S´eminai es & Cong `es
8, 2004, p. 129–148
CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL
OPERATORS AND FAITHFUL FLATNESS OF D∞
XOVER DX
by
Luis Na ´aez Maca o & An onio Rojas Le´on
Abs ac . — In hese no es we p o e he ai h ul la ness o he shea o in ini e o de
linea di e en ial ope a o s o e he shea o ini e o de linea di e en ial ope a o s
on a complex analy ic mani old. We gi e he Mebkhou -Na ´aez’s p oo based on he
con inui y o he di ision o ini e o de di e en ial ope a o s wi h espec o a na u al
opology. We ep oduce he p oo o he con inui y heo em gi en by Hause -Na ´aez,
which is simple han he o iginal p oo .
Résumé (Con inui é de la di ision des opé a eu s di é en iels e idèle pla i ude de D∞
Xsu
DX)
Dans ce cou s on d´emon e la id`ele pla i ude du aisceau d’op´e a eu s di ´e en iels
lin´eai es d’o d e in ini su le aisceau d’op´e a eu s di ´e en iels lin´eai es d’o d e ini
d’une a i´e e analy ique complexe lisse. La p eu e que nous donnons es celle de
Mebkhou -Na ´aez, qui u ilise la con inui ´e de la di ision d’op´e a eu s di ´e en iels
d’o d e ini pa appo `a une opologie na u elle. Nous ´ep oduisons la p eu e de
Hause -Na ´aez du h´eo `eme de con inui ´e, qui es plus simple que la p eu e o iginale.
In oduc ion
The shea OXo holomo phic unc ions on a complex analy ic mani old Xis he
i s na u al example o le module o e he shea o linea di e en ial ope a o s DX
on X. He e, as usual, di e en ial ope a o s ha e (locally) ini e o de . In ac , he e
is ano he na u al shea o noncommu a i e ings ex ending DX, called he shea o
linea di e en ial ope a o s o in ini e o de ,D∞
X, in oduced by Sa o. The le DX-
module s uc u e on OXex ends o a le D∞
X-module s uc u e in such a way ha
D∞
X⊗DXOX=OX.
Fo any holonomic le DX-module M, we know by he cons uc ibili y heo em o
Kashiwa a [9] (see also [12], [13]) ha he complex o holomo phic solu ions o M,
R HomDX(M,OX), is cons uc ible. The canonical DX-linea biduali y mo phism
M−→ R HomCX(R HomDX(M,OX),OX)
2000 Ma hema ics Subjec Classi ica ion. — 32C38, 32S60.
Key wo ds and ph ases. — In ini e o de di e en ial ope a o , di ision heo em.
Bo h au ho s we e pa ially suppo ed by BFM2001-3207 and FEDER.
c
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130 L. NARV´
AEZ MACARRO & A. ROJAS LE ´
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induces a D∞
X-linea mo phism
(*) D∞
X⊗DXM−→ R HomCX(R HomDX(M,OX),OX).
The local biduali y heo em o Mebkhou asse s ha (*) is an isomo phism o any
holonomic module M(see [11, 11.3] in his olume). This heo em is an essen ial
ing edien o he “ ull” Riemann-Hilbe co espondence, which es ablishes an equi -
alence be ween h ee ca ego ies: he bounded de i ed ca ego y o egula holonomic
complexes o DX-modules, he bounded de i ed ca ego y o holonomic complexes o
D∞
X-modules and he bounded de i ed ca ego y o analy ic cons uc ible complexes
(see 11.4 in loc. ci .). The shea D∞
Xdoes no ha e any known ini eness p ope ies like
DX, bu o p o e he ull Riemann-Hilbe co espondence one needs o know ha he
ex ension DX⊂D∞
Xis ai h ully la . This esul has been s a ed and p o ed o he
i s ime in [17] (see also [1]), and i s p oo depended on he mic olocal machine y.
The aim o hese no es is o gi e an elemen a y sel -con ained p oo o he ai h ul
la ness o he shea o di e en ial ope a o s o in ini e o de o e he shea o di -
e en ial ope a o s o ini e o de . The me hod we ollow is ha o [14], whose i s
s ep consis s in conside ing he ing o di e en ial ope a o s o in ini e o de as he
comple ion o he co esponding ing o ini e o de o a na u al opology, and hen
mimic Se e’s p oo o he ai h ul la ness o he comple ion o a noe he ian local
ing o e he ing i sel [18]. The essen ial echnical ool is he con inui y o he
Weie s ass-G aue -Hi onaka di ision o di e en ial ope a o s [2, 3]. We ep oduce
wi h de ail he p oo gi en in [8], which simpli ies he o iginal p oo in [14]. As a
complemen we ske ch he esul s o [15] o he case o di e en ial ope a o s wi h
polynomial coe icien s (Weyl algeb a).
We would like o hank He wig Hause o a ca e ul eading o hese no es and o
help ul sugges ions.
1. Topological s uc u e on ings o linea di e en ial ope a o s wi h
analy ic coe icien s
Le Xbe a complex analy ic mani old o pu e dimension n, coun able a in ini y.
Le us deno e by OX he shea o holomo phic unc ions and by DX he shea o linea
di e en ial ope a o s (c . [6]). Fo each open se U⊂X, he space OX(U) endowed
wi h he opology o uni o m con e gence on compac se s is a F ´eche space,i.e. a
comple e me izable locally con ex space (i is also a nuclea space, c . [5] o de ails).
The Banach open mapping heo em shows ha he p ope y o being con inuous o
aC-linea endomo phism P:OX→OXis a local p ope y. Fo ha , le {Ui}
be an open co e ing o X, ha we can ake as coun able, such ha each es ic ion
P|Ui:OX|Ui→OX|Uiis con inuous. Fo any open se U⊂X, he canonical injec ion
OX(U),→QOX(U∩Ui) is a closed inme sion by he open mapping heo em (i s
image is he ke nel o he ˇ
Cech map QOX(U∩Ui)→QOX(U∩Ui∩Uj) by he
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shea condi ion). Hence, he con inui y o P(U) : OX(U)→OX(U) comes om he
con inui y o QP(U∩Ui) : QOX(U∩Ui)→QOX(U∩Ui). As a consequence, he
p e-shea o C-linea con inuous endomo phisms o OX,Hom op(OX,OX), is ac ually
a shea .
The ollowing p oposi ion is well-known (c . [14], p op.2.1.4):
P oposi ion 1.1. — Fo any con inuous C-linea endomo phism P:OX→OXand
o any sys em (U;x1, . . . , xn)o local coo dina es o X, he e a e unique holomo phic
unc ions aα∈OX(U),α∈Nn, such ha
P|U=X
α∈Nn
aα
1
α!∂α,
wi h ∂= (∂/∂x1, . . . , ∂/∂xn)and lim|α|→∞ |aα|1/|α|= 0 uni o mly on any compac
se o U. Equi alen ly, he unc ion
(p, ξ)∈U×Cn7−→ X
α∈Nn
aα(p)ξα∈C
is holomo phic.
F om now on, we will deno e D∞
X=Hom op(OX,OX) and call i shea o in ini e
o de linea di e en ial ope a o s. F om he abo e p oposi ion we deduce ha i
coincides wi h he shea o in ini e o de linea di e en ial ope a o s de ined in [16,
17].
The ollowing p oposi ion is p o ed in [14], p op.2.1.3.
P oposi ion 1.2. — Le P:OX→OXbe a C-linea endomo phism. The ollowing
p ope ies a e equi alen :
a) Pis con inuous.
b) Fo any pai K, K0⊆Xo compac se s wi h K⊂
◦
K0, he e is a cons an
CK,K0>0such ha |P( )|K⩽CK,K0| |K0 o any holomo phic unc ion de ined
on a neighbo hood o K0.
Co olla y 1.3. — The shea DXo ( ini e o de ) linea di e en ial ope a o s is a sub-
shea (o ings) o Hom op(OX,OX).
P oo . — Le Pbe a sec ion o DXo e an open se U⊂X. Since con inui y is a
local p ope y, we can suppose ha Uis a connec ed open se o Cn. Then Padmi s
a unique exp ession
P=X
α∈Nn,|α|⩽d
aα
1
α!∂α,
whe e dis he o de o Pand he aαa e holomo phic unc ions on U. Le K, K0⊆U
be a pai o compac se s as in p oposi ion 1.2, b) and le be a holomo phic unc ion
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on a neighbo hood o K0. F om Cauchy inequali ies we deduce ha
|P( )|K=X
|α|⩽d
aα
1
α!∂α( )K
⩽X
|α|⩽d
|aα|K −|α|| |K0
whe e is he dis ance be ween Kand U−
◦
K0. By p oposi ion 1.2, we conclude ha
Pis con inuous.
De ini ion 1.4 ([14], dé .2.1.6). — Fo any open se U⊆X, he canonical opology o
D∞
X(U) o DX(U) is de ined as he locally con ex opology gi en by he semi-no ms
p(K,K0):P∈D∞
X(U)7−→ p(K,K0)(P) := sup {|P( )|K/| |K0| ∈OX(K0), 6= 0},
indexed by pai s (K, K0) o compac se s in Uwi h K⊂
◦
K0.
Fo any coo dina e sys em (U;x1, . . . , xn) in X, we can use Cauchy inequali ies as
in co olla y 1.3 and p oposi ion 1.1 o p o e ha he map
X
α∈Nn
aα(x)1
α!∂α7−→ X
α∈Nn
aα(x)yα
is an isomo phism o locally con ex ec o spaces be ween D∞
X(U) endowed wi h he
canonical opology and he space o holomo phic unc ions on U×Cnendowed wi h
he opology o uni o m con e gence on compac se s. This isomo phism depends on
local coo dina es and ca ies he space DX(U) in o he space o holomo phic unc ions
on U×Cnwhich a e polynomials wi h espec o he second ac o . Consequen ly,
D∞
X(U) is a F ´eche (and nuclea ) space and DX(U) is dense in D∞
X(U). We can w i e
hen D∞
X(U) =
DX(U).
In ac , in [14,§2] i is p o ed ha D∞
Xendowed wi h he canonical opology is a
shea wi h alues in he ca ego y o F ´eche C-algeb as.
Le us deno e by On,Dn,D∞
n he s alk a he o igin o he shea es OCn,DCn,D∞
Cn
espec i ely. Fo ρ= (ρ1, . . . , ρn), L= (L1, . . . , Ln) in (R∗
+)nle us conside he
pseudo-no m |−|L
ρ:D∞
n→R+∪{+∞} whose alue a P=Paβ∂β=Pαβ aαβxα∂β
is
(1) |P|L
ρ=X
β
|aβ|ρ|β|!Lβ=X
αβ
|aαβ|·|β|!ραLβ∈R+∪ {+∞}.
Since β!⩽|β|!⩽n|β|β!, we could also use β! ins ead |β|! in (1) o ob ain an
equi alen sys em o pseudo-no ms. Ne e heless, he choice o |β|! is o ced by he
p oo s o he majo a ions needed o ob ain he no m es ima es o heo em 2.11 (see
[14, 2.2.4] and [8]).
Le us deno e by D∞
n(ρ) he subspace o D∞
nwhe e |−|L
ρ akes ini e alues o any
L∈(R∗
+)nand le us w i e Dn(ρ) := Dn∩D∞
n(ρ). The semi-no ms |−|L
ρ,L∈(R∗
+)n,
de ine a F ´eche opology on D∞
n(ρ).
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CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 133
Following [8], we conside weigh s λ, µ ∈(N∗)nand, o eal numbe s s, > 0,
ρ=sλ= (sλ1, . . . , sλn), L= −µ= ( −µ1, . . . , −µn). When λis ixed, we deno e
|−|µ,
s:= |−|L
ρ,D∞
n(s) := D∞
n(ρ) and Dn(s):=Dn(ρ).
In he case whe e Uis an open polycylinde o Cncen e ed a 0 o poly adius
σ=sλ
0, 0 < s0⩽+∞, we ha e
D∞
Cn(U) =
0<s<s0
D∞
n(s),DCn(U) =
0<s<s0
Dn(s),
and he canonical opology o D∞
Cn(U) ( esp. DCn(U)) is he ( opological) in e se limi
o he D∞
n(s) ( esp. Dn(s)), o 0 < s < s0. In o he wo ds, he canonical opologies
o D∞
Cn(U) and DCn(U) a e gi en by he semi-no ms |−|µ,
s, 0 < s < s0, −µ0
[14], 2.2.3. The las condi ion can be ob ained wi h µ ixed and →0, o aking
= (s)<1 and µ0.
Fo ec o s P= (P1, . . . , Pq)∈(D∞
n)q, ollowing [7] we also de ine
|P|µ,
s:=
q
X
i=1
|Pi|µ,
ss−(i−1),
whe e λ∈(N∗)nis ixed.
In he abo e si ua ion, he p oduc opology on D∞
Cn(U)qand DCn(U)qis also
gi en by he semi-no ms |−|µ,
s, 0 < s < s0, −µ0.
2. The con inui y heo em
In his sec ion, we ix M1, . . . , M ∈Dq
nand a o al well o de ing <in N2n
compa ible wi h sums (c . [4, 1.3]). Whene e we speak abou he o de ing <in
N2n× {1, . . . , q}we mean he o de ing induced by <in he ollowing way:
(α, β, i)<(α0, β0, j)⇐⇒ 


(α, β)<(α0, β0)
o
(α, β) = (α0, β0) and i > j
Gi en
N= (N1, . . . , Nq) =
q
X
i=1
Niei=
q
X
i=1 X
α,β
aαβixα∂βei∈Dq
n, aαβi ∈C,
whe e {ei}i=1...q s ands o he canonical basis o Dq
nas a ee Dn-module, we deno e
by N(N), he New on diag am o N, he se o (α, β, i) in N2n× {1, . . . , q}such ha
aαβi 6= 0 and by σ(N) i s symbol,i.e. he homogeneous componen o No maximal
deg ee wi h espec o he g ading gi en by he o al deg ee in ∂:
σ(N) =
q
X
i=1 X
|β|=dX
α
aαβixα∂βei, d = degT(N) = max deg(Ni).
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The exponen o Nis exp(N) := min{(α, β, i)|aαβi 6= 0,|β|=d}, and he co espond-
ing monomial o σ(N) (and he e o e o N) is, by de ini ion, he ini ial monomial
o N.
Le (αj, βj, ij) be he exponen o Mjwi h espec o he gi en o de ing (we can
assume wi hou loss o gene ali y ha i s coe icien is 1). Also le M0
j=Mj−
xαj∂βjeij. We will deno e by F he - uple (M1, . . . , M ).
The ollowing no ion is needed in he con inui y heo em 2.6:
De ini ion 2.1. — We say ha a gi en weigh λ∈(N∗)nis adap ed o Fi o e e y
posi i e cons an K he e exis s µ∈Nnwi h µi> K, λi o e e y i= 1, . . . , n such
ha
λαj−µβj−ij< λα −µβ −i
o e e y j= 1, . . . , and e e y (α, β, i)∈N(M0
j). We say ha such a µis K-
admissible, o simply admissible, o (F, λ).
Lemma 2.2. — Fo any F= (M1, . . . , M )as abo e, he e exis s a weigh λadap ed
o F.
P oo . — Conside i s he case q= 1. Le π1, π2:N2n=Nn×Nn→Nnbe he
canonical p ojec ions. Fo e e y j= 1, . . . , and e e y β∈π2(N(Mj)) le Mj
βbe
he se o (α, β) in N(Mj) such ha αis minimal in A={α: (α, β)∈N(Mj)}
wi h espec o he componen wise o de . The se Mj
βis ini e, in ac i consis s o
he elemen s (α1, β), . . . , (αs, β), whe e {α1, . . . , αs}is he minimal se o gene a o s
o he ideal A+Nno Nn. The e o e, he se M=SjSβ(Mj
β∪ {(0, β)}) is also
ini e.
Le (σ, ρ)∈Nn×Nn=N2nbe a ec o de ining he gi en o de ing es ic ed o
he ini e se M. We claim ha λ=σis adap ed o F. Fix a posi i e cons an
K, and le pbe an in ege such ha p > max{K+|ρ|, σα +ρβ : (α, β)∈M}. Se
µ= (p, . . . , p)−ρ. We ha e hen
λα −µβ =σα +ρβ −p|β|.
We will show ha he minimum o λα −µβ o (α, β)∈N(Mj) is a ained in
he exponen o Mj. Fi s , we see ha i λα −µβ is minimal, hen (α, β)∈M.
O he wise, he e would be (α0, β)∈N(Mj), γ ∈Nn {0}such ha α=α0+γ, so
λα −µβ =σγ + (λα0−µβ)> λα0−µβ.
Fu he mo e, (α, β) mus be in N(σ(Mj)). O he wise, he e would be (α0, β0)∈
N(Mj)∩Mwi h |β0|>|β|. Then λα−µβ =σα+ρβ −p|β|⩾σα+ρβ −p(|β0|−1) >
p−p|β0|> σα0+ρβ0−p|β0|=λα0−µβ0. The e o e, min{λα−µβ : (α, β)∈N(Mj)}=
min{σα +ρβ −p|β|: (α, β)∈N(σ(Mj)) ∩M}. In his se , |β|is cons an and he
o de ing is de ined by (σ, ρ), so he minimum is a ained in he smalles elemen o
N(σ(Mj)) wi h espec o he o de ing, i.e he ini ial monomial o Mj.
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Now assume q6= 1. Le Mj=Piαβ aj
iαβxα∂βei, and de ine Mj=Piαβ |aj
iαβ|xα∂β∈
Dn. Le (αj, βj) be he exponen o Mjwi h espec o he gi en o de ing in N2n.
Le (αj, βj, ij) be he exponen o Mj. We ha e αj=αjand βj=βj. O he wise,
we would ha e (αj, βj)>(αj, βj). Le ibe such ha (αj, βj, i)∈N(Mj). Then
we ha e by de ini ion o he exponen ha (αj, βj, i)⩾(αj, βj, ij), and he e o e
(αj, βj)⩾(αj, βj), which is in con adic ion wi h he las inequali y.
By he i s pa o he p oo , he e exis s λadap ed o (M1, . . . , M ). Now gi en
a posi i e cons an K he e is µ∈Nnwi h µj> K/q and λj< µjsuch ha
λαj−µβj< λα−µβ o e e y (α, β)∈N(M0
j). Le us see ha λ=qλ is adap ed o F
and µ=qµ is K-admissible o (F, λ). Le (α, β, i)∈N(M0
j), hen (α, β)∈N(Mj),
hence λαj−µβj⩽λα −µβ by cons uc ion. Now we dis inguish wo cases:
I (α, β)>(αj, βj), hen λαj−µβj< λα −µβ. Bu λas well as µa e mul iples
o q, and he e o e λαj−µβj⩽λα −µβ −q, and λαj−µβj−ij< λαj−µβj⩽
λα −µβ −q⩽λα −µβ −i.
I (α, β) = (αj, βj), hen we mus ha e i < ij, hence λαj−µβj−ij< λαj−µβj−i=
λα −µβ −i. In ei he case, we ge he desi ed inequali y.
This comple es he p oo o he lemma.
Lemma 2.3. — Le F1, . . . , Fmbe a ini e numbe o ec o s whose coo dina es a e
in Dq
nas abo e ( hey may ha e dis inc leng hs). Then he e exis s λ∈Nnwhich is
adap ed o all o hem.
P oo . — This is a di ec consequence o he ollowing lemma applied o he ec o
cons uc ed by conca ena ion o F1, . . . , Fm.
The ollowing lemma is clea :
Lemma 2.4. — Le F=(M1, . . . , M )be a ec o in (Dq
n) and le G=(Mi1, . . . , Mik),
wi h 1⩽i1<· · · < ik⩽ . Then e e y λ∈Nnadap ed o Fis also adap ed o G.
Be o e s a ing he main heo em o his sec ion we make one u he de ini ion:
De ini ion 2.5. — Le λ∈(N∗)nbe a weigh . A basis Bo open neighbo hoods o
0∈Cnis said o be a λ-basis i i consis s o open polycylinde s o poly adius sλ o
0< s < s0, o some s0>0. We will say ha Bis adap ed o Fi i is a λ-basis o
some λadap ed o F.
F om he lemmas abo e i ollows ha we can always ind a basis o neighbo hoods
o 0 adap ed o F, and e en a basis adap ed o a ini e numbe o ec o s F1, . . . , Fm.
A e hese p elimina ies we a e eady o s a e he con inui y heo em o he di ision
o linea di e en ial ope a o s:
Theo em 2.6. — Le F= (M1, . . . , M ), wi h Mi∈Dq
nand le Qi(F;E),i= 1, . . . ,
( esp. R(F;E)) be he quo ien s ( esp. he emainde ) o he di ision o E∈Dq
nby F
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(see [4]). Then, o any weigh λ∈(N∗)nadap ed o F, he e exis s a λ-basis Bo
open neighbo hoods o 0∈Cn, such ha o e e y U∈B he C-linea mo phisms
Qi(F;−)( esp. R(F;−)) map DCn(U)qin o DCn(U)( esp. in o DCn(U)q). Fu -
he mo e,
Qi(F;−): DCn(U)q−→ DCn(U),
R(F;−): DCn(U)q−→ DCn(U)q
a e con inuous wi h espec o he canonical opology.
The p oo o heo em 2.6 will be ob ained a e some majo a ions, as in [14], [8],
and i will no be inished un il he end o 2.13. Ou ask consis s o adap ing he
p oo in [8] o he ec o case. Roughly speaking, as explained in loc. ci ., he key
poin is o app oxima e he Dn-linea map D
n→Dq
nde ined by he ini e sys em
o ec o s F= (M1, . . . , M )∈(Dq
n) ins ead o app oxima ing he sys em i sel by
hei ini ial monomials. This idea has been in oduced in [7] in he commu a i e case
o ec o s o con e gen powe se ies.
Le (αj, βj, ij) be he exponen o Mj. Fo e e y i= 1, . . . , n, le Tj:Dq
n→Dq
nbe
he C-linea map de ined by Tjxα∂βei=xα∂β+ejei. Gi en A=Pcγδxγ∂δ∈Dn, we
will deno e by Ao he map PcγδxγTδ:Dn→Dn, and A0=A−Ao(Ais conside ed
he e o be ac ing by mul iplica ion on he le ). Le also {∆1, . . . , ∆ ,∆}be he
pa i ion o N2n× {1, . . . , q}de ined by M1, . . . , M (see [3] and [4] in his olume).
We de ine now he ollowing se s Land J:
L={A∈D
n: exp(Mj) + N(Aj)⊂∆j,∀j= 1, . . . , }
J={B∈Dq
n:N(B)⊂∆}
and he linea map u:L⊕J→Dq
ngi en by u(A, B) = P
j=1 AjMj+B.
F om he di ision heo em ([4], h. 2.4.1) we see ha Land Ja e he se s whe e
quo ien s and he emainde o he di ision by M1, . . . , M a e “allowed” o lie, he
Ajand Ba e jus he quo ien s and he emainde o he di ision o u(A, B) by F
and he map uis bijec i e.
We s a by spli ing uas a sum +w1+w2, wi h
(A, B) = XAo
jxαj∂βjeij+B
w1(A, B) = XA0
jxαj∂βjeij
w2(A, B) = XAjM0
j.
The C-linea map is easily seen o be an isomo phism o C- ec o spaces, by de ini-
ion o Land J.
We ollow he no a ion in he p e ious sec ion ega ding he semino ms |−|µ,
s.
Le E∈Dq
n, and (A, B) = −1(E)∈L⊕J, wi h Aj=Pγδ aj
γδxγ∂δ. Then,
E=Pjγδ aj
γδxαj+γ∂βj+δeij+B. I we ake he | − |µ,
sno m on bo h sides, and
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keep in mind ha (αj, βj, ij) + N(Aj)⊂∆j,N(B)⊂∆ and ha he se s ∆j,∆
a e pai wise disjoin , we ge :
(2) |E|µ,
s=XjXγδ aj
γδxαj+γ∂βj+δeij
µ,
s+B
µ,
s
⩾Xjγδ aj
γδβj+δ!sλ(αj+γ)−(ij−1) −µ(βj+δ).
P oposi ion 2.7. — The e is a cons an C1>0such ha |(w1 −1)E|µ,s
s⩽C1s|E|µ,s
s
o e e y E∈Dq
nand o e e y µadmissible o (F, λ).
P oo . — Le (A, B) = −1(E), wi h Aj=Pγδ aj
γδxγ∂δ. Fi s , we ha e
|(w1 −1)E|µ,s
s=|w1(A, B)|µ,s
s=X
j
A0
jxαj∂βjeij
µ,s
s
=X
jγδ
aj
γδxγ(∂δ−Tδ)xαj∂βjeij
µ,s
s.
Expanding he inne p oduc , we ge
|(w1 −1)E|µ,s
s=X
jγδ
aj
γδxγX
0<ε⩽αj,δ αj
εδ!
(δ−ε)!xαj−ε∂βj+δ−εeij
µ,s
s
⩽X
jγδ X
0<ε⩽αj,δ
|aj
γδ|αj
εδ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)
⩽X
jX
0<ε⩽αjX
δ⩾εX
γ
|aj
γδ|2|αj|δ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε).
The e o e
|(w1 −1)E|µ,s
s
|E|µ,s
s
⩽PjP0<ε⩽αjPδ⩾εPγ|aj
γδ|2|αj|δ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)
Pjγδ |aj
γδ||βj+δ|!sλ(αj+γ)−(ij−1)−µ(βj+δ)
⩽X
jε
Pδ⩾εPγ|aj
γδ|2|αj|δ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)
Pγδ |aj
γδ||βj+δ|!sλ(αj+γ)−(ij−1)−µ(βj+δ)
=X
jε
Pδ⩾εPγ|aj
γδ|2|αj|δ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−µ(βj+δ−ε)
Pγδ |aj
γδ||βj+δ|!sλ(αj+γ)−µ(βj+δ)
⩽X
jε
Pγδ |aj
γδ|2|αj|δ!
(δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−µ(βj+δ−ε)
Pγδ |aj
γδ||βj+δ|!sλ(αj+γ)−µ(βj+δ).
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As in he p e ious case, i is also possible o ob ain a sha pe esul i we assume he
addi ional hypo hesis (?) gi en ea lie on he o de ing. The new s onge s a emen
is he ollowing:
Theo em 2.19. — I he chosen o de ing sa is ies hypo hesis (?), he map u:L⊕J→
An(C)qis a bi-con inuous isomo phism. The e a e cons an s s0>0,C > 0such ha
o e e y E=u(A, B)∈An(C)qwe ha e, o 0< < s < s0,
X
j
|Aj| −µ
s−λ|Mj| −µ
s−λ+|B| −µ
s−λ⩽C|E| −µ
s−λ.
3. Con inuous scissions
P oposi ion 3.1. — Le V⊂Cnbe an open neighbo hood o 0, i, qi⩾1,1⩽i⩽m
in ege s and Fi:D i
V→Dqi
V,1⩽i⩽ma ini e amily o DV-linea maps. The e
exis s a weigh λ∈(N∗)n, a λ-basis Bo open neighbo hoods o 0∈Cnand a amily
o con inuous scissions {σi
U:DV(U)qi→DV(U) i}U∈Bo Fi, i.e. :
Fi(U)◦σi
U◦Fi(U) = Fi(U), U ∈B,1⩽i⩽m
compa ible wi h es ic ions, i.e. σi
U|W=σi
W o W⊂U.
P oo . — Fi s , le us w i e i= (Fi)0and le us ake a G ¨
obne (o s anda d) basis
Gi={Ni
1, . . . , Ni
si}o im( i)⊂Dqi
n o each i= 1, . . . , m (see [4] in his olume) , and
conside he co esponding linea map gi:Dsi
n→Dqi
n. By sh inking Vi necessa y,
we can suppose ha giis he s alk a 0 o a linea map Gi:Dsi
V→Dqi
V. Le us
conside
τi= (Q1(Gi;−), . . . , Qsi(Gi;−)) : Dqi
n−→ Dsi
n.
Le λ∈(N∗)nbe a weigh adap ed o Gi, o e e y i= 1, . . . , m (see lemma 2.3). By
heo em 2.6, he e exis s a λ-basis Bo open neighbo hoods o 0 ∈Cn, such ha o
e e y U∈Band e e y i he C-linea map τi
U:= τi|DV(U)qi:DV(U)qi→DV(U)siis
con inuous. Fu he mo e, he ac ha Giis a G ¨
obne basis implies ha gi◦τi◦gi=
gi. By analy ic con inua ion we ob ain
Gi(U)◦τi
U◦Gi(U) = Gi(U),∀U∈B,∀i= 1, . . . , m.
Le hi:Dsi
n→D i
nbe he linea map such ha i◦hi=gi. By sh inking Vagain
i necessa y, we can suppose ha hiis he s alk a 0 o a linea map Hi:Dsi
V→D i
V.
Le σi:= hi◦τi:Dqi
n→D i
nand σi
U:= hi(U)◦τi
U:DV(U)qi→DV(U) i,U∈B,
which a e con inuous.
F om im( i) = im(gi) (Giis a G ¨
obne basis o im( i)) and gi◦τi◦gi=giwe
deduce i◦σi◦ i= i. Analy ic con inua ion gi es again Fi(U)◦σi
U◦Fi(U) = Fi(U)
o all U∈B, 1 ⩽i⩽m.
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CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 145
Co olla y 3.2. — Le V⊂Cnbe an open neighbo hood o 0, i, si, qi⩾1,i= 1, . . . , m
in ege s and
D i
V
Fi
−−−→ Dsi
V
Gi
−−−→ D i
V, i = 1, . . . , m
a ini e amily o exac sequences o DV-linea maps. The e exis s a weigh λ∈(N∗)n
and a λ-basis Bo open neighbo hoods o 0∈Cnsuch ha o any U∈Band any
i= 1, . . . , m he sequence
DV(U) iFi(U)
−−−−−−→ DV(U)siGi(U)
−−−−−−→ DV(U) i
is exac and opologically spli .
P oo . — By p oposi ion 3.1, he e exis s a weigh λ∈(N∗)n, a λ-basis Bo open
neighbo hoods o 0 ∈Cnand a amily o con inuous scissions {σi
U:DV(U)qi→
DV(U) i}U∈Bo Fi,i= 1, . . . , m compa ible wi h es ic ions. Le us w i e i=
(Fi)0, gi= (Gi)0and σi= lim
U∈Bσi
U o each i= 1, . . . , m. We ha e i= i◦σi◦ i
and
ke gi= im i= ke (1 − i◦σi).
Then, o any U∈Band any M∈ke Gi(U) we ha e
0 = M0− i(σi(M0)) = M−Fi(U)(σi
U(M))0,
and by analy ic con inua ion we deduce M∈ke (1 −Fi(U)◦σi
U)⊂im Fi(U).
P oposi ion 3.3. — Le V⊂Cnbe an open neighbo hood o 0,Mia cohe en DV-
module, i= 1, . . . , m, and
D i
V
Fi
−−−→ Dsi
V
πi
−−−→ Mi−→ 0
a ini e p esen a ion. Then, o any λ-basis Bo open neighbo hoods o 0∈Cnsuch
ha he mo phisms Fi(U)spli o U∈Band i= 1, . . . , m, he sequence
DV(U) iFi(U)
−−−−−−→ DV(U)siπi(U)
−−−−−−→ Mi(U)−→ 0
is exac o U∈Band i= 1, . . . , m.
P oo . — By sh inking Vi needed, we can suppose ha he ke nel o Fihas a good
il a ion on V(c . [6], p op. 10). Then, o any compac polycylinde K⊂Vand
any i= 1, . . . , m he sequence
(5) DV(K) iFi(K)
−−−−−−→ DV(K)siπi(K)
−−−−−−→ Mi(K)−→ 0
is exac by he Ca an-Oka heo em (c . loc. ci ., p op.11). By p oposi ion 3.1, he e
exis a weigh λ∈(N∗)nand a λ-basis Bo open neighbo hoods o 0 ∈Cnsuch ha
o any U∈Band any i= 1, . . . , m he maps Fi(U) : DV(U) i→DV(U)sispli ,
wi h scissions compa ible wi h es ic ions.
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I Kis he closu e o a U0∈B, he sequence (5) is he induc i e limi o he
sequences
DV(U00) iFi(U00)
−−−−−−−→ DV(U00)si−→ coke Fi(U00)−→ 0
wi h U00 ∈B,K⊂U00, and hence i spli s. Now, o any U∈B he sequence
DV(U) iFi(U)
−−−−−−→ DV(U)siπi(U)
−−−−−−→ Mi(U)−→ 0
is he p ojec i e limi o sequences (5), wi h K=U0⊂U,U0∈B, and consequen ly
i is exac .
4. Fai h ul la ness o D∞
Xo e DX
Fai h ul la ness o he ing o di e en ial ope a o s o in ini e o de o e he ing
o di e en ial ope a o s o ini e o de has been s a ed o he i s ime by Sa o,
Kashiwa a and Kawai in [17]. Thei p oo used mic olocal me hods. In his sec ion
we ep oduce he p oo gi en in [14], based on he con inui y o di ision o di e en ial
ope a o s s udied in he p eceden sec ions.
Theo em 4.1. — Fo any complex analy ic mani old X, he ex ension DX→D∞
Xis
ai h ully la .
P oo . — I is enough o p o e ha he ing ex ension Dn→D∞
nis ai h ully la .
Le 0 −→ M1−→ M2−→ M3−→ 0 be an exac sequence o Dn-modules. I is he s alk
a 0 o an exac sequence 0 −→ M1−→ M2−→ M3−→ 0 o DV-modules, whe e Vis an
open neighbo hood o 0. We can ind a commu a i e diag am
0 0 0
0//M1//
OO
M2//
OO
M3//
OO
0
0//D 1
V//
OO
D 2
V//
OO
D 3
V//
OO
0
0//Ds1
V//
OO
Ds2
V//
OO
Ds3
V//
OO
0
0//D 1
V//
OO
D 2
V//
OO
D 3
V//
OO
0
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CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 147
wi h exac ows and columns. By p oposi ions 3.1, 3.3 and co olla y 3.2, he e exis
a weigh λand a λ-basis o neighbo hoods o 0 such ha o any U∈B he diag am
0 0 0
0//M1(U)//
OO
M2(U)//
OO
M3(U)//
OO
0
0//DV(U) 1//
OO
DV(U) 2//
OO
DV(U) 3//
OO
0
0//DV(U)s1//
OO
DV(U)s2//
OO
DV(U)s3//
OO
0
0//DV(U) 1//
OO
DV(U) 2//
OO
DV(U) 3//
OO
0
has exac and opologically spli ing ows and columns.
Then, he co esponding diag am o opological comple ions has also exac ows
and columns. As o any open polycylinde Wwe ha e
DV(W) = D∞
V(W), we deduce
Mi(U) = D∞
V(U)⊗DV(U)Mi(U) and he exac ness o he sequence
0−→ D∞
V(U)⊗DV(U)M1(U)−→ D∞
V(U)⊗DV(U)M2(U)−→ D∞
V(U)⊗DV(U)M3(U)−→ 0
o e e y U∈B. Taking di ec limi s we ob ain he exac ness o
0−→ D∞
n⊗DnM1−→ D∞
n⊗DnM2−→ D∞
n⊗DnM3−→ 0,
and so he ex ension Dn→D∞
nis la .
To conclude, we obse e ha he quo ien opology on Mi(U), U∈B, is sepa a ed.
Then Mi(U),→
Mi(U) = D∞
V(U)⊗DV(U)Mi(U) and Mi,→D∞
n⊗DnMi. In pa icula
Mi6= 0 implies D∞
n⊗DnMi6= 0 and he ex ension Dn→D∞
nis ai h ully la .
Using he co esponding esul s o he Weyl algeb a (Theo ems 2.15 and 2.18) i
is possible o ob ain he ai h ul la ness o he ex ension An(C)→D∞
Cn(Cn), as done
in [15].
Re e ences
[1] J.E. Bj¨
o k –Rings o di e en ial ope a o s, No h Holland, Ams e dam, 1979.
[2] J. B ian¸con &Ph. Maisonobe – Id´eaux de ge mes d’op´e a eu s di ´e en iels `a une
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[3] F.J. Cas o-Jim´
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Ma h´ema ique de F ance, 1974, p. 7–32.
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AEZ MACARRO & A. ROJAS LE ´
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[6] M. G ange &Ph. Maisonobe – A basic cou se on di e en ial modules, in ´
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aez-Maca o – D´emons a ion g´eom´e ique du h´eo `eme
de cons uc ibili ´e, in Le o malisme des six op´e a ions de G o hendieck pou les D-
modules coh´e en s, pa Z. Mebkhou , T a aux en cou s, ol. 35, He mann, Pa is, 1989,
p. 248–253.
[13] , Le h´eo `eme de cons uc ibili ´e de Kashiwa a, in ´
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sys `emes di ´e en iels, I, II [10], ol. II, p. 47–98.
[14] , Le h´eo `eme de con inui ´e de la di ision dans les anneaux d’op´e a eu s di ´e en-
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ciales, Dep. ´
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[17] M. Sa o, T. Kawai &M. Kashiwa a – Mic o unc ions and pseudo-di e en ial equa-
ions, in Hype unc ions and pseudo-di e en ial equa ions, P oc. Con . Ka a a, (1971),
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L. Na ´
aez Maca o, Depa amen o de Algeb a, Facul ad de Ma em´a icas, Uni e sidad de Se illa,
E-41012 Se illa, Spain •E-mail : [email p o ec ed]
A. Rojas Le´
on, Depa men o Ma hema ics, P ince on Uni e si y, U.S.A.
E-mail : [email p o ec ed]
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