[s]hq(s)is gene a ed by hq(s)o e V[s]and annV[s]hq(s)is
he le V[s]-ideal gene a ed by he (C,O)-Lie-Rineha algeb a
Θh,q(s):= {δ−αq(s)|δ∈De (−log D)0, δ(h) = αh}.
Fo any C-algeb a map ϕ:C[s]→C[s]le us also call ϕi s i ial ex ensions
o O[s],O[s, h−1],D[s]and V[s]. Fo any q(s)∈C[s] he map
ϕ:ahq(s)∈O[s, h−1]hq(s)7→ ϕ(a)hϕ(q(s)) ∈O[s, h−1]hϕ(q(s))
( esp. ϕ:ahq(s)∈O[s]hq(s)7→ ϕ(a)hϕ(q(s)) ∈O[s]hϕ(q(s)))
is linea o e ϕ:D[s]→D[s]( esp. o e ϕ:V[s]→V[s]):
ϕP(s)hq(s)=ϕ(P(s))hϕ(q(s)), P(s)∈D[s].
8
In pa icula , ϕD[s]hq(s)⊂D[s]hϕ(q(s)).
Fo any le D[s]-module M, le us call ϕ∗(M) := D[s]⊗ϕM he scala
ex ension associa ed wi h ϕ:D[s]→D[s]. In ϕ∗(M)one has (Pϕ(Q)) ⊗m=
P⊗(Qm) o m∈Mand P, Q ∈D[s]. In a simila way we de ine ϕ∗(M) :=
V[s]⊗ϕM o any le V[s]-module M.
Since ϕΘh,q(s)= Θh,ϕ(q(s)), he map
eϕ:ϕ∗O[s]hq(s)=V[s]⊗ϕO[s]hq(s)−→ O[s]hϕ(q(s))
induced by ϕ:O[s]hq(s)→O[s]hϕ(q(s)) is an isomo phism o le V[s]-modules:
V[s]⊗ϕO[s]hq(s)≃V[s]⊗ϕV[s]/V[s]·Θh,q(s)≃
V[s]/V[s]·ϕΘh,q(s)≃V[s]/V[s]·Θh,ϕ(q(s)) ≃O[s]hϕ(q(s)).
I is clea ha ϕannD[s]hq(s)⊂annD[s]hϕ(q(s)). I mo eo e ϕis an au o-
mo phism, his inclusion becomes an equali y. This shows he ollowing lemma.
(2.6) Lemma. I ϕ:C[s]→C[s]is an au omo phism, hen he map
eϕ:ϕ∗D[s]hq(s):= D[s]⊗ϕD[s]hq(s)−→ D[s]hϕ(q(s))
induced by ϕ:D[s]hq(s)→D[s]hϕ(q(s)) is an isomo phism o le D[s]-modules.
(2.7) P oposi ion. Assume ha ϕ:C[s]→C[s]is an au omo phism o
C-algeb as. Then, he ollowing p ope ies a e equi alen o he p ope ies o
P oposi ion (2.2):
(a’) The canonical map D[s]
L
⊗V[s]O[s]hϕ(s)−→ D[s]hϕ(s)is an isomo phism
in he de i ed ca ego y o le D[s]-modules.
(b’) annD[s]hϕ(s)is he le D[s]-ideal gene a ed by Θh,ϕ(s)and he complex
D[s]⊗V[s]Sp•
Θh,ϕ(s),V[s]is exac in deg ees 6= 0, whe e Sp•
Θh,ϕ(s),V[s]is
de ined in a comple ely simila way o Sp•
Θh,V[s].
P oo . Since ϕis an au omo phism, he unc o s ϕ∗a e exac . On he o he
hand we ob iously ha e ϕ∗D[s]⊗V[s]−≃D[s]⊗V[s]ϕ∗(−)and so
ϕ∗D[s]
L
⊗V[s]O[s]hs≃D[s]
L
⊗V[s]ϕ∗(O[s]hs)≃D[s]
L
⊗V[s]O[s]hϕ(s).
The equi alence be ween (a’) and p ope y (a) in P oposi ion (2.2) comes om
Lemma (2.6). The equi alence be ween (a’) and (b’) comes om he ac ha
Sp•
Θh,ϕ(s),V[s]is a ee esolu ion o he le V[s]-module O[s]hϕ(s). Le us also
no ice ha Sp•
Θh,ϕ(s),V[s]≃ϕ∗Sp•
Θh,V[s]. Q.E.D.
9
explained in [9, § 3.1], we conside he loga i hmic Be ns ein-Kashiwa a module
E[s]hsinside he me omo phic connec ion E[s, h−1]hs. Co olla y 3.2 in [9] ells
us ha he canonical map
DX[s]
L
⊗V
X[s]E[s]hs→DX[s]E[s]hs
is an isomo phism in Db
(D[s]).
Mo e gene ally, o any q(s)∈C[s]we conside he q(s)-Be ns ein module
associa ed wi h E
E[s]hq(s):= E[s]⊗O[s]O[s]hq(s)⊂E[s, h−1]hq(s):= E[s, h−1]⊗O[s]O[s, h−1]hq(s)
as in De ini ion (2.5). In he same way as in Lemma (2.6) and P oposi ion (2.7)
we p o e ha o any au omo phism o C-algeb as ϕ:C[s]→C[s] he canonical
map
ϕ∗D[s]E[s]hq(s):= D[s]⊗ϕ(D[s]E[s]hs)→D[s]E[s]hϕ(q(s)) (3)
induced by ϕ:O[s, h−1]hq(s)→O[s, h−1]hϕ(q(s)) is an isomo phism o le D[s]-
modules. Also, he canonical map
D[s]
L
⊗V[s]E[s]hϕ(s)−→ D[s]E[s]hϕ(s)
is an isomo phism in he de i ed ca ego y o le D[s]-modules. As in Co olla y
(3.6) we ob ain a canonical isomo phism
DD[s]E[s]hϕ(s)≃D[s]E∗[s]h−ϕ(s)−1.
Recall ha he Be ns ein-Sa o polynomial o Eis de ined as he minimal poly-
nomial o he ac ion o son he quo ien QE:= D[s]E[s]hs/D[s]E[s]hs+1 and
i is deno ed by bE(s)[9, Rema k 3.5]. The exis ence o a non-ze o bE(s)is a
s aigh o wa d consequence o he exis ence o non- i ial Be ns ein-Sa o unc-
ional equa ions wi h espec o sec ions o holonomic D-modules ([21, Theo em
2.7]; see also [25]).
Now we a e eady o s a e and p o e he announced ex ension o Theo em
(4.1) o he case o a bi a y loga i hmic connec ions.
(5.1) Theo em. Le (D, 0) ⊂(Cd,0) be a ge m o a ee di iso o linea
Jacobian ype wi h educed equa ion h: (Cd,0) →(C,0) and Ea ge m a 0o
in eg able loga i hmic connec ion wi h espec o D. Then he Be ns ein-Sa o
polynomials o Eand o i s dual E∗a e ela ed by he equali y
bE(s) = ±bE∗(−s−2).
P oo . We p oceed as in he p oo o Theo em (4.1). Le us conside he
exac sequence o le D[s]-modules
0→D[s]E[s]hs+1 →D[s]E[s]hs→QE→0.
16

By applying he duali y unc o Dwe ob ain a iangle
D(QE)→D(D[s]E[s]hs)→DD[s]E[s]hs+1+1
→
in which he second a ow co esponds o he inclusion D[s]E∗[s]h−s−1→
D[s]E∗[s]h−s−2,D(QE)is concen a ed in deg ee 1and he e is an exac se-
quence o le D[s]-modules
0→D[s]E∗[s]h−s−1→D[s]E∗[s]h−s−2→D1(QE)→0.
Le ϕ:C[s]→C[s]be he au omo phism o C-algeb as de e mined by ϕ(s) =
−s−2and le us conside he exac sequence o le D[s]-modules
0→D[s]E∗[s]hs+1 →D[s]E∗[s]hs→QE∗→0.
F om (3) we deduce an isomo phism ϕ∗(QE∗)≃D1(QE)and so he minimal
polynomial o he ac ion o son D1(QE)is ϕ(bE∗(s)) = bE∗(−s−2). On he
o he hand, he ac ion o son D1(QE)is annihila ed by bE(s)and we conclude
ha bE(s)is a mul iple o bE∗(−s−2).
In a symme ic way we deduce ha bE∗(s)is a mul iple o bE(−s−2), o
equi alen ly bE∗(−s−2) is a mul iple o bE(s), and so bE(s) = ±bE∗(−s−2).
Q.E.D.
6 Open ques ions
The s a ing poin o his pape has been [16] and he obse ed e ec o duali y
on he Be ns ein-Sa o polynomial o some examples o in eg able loga i hmic
connec ions wi h espec o quasi-homogeneous plane cu es [29].
In [16], he au ho s p o ed ha he Be ns ein-Sa o polynomial o any egula
special linea ee di iso (in pa icula , any educ i e linea ee di iso ), and o
any educ i e p ehomogeneous de e minan ( hese a e he non- educed e sion
o linea ee di iso s) ha e he symme y p ope y b(s) = ±b(−s−2) (see
Theo ems 3.5 and 5.5 and he de ini ion o he in ol ed no ions in [16]). These
esul s and ou heo em (4.1) o e lap, bu hey a e logically independen . On
one hand, he esul s in [16] only apply o in a ian s o some p ehomogeneous
ec o spaces. On he o he hand, ou heo em (4.1) canno co e ei he he case
o non- educed educ i e p ehomogeneous de e minan s, since i only applies o
educed equa ions, o he case o educ i e linea ee di iso , since he e a e
examples o such di iso s which a e no o di e en ial linea ype. Namely, D.
And es and J. Ma ín-Mo ales ha e ecen ly s udied he example D={h=
0} ⊂ M3,4=C3×4in [15, P oposi ion 7.12] by using he echniques in [1] and
ha e ound di e en ial ope a o s o o de 2annihila ing hswhich a e no in
ann(1)
D[s]hs=D[s] annV[s]hs.
All he examples o ee di iso s gi en in [27] a e o linea Jacobian ype and
so hey a e co e ed by Theo em (4.1).
(6.1) Ques ion. Is he e a common gene aliza ion o Theo em (4.1) and
Theo ems 3.5 and 5.5 in [16]?
17
(6.2) Ques ion. The educed Be ns ein-Sa o polynomial eb(s) = b(s)
s+1 o any
quasi-homogeneous polynomial h:Cd→Cwi h an isola ed singula i y a he
o igin sa is ies he symme y eb(s) = ±eb(−s−d). This is a consequence o a
celeb a ed esul o Malg ange [23], namely ha he oo s o eb(s) o any isola ed
singula i y a e he eigen alues o he connec ion on he sa u a ed B iesko n
la ice which equals he usual one p ecisely in he quasi-homogeneous case. In
his case he spec um and he oo s o eb(s)a e (up o an o e all shi ) he same,
hence he symme y o he spec um gi es he symme y o he oo s o eb(s).
In he non-quasi-homogeneous isola ed singula i y case, he spec um is di -
e en om he oo s o eb(s), so ha he symme y o he spec um does no
imply he symme y o he oo s o eb(s).
Le us no ice ha in he quasi-homogeneous isola ed singula i y case, he
oo s o eb(s)can be explici ly lis ed in e ms o he weigh s o a iables (c . [38,
Co olla y 3.9]) and he symme y can be checked di ec ly. Also, R. A cadias
and he au ho obse ed ha his symme y can be ob ained by using D-module
duali y heo y.
The in e sec ion o Theo em (4.1) and he symme y p ope y o quasi-
homogeneous isola ed singula i ies is he case o quasi-homogeneous plane cu es.
Quasi-homogeneous isola ed singula i ies a e o linea Jacobian ype and hei
Jacobian ideal a e a comple e in e sec ion, and so Cohen-Macaulay. On he
o he hand, he Jacobian ideal o a singula ee di iso is also Cohen-Macaulay
o codimension 2.
By means o M. Sai o’s o mula o he educed b- unc ion o he Thom-
Sebas iani join o wo ge ms, we can gi e, o any e= 2,...,d, non- i ial
examples o i educible ge ms h: (Cd,0) →(C,0) such ha he ollowing p op-
e ies hold: (a) he Jacobian ideal Jis o linea ype; (b) Jis Cohen-Macaulay;
and (c) he equali y ebh(s) = ±ebh(−s−e)holds o e=codimension o he
singula locus o {h= 0}. Ne e heless, (a) + (b) 6⇒ (c) as shown in Example
(6.3).
The ques ion o inding gene al c i e ia on himplying p ope y (c) and ex-
ending he known ex eme cases o quasi-homogeneous isola ed singula i ies
(e=d) and ee di iso s o linea Jacobian ype (e= 2) seems in e es ing and
emains open.
I owe Ka i Vilonen he sugges ion o checking he ollowing example.
(6.3) Example. Le Xbe he ec o space o squa e n×ncomplex ma ices
and h= de : X→C. I is well known ha he Be ns ein-Sa o polynomial o
his gi en by bh(s) = (s+ 1)(s+ 2) ···(s+n)and so ebh(s) = (s+ 2) ···(s+n).
The Jacobian ideal Jo his gene a ed by all he (n−1) ×(n−1) mino s o he
gene ic ma ix (xij). In pa icula he singula locus o D=h−1(0) consis s o
ma ices o ank ≤n−2,dim Dsing =n2−4and e= codim Dsing = 4. We
know ha Jis o linea ype [19] and Cohen-Macaulay [3, page 25]. Howe e ,
he equali y ebh(−s−4) = ±ebh(s)only holds o n= 2.
(6.4) Ques ion. Co olla y (4.2) applies o locally quasi-homogeneous ee
di iso s a e [6, Theo em 5.6]), in pa icula o ee hype plane a angemen s.
Howe e , he Be ns ein-Sa o polynomial o a non- ee hype plane a angemen
does no sa is y in gene al he symme y b(s) = ±b(−s−2). Fo ins ance, o
18
h=x1x2x3(x1+x2+x3) = 0 we ha e bh(s) = (s+ 1)3(s+ 3/2)(s+ 3/4)(s+
5/4), and bh(s)6=±bh(−s−2). In ac , his symme y p ope y ails in many
o he examples o non- ee hype pane a angemen s. Can we cha ac e ize he
hype plane a angemen s whose Be ns ein-Sa o polynomial sa is y he abo e
symme y p ope y?
Appendix A
This appendix con ains some basic no ions and esul s abou Lie-Rineha al-
geb as and hei modules, a simple p oo o he associa i i y law in [8] (see
Theo em (A.11)) and a de ailed p oo o he duali y heo em in [8] (see The-
o em (A.32)). To be b ie , we ha e included only he s a emen s and esul s
s ic ly needed o he p oo o he ci ed esul s and o P oposi ions (A.15) and
(A.16). The e a e a ian s o all hese esul s which a e le up o he eade
(see Rema k (A.17)).
Le k→Abe a homomo phism o commu a i e ings. Le us deno e by
De k(A) he A-module o k-linea de i a ions λ:A→A, which is a le sub-A-
module o Endk(A)closed by he b acke [−,−].
A Lie-Rineha algeb a o e (k, A)(o a (k, A)-Lie algeb a in [30]) is an
A-module Lendowed wi h a k-Lie algeb a s uc u e and an A-linea map ρ:
L→De k(A), called ancho map, which is also a mo phism o Lie algeb as and
sa is ies
[λ, aλ′] = a[λ, λ′] + ρ(λ)(a)λ′
o λ, λ′∈Land a∈A. To simpli y, we w i e λ(a) := ρ(λ)(a) o λ∈Land
a∈A.
Le (L, ρ),(L′, ρ′)be wo Lie-Rineha algeb as o e (k, A). A map o Lie-
Rineha algeb as :L→L′is an A-linea map which is also a k-Lie algeb a
map and such ha ρ′◦ =ρ.
I is clea ha De k(A)is a Lie-Rineha algeb a o e (k, A)wi h he iden i y
as ancho map, and ha o any Lie-Rineha algeb a (L, ρ),ρis a map o Lie-
Rineha algeb as.
Le Lbe a Lie-Rineha algeb a o e (k, A). A le L-module is an A-module
Mendowed wi h a k-bilinea ac ion (λ, m)∈L×M→λm ∈Msuch ha
(aλ)m=a(λm),[λ, λ′]m=λ(λ′m)−λ′(λm), λ(am) = a(λm) + λ(a)m
o all a∈A,λ, λ′∈Land m∈M.
The A-module Abecomes a le L-module wi h he ac ion (λ, a)∈L×A7→
λ(a)∈A.
A igh L-module is an A-module Q, whe e he mul iplica ion by elemen s
o Ais w i en on he igh , endowed wi h a k-bilinea ac ion (q, λ)∈Q×L→
qλ ∈Qsuch ha
q(aλ) = (qa)λ, q[λ, λ′] = (qλ)λ′−(qλ′)λ, (qa)λ= (qλ)a−qλ(a)
o all a∈A,λ, λ′∈Land q∈Q.
19
Le Lbe a Lie-Rineha algeb a o e (k, A)and U=U(L)i s en eloping
(o uni e sal) algeb a (see [30]). I is endowed wi h an injec i e ing map A≃
U0֒→Uwi h k→Ucen al, and a le A-linea map L→U. Mo eo e , Uis
gene a ed as a ing by Aand he image o L, and i ca ies a canonical il a ion
(U ) ≥0(U is gene a ed as le o igh A-module by all p oduc s o elemen s
o Lo leng h ≤ ) such ha g Uis a commu a i e A-algeb a.
F om he uni e sal p ope y o U(see [17, pgs. 63-64]) we ha e he ollowing:
I Mis an A-module, o gi e a le L-module s uc u e on Mis equi alen o
ex ending i s A-module s uc u e o a le U-module s uc u e. Simila ly, i Qis
an A-module, o gi e a igh L-module s uc u e on Qis equi alen o ex ending
i s A-module s uc u e o a igh U-module s uc u e.
The su jec ion p∈U7→ p·1∈Ainduces a canonical isomo phism o le
U-modules U/U ·L≃A.
(A.1) Example. (a) I Xis a complex smoo h mani old, p∈X,k=Cand
A=OX,p is he ing o ge ms a po holomo phic unc ions, hen he en eloping
algeb a o he Lie-Rineha algeb a De C(OX,p)is he ing DX,p o ge ms a p
o linea di e en ial ope a o s wi h holomo phic coe icien s in X.
(b) I Xis a complex smoo h mani old, D⊂Xis a ee di iso , p∈D,
k=C,A=OX,p and L=De (−log D)pis he Lie-Rineha algeb a o ge ms
o loga i hmic ec o ields wi h espec o D, hen he en eloping algeb a o L
is he ing o ge ms o loga i hmic di e en ial ope a o s DX,p(−log D), which
coincides wi h he sub ing o DX,p gene a ed by OX,p and De (−log D)p(see
[4, p op. 2.2.5]).
(A.2) In e nal ope a ions: In wha ollows, M, M1, M2,... will deno e le
U-modules and Q, Q1, Q2,... igh U-modules. I is well known ha he A-
modules M1⊗AM2,HomA(M1, M2)and HomA(Q1, Q2)( esp. Q1⊗AM1and
HomA(M1, Q1)) ha e na u al le ( esp. igh ) U-module s uc u es (c . [18,
§2]).
(A.3) The na u al A-linea maps M1⊗AM2≃M2⊗AM1, A ⊗AM2≃M2≃
HomA(A, M2),[M1⊗AM2]⊗AM3≃M1⊗A[M2⊗AM3]a e le U-linea , and
he na u al A-linea maps Q1⊗AA≃Q1≃HomA(A, Q1),[Q1⊗AM1]⊗AM2≃
Q1⊗A[M1⊗AM2]a e igh U-linea .
The p oo o he ollowing lemmas is s aigh o wa d.
(A.4) Lemma. The na u al isomo phisms o A-modules
HomA(M1⊗AM2, M3)≃HomA(M1,HomA(M2, M3)),(4)
HomA(Q1⊗AM1, Q2)≃HomA(M1,HomA(Q1, Q2)) (5)
a e le U-linea .
(A.5) Lemma. The na u al A-linea map Q1⊗AHomA(Q1, Q2)→Q2is igh
U-linea . Mo eo e , i is an isomo phisms i Q1is a ee A-module o ank 1.
I M1is a p ojec i e A-module o ini e ank, we deno e M∗
1= HomA(M1, A).
(A.6) Lemma. Assume ha M1is a p ojec i e A-module o ini e ank. Then,
he na u al map M∗
1⊗AM2→HomA(M1, M2)is an isomo phism o le U-
modules.
20
(A.7) Lemma. The na u al A-linea map M1→HomA(Q1, Q1⊗AM1)is le
U-linea . Mo eo e , i is an isomo phism i Q1is a ee A-module o ank 1.
(A.8) Lemma. Fo any le U-modules Mand M′( esp. o any igh U-
modules Qand Q′), a map h∈HomA(M, M′)( esp. h∈HomA(Q, Q′)) is
U-linea i and only i λh = 0 o all λ∈L. Consequen ly he e a e canoni-
cal isomo phisms HomU(M, M′)≃HomU(A, HomA(M, M′)),HomU(Q, Q′)≃
HomU(A, HomA(Q, Q′)).
(A.9) Lemma. The U-linea isomo phism (4) in Lemma (A.4) and Lemma
(A.8) induce a k-linea isomo phism
β: HomU(M1⊗AM2, M3)−→ HomU(M1,HomA(M2, M3)).
(A.10) Lemma. The U-linea isomo phism (5) in Lemma (A.4) and Lemma
(A.8) induce a k-linea isomo phism
γ: HomU(Q1⊗AM1, Q2)≃
−→ HomU(M1,HomA(Q1, Q2)).
Assume now ha L0→Lis a map o Lie-Rineha algeb as o e (k, A),
which induces a ing map U0=U(L0)→U=U(L)be ween hei en eloping
algeb as.
Gi en a igh U-module Qand a le U0-module M0, we know ha Q⊗AM0
and Q⊗A[U⊗U0M0]ha e na u al igh module s uc u es o e U0and U,
espec i ely, and ha he map
σ:Q⊗AM0→Q⊗A[U⊗U0M0], σ(q⊗m) = q⊗[1 ⊗m]
is U0-linea .
The ollowing heo em gi es he associa i i y law needed in he p oo o he
duali y heo em (A.32). The p oo we gi e he e simpli ies he o iginal p oo in
[8].
(A.11) Theo em. The U-linea map
τ: [Q⊗AM0]⊗U0U→Q⊗A[U⊗U0M0]
induced by σis an isomo phism o igh U-modules.
P oo . I is enough o p o e ha o any igh U-module Q′, he induced map
τ∗: HomU(Q⊗A[U⊗U0M0], Q′)−→ HomU([Q⊗AM0]⊗U0U, Q′)
is an isomo phism, and o ha we conside he ollowing commu a i e diag am
HomU([Q⊗AM0]⊗U0U, Q′)τ∗
←−−−− HomU(Q⊗A[U⊗U0M0], Q′)
α∗
1

y≃γ

y≃by (A.10)
HomU0(Q⊗AM0, Q′) HomU(U⊗U0M0,HomA(Q, Q′))
≃by (A.10)

yγ0≃

yα∗
2
HomU0(M0,HomA(Q, Q′)) HomU0(M0,HomA(Q, Q′)).
21

whe e α1:Q⊗AM0−→ [Q⊗AM0]⊗U0Uis he na u al ( igh ) U0-linea map
de ined by α1(ξ) = ξ⊗1 o ξ∈Q⊗AM0and α2:M0−→ U⊗U0M0is he
na u al (le ) U0-linea map de ined by α2(m) = 1 ⊗m o m∈M0. Q.E.D.
Le us no ice ha Q⊗AUhas wo igh U-module s uc u es: he i s one
comes by scala ex ension A→U om he A-module s uc u e on Q(we o ge
he e he igh U-module s uc u e on Q):
(q⊗p)◦p′:= q⊗(pp′), q ∈Q, p, p′∈U,
and he second one comes by (A.2) om he igh U-module s uc u e on Qand
he le U-module s uc u e on U:
(q⊗p)⋆ a := (qa)⊗p=q⊗(ap), q ∈Q, p ∈U, a ∈A,
(q⊗p)⋆ λ := (qλ)⊗p−q⊗(λp), q ∈Q, p ∈U, λ ∈L.
I is clea ha (ξ◦p)⋆ a = (ξ ⋆ a)◦p,(ξ◦p)⋆ λ = (ξ ⋆ λ)◦p o any ξ∈Q⊗AU,
p∈U,a∈Aand λ∈L, and so (ξ◦p)⋆ p′= (ξ ⋆ p′)◦p o any ξ∈Q⊗AU,
p, p′∈U.
Le us also no ice ha bo h s uc u es induce by scala es ic ion wo di -
e en A-module s uc u es on Q⊗AU: o he i s one (q⊗p)◦a=q⊗(pa),
and o he second one (q⊗p)⋆ a = (qa)⊗p=q⊗(ap).
(A.12) Co olla y. Unde he abo e condi ions, he e is a unique U-linea
map τ: (Q⊗AU, ◦)→(Q⊗AU, ⋆)such ha τ(q⊗1) = q⊗1 o all q∈Q.
Mo eo e , τis an isomo phism, τ(ξ ⋆ p) = τ(ξ)◦p o all ξ∈Q⊗AUand o
all p∈U, and τis in olu i e.
P oo . Le us conside he A-linea map σ:Q→Q⊗AUde ined by σ(q) =
q⊗1, whe e we conside on Q⊗AU he A-module s uc u e induced by he
second igh U-module s uc u e. The map σinduces a map o igh U-modules
τ: (Q⊗AU, ◦)→(Q⊗AU, ⋆)
wi h τ(q⊗p) = (q⊗1) ⋆ p. By applying Theo em (A.11) o L0= 0,U0=A
and M0=Awe deduce ha τis an isomo phism.
Fo any q∈Qand any a∈Awe ha e τ(q⊗a) = τ((q⊗1)◦a) = (q⊗1)⋆a =
q⊗a, and so τ2(q⊗a) = q⊗a.
Le (U ) ≥0be he canonical il a ion on U:U is he le (o igh ) A-
module gene a ed by Aand L . Le (Γ ) ≥0be he il a ion on Q⊗AUinduced
by (Q⊗AU ) ≥0. I is clea ha Γ ◦Us⊂Γ +s.
I is no di icul o p o e ha τ(ξ⋆p) = τ(ξ)◦pby induc ion on deg ξ+deg p.
The in olu i i y o τcomes om he ac ha τ2(q⊗p) = τ((q⊗1) ⋆ q) =
(q⊗1)◦p=q⊗p. Q.E.D.
(A.13) Co olla y. I Mis a ee le U-module and Qis a igh U-module
which is ee o e A, hen Q⊗AMis a ee igh U-module.
P oo . Since Mis ee, M≃ ⊕i∈IU o some se I, and Q⊗AM≃ ⊕i∈I(Q⊗A
U, ⋆), bu (Q⊗AU, ⋆)≃(Q⊗AU, ◦)is a ee igh U-module because Qis a
ee A-module. Q.E.D.
22
(A.14) Co olla y. Assume ha Qis a igh U-module which is ee o e A.
Then, o any uppe bounded complex M•
0o le U0-modules he e is a canonical
isomo phism
τ: [Q⊗AM•
0]
L
⊗U0U→Q⊗A[U
L
⊗U0M•
0]
in he uppe bounded de i ed ca ego y o igh U-modules.
P oo . Le L•→M•
0be a ee esolu ion. Since Qis a ee A-module,
we ha e an isomo phism Q⊗AL•≃
−→ Q⊗AM•
0in he uppe bounded de i ed
ca ego y o igh U0-modules, and we can apply Co olla y (A.13) o deduce ha
Q⊗AL•is an uppe bounded complex o ee igh U0-modules. So we ha e
[Q⊗AM•
0]
L
⊗U0U≃[Q⊗AL•]
L
⊗U0U≃[Q⊗AL•]⊗U0U
and Q⊗A[U
L
⊗U0M0]≃Q⊗A[U⊗U0L•]. To conclude, we apply Theo em
(A.11). Q.E.D.
Le Mbe a le U-module and N0an A-module. The module U⊗AN0is,
by scala ex ension, a le U-module: p(q⊗n) = (pq)⊗n,p, q ∈U, n ∈N. Le
us deno e by [U⊗AN0]⊗′
AM he enso p oduc wi h espec o he A-module
s uc u e on U⊗AN0induced by i s le U-module s uc u e:
a([q⊗n]⊗′m) = (a[q⊗n]) ⊗′m= [(aq)⊗n]⊗′m= [q⊗n]⊗′(am)
o a∈A, q ∈U, n ∈N0, m ∈M. I has a le U-module s uc u e gi en by
(A.2):
λ([q⊗n]⊗′m) = [(λq)⊗n]⊗′m+[q⊗n]⊗′(λm), λ ∈L, q ∈U, n ∈N0, m ∈M.
The map
σ′:N0⊗AM→[U⊗AN0]⊗′
AM, σ′(n⊗m) = [1 ⊗n]⊗′m
is A-linea and induces a U-linea map τ′:U⊗A[N0⊗AM]→[U⊗AN0]⊗′
AM
wi h τ′(p⊗[n⊗m]) = p·σ′(n⊗m).
(A.15) P oposi ion. Unde he abo e hypo heses, he map
τ′:U⊗A[N0⊗AM]→[U⊗AN0]⊗′
AM
is an isomo phism o le U-modules.
P oo . I can be done in a comple ely simila way o he p oo o heo em
(A.11), bu we use lemma (A.9) ins ead o lemma (A.10). Q.E.D.
In he special case whe e N0=A, we ob ain a eache s a emen . The le
U-modules U⊗AMand U⊗′
AMa e endowed wi h compa ible igh U-module
s uc u es (A.2):
(q⊗m)a=q⊗(am) = (qa)⊗m, (q⊗m)λ= (qλ)⊗m−q⊗(λm),
(q⊗′m)p= (qp)⊗′m( emembe ha (aq)⊗′m=q⊗′(am))
o a∈A, λ ∈L, p, q ∈U, m ∈M.
23
(A.16) P oposi ion. The le U-linea map τ′:U⊗AM→U⊗′
AMde ined
by τ′(p⊗m) = p·(1 ⊗′m)is also igh U-linea and so i is an isomo phism o
(U;U)-bimodules.
P oo . The ac ha τ′is an isomo phism o le U-modules comes om
P oposi ion (A.15). The ac ha τ′is igh U-linea can be p o en in a simila
way o Co olla y (A.12). Q.E.D.
(A.17) Rema k. P oposi ion (A.15) is a pa icula case o a second asso-
cia i i y law U⊗U0[N0⊗AM]≃[U⊗U0N0]⊗′
AMwi h L0= 0 and U0=A.
Ac ually, he e is a hi d associa i i y law [Q0⊗AM]⊗U0U≃[Q0⊗U0U]⊗AM.
Bo h laws and hei co esponding co olla ies can be p o ed in a simila way o
Theo em (A.11). Fo ha one needs some a ian s o Lemmas (A.4) – (A.10).
We do no need hese esul s in his pape and we skip hei p oo .
F om now on, we assume ha Lis a Lie-Rineha algeb a o e (k, A)which
is a ee A-module o ank d. By he Poinca é-Bi kho -Wi heo em [30, h.
3.1] we know ha Sym L≃g U.
Le us deno e ΩL=L∗= HomA(L, A),Ωn
L=VnΩL≡HomA(VnL, A) o
n= 0,...,d, and ωL= Ωd
L.
(A.18) Fo each le U-module Ewe de ine he Ca an-Eilenbe g-Che alley-
Rineha -Spence Sp•
L(E)complex associa ed wi h Eas [9, 1.1.2]:
Sp−n
L(E) = U⊗A
n
^L⊗AE, n = 0,...,d
whe e he le U-module s uc u e comes exclusi ely om he i s ac o Uo
he enso p oduc , he di e en ial d−n: Sp−n
L(E)→Sp−n+1
L(E)is gi en by:
d−1(P⊗λ⊗e) = (Pλ)⊗e−P⊗(λe),
d−n(P⊗(λ1∧ · · · ∧ λn)⊗e) =
n
X
i=1
(−1)i−1Pλi⊗(λ1∧ · · · b
λi· · · ∧ λn)⊗e
−
n
X
i=1
(−1)i−1P⊗(λ1∧ · · · b
λi· · · ∧ λn)⊗(λie)
+X
1≤i<j≤n
(−1)i+jP⊗([λi, λj]∧λ1∧ · · · b
λi···c
λj· · · ∧ λn)⊗e, 2≤n≤d,
and he augmen a ion is P⊗e∈U⊗AE= Sp0
L(E)7→ d0(P⊗e) := P·e∈E.
In he case E=A he abo e complex coincides wi h he complex de ined in
[30, §4], which will be simply deno ed by Sp•
L.
Le us deno e by g
Sp•
Land g
Sp•
L(E) he augmen ed complexes Sp•
L→Aand
Sp•
L(E)→E espec i ely.
We quo e he ollowing esul (c . [9, P oposi ion 1.7]):
(A.19) P oposi ion. Unde he abo e hypo heses, i Eis ee o e A, hen
Sp•
L(E)is a ee esolu ion o he U-module E.
Fo any le U-module M, he complex HomU(Sp•
L, M)is canonically isomo -
phic o he “de Rham” complex Ω•
L(M), whe e Ωn
L(M) := HomA(VnL, M)≡
24
HomA(VnL, A)⊗AM= Ωn
L⊗AM,n= 0,...,d, and he di e en ial d:
Ωn−1
L(M)→Ωn
L(M)is gi en by
(dα)(λ1∧ · · · ∧ λn)) =
n
X
i=1
(−1)i−1λiα(λ1∧ · · · b
λi· · · ∧ λn)
+X
1≤i<j≤n
(−1)i+jα([λi, λj]∧λ1∧ · · · b
λi···c
λj· · · ∧ λn).
Fo any λ∈Land any nwe ha e he Lie de i a i e Lλ: Ωn
L(M)→Ωn
L(M)
(c . [30], p op. 6.3) de ined by
(Lλα)(λ1∧ · · · ∧ λn) = λ·α(λ1∧ · · · ∧ λn)−
n
X
i=1
α(λ1∧ · · · ∧ [λ, λi]∧ · · · ∧ λn).
The p oo o he ollowing p oposi ion can be ound in [18], p op. 2.8 and
h. 2.10 (see also [7, P oposi ion 3.1]). I is a gene aliza ion o he well known
co esponding esul s in D-module heo y.
(A.20) P oposi ion. Unde he abo e hypo heses, he ollowing p ope ies
hold:
(1) The ac ion (α, λ)∈ωL×L7→ −Lλα∈ωLde ines a igh U-module
s uc u e on ωL.
(2) The complex Ω•
L(U)can be augmen ed h ough he map α⊗p∈Ωd
L⊗AU7→
αp ∈ωL= Ωd
Land i becomes a ee esolu ion o he igh U-module ωL.
(A.21) Co olla y. The complex o igh U-modules RHomU(A, U)is con-
cen a ed in deg ee dand i s d-cohomology Ex d
U(A, U)is canonically isomo phic
o he igh U-module ωL.
P oo . I is a consequence o he abo e p oposi ions and he canonical iso-
mo phisms RHomU(A, U)≃HomU(Sp•
L, U)≃Ω•
L(U). Q.E.D.
(A.22) De ini ion. The igh L-module ωLis called he dualizing module o
he Lie-Rineha algeb a L.
(A.23) P oposi ion. Wi h he abo e no a ions, we ha e a canonical isomo -
phism o le U-modules g
Sp•
L(E)≃g
Sp•
L⊗′
AEwhe e he enso p oduc ⊗′
Ais
aken wi h espec o he A-module s uc u e on each Sp−n
Linduced by i s le
U-module s uc u e.
P oo . F om P oposi ion (A.15) he e a e canonical isomo phisms
α−n: Sp−n
L(E) = U⊗A n
^L⊗AE!≃ U⊗A
n
^L!⊗′
AE= Sp−n
L⊗′
AE
o n= 0,...,d. To p o e he commu a i i y o he diag ams
U⊗An
VL⊗AEα−n
−−−−→ U⊗A
n
VL⊗′
AE
d−n

y

yd−n⊗′Id
U⊗An−1
VL⊗AEα−n+1
−−−−→ U⊗A
n−1
VL⊗′
AE
25