a Xi :ma h/0505465 1 [ma h.AG] 23 May 2005
A la ness p ope y o il e ed D-modules
F.J. Cas o-Jim´enez and M. G ange
Abs ac
Le Mbe a cohe en module o e he ing DXo linea di e en ial ope a o s on an analy ic mani old
Xand le Z1,· · · , Zkbe kge ms o ans e se hype su aces a a poin x∈X. The Malg ange-Kashiwa a
V- il a ions along hese hype su aces, associa ed wi h a gi en p esen a ion o he ge m o Ma x, gi e ise
o a mul i il a ion U•(M) o Mxas in Sabbah’s pape [9] and o an analy ic s anda d an in a way simila
o [3]. We p o e he e ha his s anda d an is adap ed o he mul i il a ion, in he sense o C. Sabbah. This
esul comple es he p oo o he exis ence o an adap ed an in [9], o which he use o [8] is no possible.
1 In oduc ion
Le us conside a cohe en module Mo e he ing DXo di e en ial ope a o s on an analy ic mani old X.
Fo any smoo h hype su ace Zo X, Malg ange and Kashiwa a de ined a il a ion along Z o DXand
he no ion o a good il a ion o M. Gi en a se o ans e se smoo h hype su aces Z1,...,Zk, Sabbah
conside ed in [9] mul i il a ions o Mindexed by k-uples o ela i e in ege s. To be p ecise he deal wi h linea
combina ions o e Q+o he il a ions V(j)along each hype su ace Zjand wi h e inemen s VΓo he o iginal
mul i il a ion associa ed wi h each a ional polyed al simplicial cone in he posi i e quad an o (Qk)⋆. The
o iginal mul i il a ion is he one which co esponds o he case Γ = Nk.
The aim o his pape is o cla i y he la ness p ope ies which appea in [9], namely o p o e he exis ence
o a an E, such ha o any cone Γ in his an, he Rees module o he il a ion VΓis la o e he a ine
ing AΓo he o ic blowing-up o Ckassocia ed wi h he an E. Such a an is called an adap ed an in [9].
The eason o his cla i ica ion is ha he p oo in [9] depends on he appendix [8] in which he main ool is
a di ision heo em in Rees ings o di e en ial ope a o s which is no co ec as s a ed. Indeed, he in ini e
p ocess ha i s p oo sugges s, would equi e monomials o unbounded deg ees in he di e en ial a iables o
he ing o ope a o s.
One o he main consequences o he exis ence o an adap ed an as de eloped in [9] is ha we hus comple e
he p oo o he exis ence o non i ial unc ional equa ions o Be ns ein-Sa o ype o a k–uple o unc ions,
ollowing he a gumen o Sabbah in [9]. I should be emphasized he e ha his p oblem in ol es a DX×Ck-
module na u ally associa ed wi h a k-uple o unc ions on X, and ha in his si ua ion a mul i il a ion, along
he ans e se hype su aces j= 0 due o he ac o Ck, appea s in a na u al way.
We mus no e he e ha he p oo o he exis ence o Be ns ein-Sa o equa ions has al eady been comple ed
in Bahloul’s pape [4], by a di e en me hod which a oids he e e ence o a la ness p ope y in ol ed in he
no ion o an adap ed an. Bahloul uses ins ead he analy ic s anda d an as de ined in [3], and desc ibed also
in [5]. I is he e o e no comple ely su p ising ha he adap ed an wan ed in [9] u ns ou in ac o be
he analy ic s anda d an. This emphasizes he in e es o Bahloul’s p oo which also has he ad an age o
being cons uc i e. This cons uc i eness is aluable in an algeb aic se ing also. In bo h p oo s, Bahloul’s
and Sabbah’s , he la e which is comple ed by his pape , he main s ep is he p oo o he goodness o he
so-called sa u a ed il a ion. The de ails a e gi en in sec ion 2.3.
The s a ing poin o he p oo o ou main heo em (Theo em 4.3) is a c i e ion o la ness by M. He mann
and U. O banz (see [7]), o g aded modules o e g aded ings, whe e he g ading is indexed by an a bi a y
commu a i e g oup. In he s a emen o his esul he e is no e e ence o any ini eness p ope y. In ou
case he indexing g oup will be Zk, and he ing a conical sub ing o he ing o Lau en polynomials in k
a iables wi h he ob ious mul ig ading by monomials. The main ing edien o he p oo o Theo em 4.3 is hen
1
he exis ence o a simul aneous L–s anda d basis o a submodule N ⊂ D o all Lin a cone o he analy ic
s anda d an o N(see [3]).
The plan o he pape is as ollows. In sec ion 2, we ecall wi h mo e de ails wha V-mul i il a ions, and
Rees modules a e, and he e inemen s o hese mul i il a ions wi h espec o a a ional polyhed al simplicial
cone. We end his sec ion by de ining he ibe a ze o o hese Rees modules seen as modules o e he ing o
an a ine o ic blowing up. We aim o p o e ha his is he ibe o a la de o ma ion i he cone is aken in
he analy ic s anda d an.
In sec ion 3 we ecall he no ion o an analy ic s anda d an as de eloped in [3], and we ske ch he easy
gene alisa ion which we need om he cyclic case ea ed in [3] o he gene al case.
In sec ion 4 we inally p o e ha he analy ic s anda d an is an adap ed an in he sense o [9].
Acknowledgemen s.- Du ing he p epa a ion o his pape bo h au ho s ha e been pa ially suppo ed by Acci´on
In eg ada-Picasso HF2004-0117. Fi s au ho has been also pa ially suppo ed by MTM2004-01165. Bo h
au ho s a e g a e ul o he D´epa emen de Ma h´ema iques e Applica ions o he ´
Ecole No male Sup´e ieu e
(Pa is) whe e he i s was in i ed du ing he p epa a ion o he inal pa o his wo k. They also hank Jeni e
G ange o he p oo eading.
2 Mul i il a ions, Rees ings and Rees modules
Le us deno e by O=C{x1,...,xn} he complex con e gen powe se ies ing and by D he ing o ge ms o
linea di e en ial ope a o s wi h holomo phic coe icien s (i.e. D=O[∂] = O[∂1, . . . , ∂n] whe e ∂iis he pa ial
de i a i e wi h espec o xiand he p oduc in Dis de ined by he Leibni z’s ule: ∂ia=a∂i+∂i(a), o each
a∈ O).
An ope a o Pin Dcan be w i en as
P=X
β∈Nn
pβ(x)∂β=X
α,β∈Nn
pαβxα∂β
whe e he i s sum is ini e, α= (α1,...,αn), β = (β1,...,βn), xα=xα1
1···xαn
n,∂β=∂β1
1···∂βn
n,pβ(x)∈ O
and pαβ ∈C.
Fo each i= 1,...,n, le us emembe ha he V- il a ion on D, wi h espec o he hype su ace xi= 0,
was de ined by Malg ange and Kashiwa a as:
V(i)
ℓ(D) = V(i)
ℓ={P=X
α,β∈Nn
pαβxα∂β∈ D | βi−αi≤ℓ}
o each ℓ∈Z. The amily (V(i)
ℓ)ℓ∈Zis an inc easing exhaus i e il a ion on he ing D. Fo i= 1, he
associa ed g aded ing
g V(1) (D) = M
ℓ
V(1)
ℓ
V(1)
ℓ−1
is isomo phic o he ing C{x2, . . . , xn}[x1, ∂1, ∂2,...,∂n], g aded by he so-called V(1)–g adua ion, whe e he
homogeneous elemen s o V(1)–deg ee ℓa e
X
α,β∈Nn;β1−α1=ℓ
pαβxα∂β.
We ha e simila desc ip ions o i= 2,...,k.
We can also conside a V(i)- il a ion on he ee module D jus by de ining V(i)
ℓ(D ) = (V(i)
ℓD) and
mo e gene ally, o any ec o m= (m1,...,m )∈Z , one can de ine he shi ed il a ion V(i)[m]ℓ(D ) =
⊕
j=1V(i)
ℓ−mjD. Such a il e ed module is called a V(i)- il e ed ee module. All he D-modules conside ed will be
le modules.
2
De ini ion 2.1 Le Mbe a D-module. We say ha a il a ion U(i)(M)indexed by Zis a good V(i)- il a ion
i he e exis s a p esen a ion M=D
Nas a quo ien by a le submodule No D , and a weigh ec o msuch
ha U(i)
ℓ(M) = π(V(i)[m]ℓ(D )) whe e πis he p ojec ion D → M.
2.1 V-mul i il a ions and VΓ-mul i il a ions
Le us ix an in ege ksuch ha 1 ≤k≤n.
Fo each s= (s1,...,sk)∈Zk, we shall deno e Vs(D) = Tk
i=1 V(i)
si(D). The amily {Vs(D)}s∈Zkde ines a
mul i- il a ion on he ing D. To simpli y we shall say ha V•(D) is a k– il a ion on Do e en, i no con usion
is possible, a il a ion on D.
Le us conside a a ional simplicial cone Γ in he i s quad an o he dual space (Qk)∗=HomQ(Qk,Q).
We deno e by ˇ
Γ he dual cone o Γ, i.e.
ˇ
Γ = {a∈Qk|γ(a)≥0,∀γ∈Γ}.
We associa e wi h such a cone Γ he a ine a ie y, deno ed by SΓ, wi h coo dina e ing equal o C[ˇ
Γ∩Zk]. We
will deno e AΓ=C[ˇ
Γ∩Zk] and A=C[Nk]. We deno e by L(Γ) he se o p imi i e elemen s in he 1-skele on
o Γ.
The mul i il a ion VΓon Dis de ined as ollows: Fo each s∈Zkwe de ine
VΓ
s(D) = X
σ∈Zk|L(σ)≤L(s);∀L∈L(Γ)
Vσ(D).
No ice ha he sum is indexed by σ∈s−ˇ
Γ and ha we ha e he inclusion Vs(D)⊂VΓ
s(D). The amily VΓ
•(D)
is a mul i il a ion o he ing D, indexed by s∈Zk. This means ha :
VΓ
s(D).V Γ
s′(D)⊂VΓ
s+s′(D) and [VΓ
s(D) = D.
We may de ine, in a simila way o he case o one il a ion, he no ion o a ee mul i- il e ed module and
ha o a good mul i il a ion o a ini ely gene a ed D-module M. Fo ha pu pose we chose a shi mul i ec o
n= (n(1),··· , n( ))∈(Zk) called also a shi ma ix, wi h columns n(i)∈Zk, and a p esen a ion M=D
No
M.
De ini ion 2.2 The mul i il e ed ee module associa ed wi h nis he module D endowed wi h he mul i il a ion
indexed by s∈Zk, and de ined as:
V[n]s(D ) =
M
i=1
Vs−n(i)(D)
De ini ion 2.3 A good mul i il a ion o M, is a il a ion indexed by s∈Zk, o he ype
Us(M) = π(V[n]s(D )) = V[n]s(D ) + N
N
o some p esen a ion π:D → M o M.
In he ob ious sense hese mul i il a ions a e compa ible wi h he mul i il e ed s uc u e on he ing D.
Rema k ha he i- h gene a o eio D is hen o mul ideg ee n(i)∈Zk. We may also endow Nwi h he
induced il a ion Us(N) = V[n]s(D )∩ N, so ha we also ha e
Us(M) = V[n]s(D )
Us(N).
We may obse e ha he mul i il a ion on he ee module D is de ined as he in e sec ion o he V(i)–
il a ion wi h espec o he ow ec o s o n,ni= (n(1)
i,··· , n( )
i)∈Z , ha is
V[n]s(D ) = V(1)[n1]s1(D )∩ · · · ∩ V(k)[nk]sk(D ),
3
bu he analogue o M, wi h espec o he good V(i)- il a ions as de ined in de ini ion 2.1 is no ue since
he inclusion:
Us(M) = V(1)[n1]s1(D )∩ · · · ∩ V(k)[nk]sk(D ) + N
N⊂
k
i=1
V(i)[ni]si(D ) + N
N=
k
i=1
U(i)
si(M)
may be s ic .
Fo each good il a ion he e is an associa ed Γ- il a ion compa ible wi h he mul i il a ion VΓ
•(D) on D:
Fo each s∈Zkle us conside
UΓ
s(M) = X
σ∈Zk|L(σ)≤L(s);∀L∈L(Γ)
Uσ(M).
The mul i il a ion UΓ
•(M) is a good il a ion wi h espec o VΓ
•(D).
The goodness means he e ha we can e i y, wi h he p esen a ion o Mas abo e, ha UΓ
s(M) =
π(V[n]Γ
s(D )), is s ill he quo ien o a il a ion on he ee module D which is a di ec sum o con enien
shi s o he il a ion VΓ
•on he ing D. Mo e p ecisely he in ol ed il a ion V[n]Γ
•is de ined by
V[n]Γ
s(D ) = X
σ∈Zk|L(σ)≤L(s);∀L∈L(Γ)
V[n]σ(D )( =
M
i=1
(VΓ
s−n(i)(D))
and i emains o be ema ked ha :
UΓ
s(M) = X
σ∈Zk|L(σ)≤L(s);∀L∈L(Γ)
Uσ(M) = X
σ∈Zk|L(σ)≤L(s);∀L∈L(Γ)
V[n]σ(D ) + N
N=V[n]Γ
s(D ) + N
N
so ha he goodness o he il a ion U•(M) implies he goodness o UΓ
•(M).
2.2 Rees ings and Rees modules.
2.2.1 De ini ion o Rees ings RV(D),RΓ(D)and o ela ed Rees modules
The Rees ing associa ed wi h he V-mul i il a ion on Dis de ined by:
RV(D) = M
s∈Zk
Vs(D)us
whe e u= (u1,...,uk) a e a iables and he p oduc in he Rees ing is induced by he na u al p oduc in he
Lau en polynomial ing D[u±1] = D[u1, u−1
1,...,uk, u−1
k].
By de ini ion he Rees ing RV(D) is a g aded C-algeb a wi h alues g oup Zk, whose homogeneous elemen s
a e P us o P∈Vs(D) and s∈Zk.
Simila ly gi en a D-module M=D /Nand a shi ma ix n, we de ine om he good il a ions V[n]s(D ),
and Us(M), he Rees modules
RV[n](D ) = ⊕s∈ZkV[n]s(D )us
RU(M) = M
s∈Zk
Us(M)us.
Bo h ha e a na u al s uc u e o mu ig aded le RV(D)-module and we shall conside RU(M) as a sub-g oup
o M[u±1], he Lau en polynomials wi h coe icien s in M.
As Nis a le submodule o D , we can also conside on N he induced V- il a ion:
Vs(N) := N ∩ Vs[n](D )
4
o each s∈Zkas de ined be o e (see 2.1). The abelian g oup
RV(N) := M
s∈Zk
Vs(N)us
is in ac an homogeneous submodule o RV[n](D ). The RV(D)–module RU(M) is na u ally isomo phic o
he quo ien RV[n](D )/RV(N).
De ini ion 2.4 We call he module RU(M) he Rees module associa ed wi h he il a ion U•(M).
Le us now conside he il a ion VΓ
•(D) as in 2.1. Then he Rees ing associa ed wi h his il a ion is
de ined in a simila way as
RΓ(D) = M
s∈Zk
VΓ
s(D)us⊂ D[u±1]
he p oduc being induced by he one o D[u±1].
I M=D
Nis a ini ely gene a ed D-module and U•(M) is a good il a ion on Mwi h espec o V•(D)
hen, o any cone Γ as in 2.1, he abelian g oup
RΓ(M) = M
s∈Zk
UΓ
s(M)us⊂ M[u±1]
is a le g aded RΓ(D)-module.
De ini ion 2.5 We call he module RΓ(M) he Rees module associa ed wi h he il a ion U•(M)and he cone
Γ.
The ac ha a il a ion is good in he sense o p e ious sec ion 2.1 is hen equi alen o he ac ha he
Rees module o Mis ini ely gene a ed o e RV(D). Simila ly he goodness o UΓ
•(M) is equi alen o he ac
ha he Rees module RΓ(M) as de ined abo e, is ini ely gene a ed o e RΓ(D). Since RΓDis a Noe he ian
ing, his implies in pa icula an A in ype p ope y: I M′⊂ M is a submodule o a D-module Mendowed
wi h a good il a ion U•(M) hen he induced il a ion on M′, namely UΓ
•(M)∩ M′is good.
The ing RΓ(D) con ains as a sub ing he C–algeb a AΓ=C[ˇ
Γ∩Zk]⊂C[u, u−1], all hese ings being
included in D[u, u−1].
Theo em 2.6 The e is a an Ein (Qk)∗
+such ha o each cone Γ∈ E he Rees module RΓ(M)is la o e
AΓ.
We will p o e a mo e p ecise o m o his esul we in sec ion 4, see heo em 4.3, a e ha ing ecalled in
sec ion 3 he no ion o an analy ic s anda d an.
2.2.2 The ibe a 0
Le us deno e by m he ideal o RV(D) gene a ed by (u1,...,uk). This is a wo sided ideal since i is gene a ed
by cen al elemen s. The ibe a ze o o RV(D) ( esp. o RV[n](D )) is by de ini ion he quo ien ing ( esp.
he quo ien module),
RV(D)
m( esp. RV[n](D )
m.RV[n](D )).
Mo e gene ally he ibe a ze o o RU(M) = RV[n](D )
RV(N)is he quo ien
RU(M)
mRU(M)
which is na u ally isomo phic o he quo ien
RV[n](D )
RV(N) + m.RV[n](D ).
5

No ice ha he ibe a 0 o he module RU(M) can be ze o o a non-ze o RU(M), as shown in he ollowing
example. Le us deno e by I he p incipal ideal o Dgene a ed by he di e en ial ope a o P= 1 + x2
1∂1and
le us suppose k≥1. Then RV(I) is he p incipal ideal o RV(D) gene a ed by P u0
1= 1 + (x2
1∂1u−1
1)u1and
he ibe a ze o o RV(D/I) is hen ze o since RV(I) + m.RV(D) con ains P−(x2
1∂1u−1
1)u1= 1, so ha i is
equal o RV(D).
Simila ly o any k-dimensional cone Γ he C–algeb a AΓ=C[ˇ
Γ∩Zk] has a maximal ideal
mΓ=C[ˇ
Γ∩Zk {0}]
because ˇ
Γ is s ic ly con ex. Then we de ine he ibe a ze o o he Rees module RΓ(M) as RΓ(M)
mΓRΓ(M).As is
explained in [9], his ibe is a module o e he ing RΓ(D)
mΓRΓ(D)≃g ΓD.
2.2.3 Desc ip ion o Rees ings RV(D)and RΓ(D)
I is use ul o desc ibe he Rees ing RV=RV(D) ( esp. RΓ(D)) as sub ings o he ing o ela i e di e en ial
ope a o s DCn×Ck/Ck( esp DCn×SΓ/SΓ). This is no hing bu an explici e sion o he in e p e a ion o a Rees
ing as a ing o ela i e di e en ial ope a o s on he de o ma ion o Y×0Γ o i s no mal cone in Cn×SΓ,see
[9]. We deno e by SΓ he algeb aic a ie y associa ed o he ing AΓ.
Le us de ine i s
A=C{X′U, X′′}[X′, U, ∆] = C{X1U1,...,XkUk, Xk+1,...,Xn}[X′, U, ∆]
whe e U= (U1,...,Uk), X′= (X1,...,Xk), X′′ = (Xk+1,...,Xn), X= (X1, . . . , Xn), ∆ = (∆1,...,∆n) a e
new a iables sa is ying he ollowing ela ions, o i= 1,...,n:
∆iXi=Xi∆i+ 1
he o he ela ions being i ial.
The ing Ais g aded wi h Zkas a alues g oup (we will say ha Ais a Zk–g aded ing) , namely we ha e
A=M
s∈Zk
As
whe e Asis he se
As={X
α,β∈Nn;σ∈Nk
αβσXα∆βUσ∈ A | αβσ ∈C{X′U, X′′}, σi+βi−αi=si;i= 1,...,k}.
P oposi ion 2.7 The e is an isomo phism o Zk–g aded ings
i=iV:RV(D)→ A
de ined by:
•i(uj) = Uj o j= 1,...,k.
•i(xju−1
j) = Xj o j= 1, . . . , k and i(xj) = Xj o j=k+ 1,...,n.
•i(∂juj) = ∆j o j= 1,...,k and i(∂j) = ∆j o j=k+ 1,...,n.
6
P oo .I is clea ha iis injec i e and ha , by he o mula i(xα∂βus) = Xα∆β(Πk
i=1Usi+αi−βi
i), i(Vs(D)us) =
As o all s∈Zk.
Le us gi e now he same desc ip ion o he ing RΓ(D). We shall do i in he only case o in e es o us,
when he cone Γ is basic, which means ha i i is de ined by kindependen linea o ms {L1,...,Lk}gene a ing
he la ice Zk. Fo he sake o simplici y le us suppose k=n.
In his si ua ion we a e going o make a change o a iables in o de o w i e AΓas a polynomial ing.
Fo any iwe can w i e Li= (ℓi1,...,ℓik) and we can suppose by a sui able o de ing o he linea o ms Li
ha he de e minan o he ma ix L= (ℓij) equals 1.
Le us w i e L′ o he in e se ma ix o L, hen he columns {C1,...,Ck}o L′ o m a basis o he dual
cone ˇ
Γ.
The amily uC1,...,uCkgene a es he C–algeb a AΓ=C[ˇ
Γ∩Zk]. He e we w i e uCj=uc1j
1···uckj
kwhe e
he cij’s a e he en ies o he column Cj.
We will w i e Wi=UCi o i= 1,...,k. Simila ly, we ha e Uj=WCj(L) o j= 1,...,k whe e Cj(L) is
he j− h column o he ma ix L. In sec ion 4 we will only use he column Cj=Cj(L′).
Le us ema k he e ha i k < n hen we can s ill de ine W, he ela ionship be ween Wand Ubeing exac ly
he same.
P oposi ion 2.8 The ing RΓ(D)is isomo phic o he sub ing o A′=C{X, W}[∆], which con ains all
polynomials and in which only con e gen powe se ies wi h espec o he amily o monomials XiUi=XiWCi(L)
o i= 1,···k, and Xk+1,··· , Xn, in he gene al case k≤na e allowed.
P oo .A monomial Xα∆βUs−β+αcan be w i en in e ms o X, ∆, W as Xα∆βWL(s−β+α).All ha emains
o be done is o enume a e he monomials de i ed om he powe se ies a iables x1,··· , xn
The ing AΓis iden i ied o he sub ing C[W](= C[W1,...,Wk]) ⊂ RΓ(D) and his inclusion is la . We
shall p o e his s a emen in 4.2.
The ibe o RΓ(D) a he o igin is by de ini ion RΓ(D)⊗C[W]
C[W]
(W)which is isomo phic o a Weyl algeb a,
namely he Weyl algeb a C[X, ∆]. This Weyl algeb a is endowed wi h a Zk–g adua ion by weigh (Xi) = −ǫi
and weigh (∆i) = ǫi o i= 1, . . . , k whe e ǫiis he ec o in Zkwhose j− h coo dina e is δij.
2.3 Mul i il a ions and Be ns ein-Sa o unc ional equa ions
Le us de ail he unc ional equa ion p oblem aised in he in oduc ion. I has al eady been ema ked ha
i gi es ise o a si ua ion whe e a mul i il a ion along ans e se hype su aces comes ou in a na u al way.
These equa ions a e o he ype
P(λ) λ=b(λ) λ1+1
1··· λk+1
k
whe e λ= (λ1,··· , λk) is a k-uple o inde e mina es, and = ( 1,··· , k) is a k-uple o analy ic unc ions on
X. They a e na u ally w i en in he module OXλ1,··· , λk,1
1··· k λ, endowed wi h a s uc u e o a DX[λ]-
module, which can be ex ended in a way disco e ed by B. Malg ange o a DX×Ck-module s uc u e wi h an
ac ion o he 2k a iables j, ∂ jsuch ha λj=−∂ j j.The module M ha we ha e hen o conside is he
module gene a ed o e he ing DX×Ckby λ, wi h i s na u ally de ined mul i il a ion V•(DX×Ck)· λalong
he hype su aces 1= 0,··· , k= 0. The p oo o Sabbah in [9] can be ske ched as ollows: le us de ine o
any linea o m L∈(Qk)⋆wi h posi i e a ional coe icien s ℓj≥0, a il a ion o UL
•(M), associa ed wi h
he linea combina ion PℓjV(j)o he basic V- il a ions (see no a ions a 2). P ecisely we se (see also he
no a ion 3.1 below):
VL
•(DX×Ck) = {P=X
µ,ν∈Nk
Pµν(x, ∂x) µ∂ν
,Xℓi(νi−µi)≤ •}
UL
•(M) = VL
•(DX×Ck)· λ
Then he exis ence o a unc ional equa ion, in which b(λ) is a p oduc o a ine o ms L(λ) + ccomes ou om
he ollowing h ee s eps:
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1) The e a e Be ns ein-Sa o polynomials bL(λ), ela i e o each L, wi h unc ional equa ions
bL(λ) λ∈VL
<0(DX×Ck)· λ.
2) The sa u a ed il a ion
Us(M) =
L∈(Qk)⋆
UL
L(s)(M)
can be de ined by using only a ini e numbe o ixed linea o ms.
3) The sa u a ed il a ion is good which is equi alen o he exis ence o a k-uple o in ege s κ, such ha
∀s∈Nk, Us(M)⊂Us(M)⊂Us+κ(M).
I is s ep 2) which, in he p oo in [9], makes an essen ial use o he no ion o an adap ed an, whose exis ence
is p o ed in his pape , see heo em 2.6. In Bahloul’s pape [4], his s ep is made by cons uc i e me hods which
do no use he la ness p ope y.
3 The analy ic s anda d an
In his sec ion we will summa ize he main esul s o [3]. Since hese esul s a e only gi en o a module D/I
o e Dwhich is a quo ien by an ideal we will hen show b ie ly how o adap hem o a module o he ype
D
N. Le us ema k ha he an used in his pape is ob ained by a es ic ion o he an in [3] o a subse o
linea o ms o he se Ude ined below, namely he linea combina ions o he il a ions V(i).
Le Ube he se o linea o ms Λ : R2n→R, Λ(α, β) = Pn
i=1 eiαi+Pn
i=1 iβiwi h ei+ i≥0 and ei≤0
o i= 1,...,n. I
P=X
αβ
pαβxα∂β
is an elemen in Dwe de ine o dΛ(P) – he Λ-o de o P– o be he maximal alue o Λ(α, β) o α, β such ha
pαβ 6= 0.
The Λ- il a ion FΛ,•(D) is de ined by
FΛ,ℓ =FΛ,ℓ(D) = {P∈ D | o dΛ(P)≤ℓ}
o any ℓ∈R. We will w i e FΛ,<ℓ(D) := {P∈ D | o dΛ(P)< ℓ}. I ei= 0, i= 1 o i= 1,...,n he
co esponding Λ– il a ion is no hing bu he usual il a ion by he o de o di e en ial ope a o s. We shall
deno e i simply F•(D).
Le us ecall ha we ha e ixed k≤n. Fo each linea o m L∈(Qk)∗
+(i.e. he coe icien s o La e
non-nega i e) we deno e by e
L he linea o m on R2nde ined by e
L(α, β) = L(β)−L(α) whe e L(α1,...,αn) =
L(α1,...,αk).
No a ion 3.1 We de ine he il a ion VL
•on D, indexed by he se o alues L(Zk), as:
VL
ℓ(D) = Fe
L,ℓ(D) = {P∈ D | o de
L(P)≤ℓ}
Le us ix L∈(Qk)∗
+. The g aded ing associa ed o he il a ion VL=Fe
Lon D, is by de ini ion
g e
L(D) = M
ℓ∈e
L(Z2n)
Fe
L,ℓ
Fe
L,<ℓ
.
I no con usion is possible we shall w i e simply o dLand g L(D) ins ead o o de
Land g e
L(D).
The g aded ing g L(D) is a ing o di e en ial ope a o s and i s s uc u e is he ollowing: suppose he
coe icien s o he o m La e (e1,...,ek) and suppose we also ha e o de ed he a iables o ha e ei>0 o
1≤i≤ℓ o some ℓ≤k.
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Then he g aded ing g L(D) is isomo phic o he ing
C{xℓ+1,...,xn}[x1,...,xℓ, ∂1,...,∂n].
In his ing he g adua ion is induced by he weigh s weigh (xi) = −ei,weigh (∂i) = ei o 1 ≤i≤ℓand
weigh (xj) = weigh (∂j) = 0 o he wise. The e is only a ini e numbe o ypes o hese ings, one o each
pa i ion o {1,...,k}in o wo se s.
No a ion 3.2 Fo each P∈ D and o each d∈L(Zk)wi h o dL(P)≤dwe deno e by σL
d(P)– he L–symbol
o Po o de d– he class o Pin Fe
L,d/Fe
L,<d. The p incipal symbol o Pis by de ini ion σL(P) = σL
d(P)i
d= o dL(P). Fo P, Q ∈ D we ha e σL(PQ) = σL(P)σL(Q). Fo any le ideal Iin Dwe deno e by g L(I)
he g aded ideal o g L(D)gene a ed by he se {σL(P)|P∈I}. We se σL(0) = 0.
We deno e by D[ ] he C-algeb a O[∂, ] = C{x1,...,xn}[∂1,...,∂n, ] wi h ela ions ( being a new a iable)
•[ , a] = [ , ∂i] = [a, b] = [∂i, ∂j] = 0,
•[∂i, a] = ∂a
∂xi ,
o a, b ∈ O and i= 1,...,n.
The ing D[ ] is isomo phic o he Rees ing associa ed wi h he o de il a ion F•on D. Since his Rees
ing is by de ini ion
RF(D) = M
ℓ∈Z
Fℓ(D) ℓ⊂ D[ ] = D ⊗CC[ ]
o a new a iable , we can de ine an isomo phism o g aded ings ι:D[ ]→ RF(D) by ι(a) = a=a 0, ι(∂j) =
∂j , and ι( ) = 1 = .
The na u al g aded s uc u e o RF(D) can be ansla ed on D[ ]. An homogeneous elemen o deg ee d∈Z
in D[ ] is no hing bu an exp ession X
ℓ+|β|=d
aℓ β∂β ℓ
o some aℓ β ∈ O.
Fo P=Pβpβ(x)∂β∈ D, he elemen P o d(P)∈ RF(D) is called he homogeniza ion o P, whe e o d(P)
is he usual o de o P. I is use ul o see P o d(P)as an elemen o D[ ]. The homogeniza ion o Pis hen
deno ed by h(P) and we ha e
h(P) = X
β
pβ(x)∂β d−|β|
o d= o d(P) he usual o de o P.
Fo each linea o m L∈(Qk)∗
+we can de ine in a na u al way a il a ion VL
•(D[ ]) on D[ ], he L–o de
–deno ed o dL(R)– o an elemen
R=X
ℓαβ
ℓαβxα∂β ℓ
being he maximal alue o L(β)−L(α) o ℓαβ 6= 0. The associa ed g aded ing is deno ed by g L(D[ ]). We
ha e a na u al ing isomo phism om g L(D[ ]) on o g L(D)[ ], whe e wi h he no a ions as be o e we ha e
[∂i, xj] = δij ,δij being he K onecke symbol.
Fo each le ideal Iin Dwe deno e by h(I) he le homogeneous ideal o D[ ] gene a ed by {h(P)|P∈I}.
As in he case o D, we deno e by g L(h(I)) he homogeneous ideal –in g L(D[ ])– gene a ed by he se o
p incipal symbols σL(G) o he elemen s Gin h(I).
The main esul o [3] is he ollowing
Theo em 3.3 Le Ibe a non-ze o le ideal o Dand le h(I)be he associa ed homogenized ideal in D[ ]. Then
he e exis s a pa i ion Eo Uin o con ex a ional polyhed al cones such ha o any Γ∈ E he ideals g Λ(h(I))
and g Λ(I)do no depend on Λ∈Γ.
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