Filtered restriction of geometric D-modules

0.2 MB·English

by  Rémi Arcadias

Filtered restriction of geometric D-modules
2
1 R´emi Arcadias
0
2 Depto. de A´lgebra, Fac. Matem´aticas, Campus Reina Mercedes
41012 Sevilla (Spain)
n
a E-mail: [email protected]
J

0 January 23, 2012
2

]
G
Introduction
A
. Let D (resp. D′) be the ring of germs at the origin of linear partial differential
h
t operators with analytic coefficients on Cn (resp. Cn × Cp). We take a sys-
a tem of coordinates (x ,...,x ) on Cn and (x ,...,x ,t ,...,t ) on Cn ×Cp
m 1 n 1 n 1 p
and we denote by i : Cn → Cn ×Cp the inclusion x → (x,0). Let M be a
[ finitely generated D′-module specializable along Y = {t1 = ··· = tp = 0}. The

1 D-module theoretic restriction i∗M of M along Y is defined as the complex
v (D′/ t D′)⊗LM,realizedasaKoszulcomplex(M⊗ΛiCp) ,whosecohomol-
i i
9
ogy gProups are finitely generated over D. T. Oaku and N. Takayama show in
5
[10] how to realize this complex by a complex composed of finitely generated
2
4 D-modules. Let V(D′) denote the V-filtration of Malgrange-Kashiwara along
. Y, and let k be the maximal integral root of the b-function of M along Y.
1 1
Then(M⊗ΛiCp) is quasi-isomorphicto the sub-complex (V (M)⊗ΛiCp) .
0 i k1+i i
2 Then a free resolution of i∗M can be computed as follows: let
1
: ···→L →L →M →0 (1)
v 1 0
i
X beafreeresolutionofM adaptedtotheV-filtration. Then(V (M)⊗ΛiCp)
k1+i i
r is isomorphic, in the derived category of D-modules, to the complex
a
V (L ) V (L )
···→ k1 1 → k1 0 →0,
t V (L ) t V (L )
i k1+1 1 i k1+1 0
P P
denoted by (V (L )/( t V (L ))) .
k1 i j j k1+1 i+1 i
Ontheotherhand,PthemostclassicalfiltrationinD-moduletheoryisthatby
theorder,denotedby(F (D)). Onedefinesthenotionofaminimalfilteredfree
d
resolution of a F-filtered D-module, thus one has the notion of Betti numbers
of such a module. We remark that the notion of Betti numbers can be defined
as well for complexes of D-modules in the derived category of F-filtered D-
modules DF(D). Furthermore, T. Oaku and N. Takayama[11] and M. Granger
and T. Oaku[6] define the notion of a minimal bifiltered free resolution of a
1
(F,V)-bifiltered D′-module. With the help of this theory, we establish that the
isomorphism
(V (M)⊗ΛiCp) ≃(V (L )/( t V (L )))
k1+i i k1 i j k1+1 i+1 i
Xj
can be seen in the category DF(D) (Proposition 3.1).
AfurtherpropertyoncomplexesofF-filteredD-modulesisthatofstrictness,
a property any filtered free resolution of a F-filtered module satisfies. In the
case where p =1 and t: M →M is injective, we give conditions such that the
complex
(V (M)⊗ΛiCp) =(0→V (M)→t V (M)→0)
k1+i i k1+1 k1
is strict (Proposition 3.3). In that case the complex
(V (L )/(tV (L )))
k1 i k1+1 i+1 i
becomes a F-filtered free resolution of the module M/tM. This latter fact,
suggested by Toshinori Oaku, was the original motivation of this paper.
In the last section, we apply our results to the algebraic local cohomology
module N = O[1/f]/O, seen as a D-module. Here f is a quasi-homogeneous
polynomial with an isolated singularity at the origin. N is endowed with the
good F-filtration
F (N)= F (D)[1/fj],
d d−j−1
1≤Xj≤k
which takes into account the order of the pole f, where −k is the least integral
root of the Bernstein-Sato polynomial associated with f. By using the restric-
tion,wegiveaminimalpresentationassociatedwiththisdata(Proposition4.3).
1 Free resolutions of D-modules
LetD =C{x ,...,x }[∂ ,...,∂ ]denotetheringofgermsattheoriginoflinear
1 n 1 n
partialdifferentialoperatorswithanalyticcoefficientsonCn. Itisendowedwith
a filtration by the order, denoted by (Fd(D))d∈Z.
LetM beafinitelygeneratedleftmoduleoverD. AnF-filtration(Fd(M))d∈Z
onM isanexhaustivesequenceofsubspacessatisfyingFd(D)Fd′(M)⊂Fd+d′(M)
for any d,d′. It is called a good F-filtration if moreoverthere exist f ,...,f ∈
1 r
M and a vector shift n=(n ,...,n ) such that for any d,
1 r
F (M)= F (D)f .
d d−ni i
X
For example, we denote by Dr[n] the free module Dr with basis e ,...,e en-
1 r
dowed with the filtration
F (Dr[n])= F (D)e .
d d−ni i
X
2
Thus a good F-filtration of M is defined by a surjective map Dr[n]→M.
One then defines the notion of a F-filtered free resolution: it is an exact
sequence
···→L →L →M →0 (2)
1 0
which induces for each d an exact sequence of vector spaces
···→F (L )→F (L )→F (M)→0.
d 1 d 0 d
Following T. Oaku and N. Takayama[11] and M. Granger and T. Oaku[6], we
are able to define the notion of a minimal F-filtered free resolution. To that
end we introduce, following F.J. Castro-Jim´enez and L. Narva´ez-Macarro[4],
the homogenization ring RD= F (D)Td. It is isomorphic to the ring
d
L
D(h) =C{x}[∂ ,...,∂ ,h]
1 n
satisfying for any i, ∂ x −x ∂ = h. The ring D(h) is graded by the order in
i i i i
∂,h. The homogenization of a F-filtered module M is defined by
RM = F (M)Td.
d
Md
In particular we have an isomorphism
R(Dr[n])≃(D(h))r[n]
where at the right hand side the vector shift n refers to the grading of the
module. The F-filtered free resolution (2) induces a graded free resolution
···→RL →RL →RM →0. (3)
1 0
Thisresolutioniscalledminimal ifalltheentriesofthematricesrepresentingthe
maps RL → RL belong to the maximal two-sided ideal of D(h) generated
i i−1
by (x ) ,(∂ ) ,h. The resolution (2) is called minimal if it induces a minimal
i i i i
graded free resolution (3). There exists a minimal F-filtered free resolution of
M, unique up to F-filtered isomorphism (see [6]).
Let us denote now D = C{x ,...,x ,t ,...,t }[∂ ,...,∂ ,∂ ,...,∂ ]
x,t 1 n 1 p 1 n t1 tp
the ring of germs of linear partial differential operators with analytic coeffi-
cients on Cn×Cp. It is endowed with the F-filtration by the order (F (D )).
d x,t
Another filtration is the so-called V-filtration of B. Malgrange and M. Kashi-
wara,denoted by (Vk(Dx,t))k∈Z, defined as follows: let us define the V-order of
a monomial
ordV(λxαtµ∂β∂ν)= ν − µ ,
t i i
X X
with λ ∈ C\0, then V (D ) denote the set of operators whose development
k x,t
only contains monomials xαtν∂β∂µ having order at most k.
t
We have also the notion of a V-filtration of a D -module: that is an ex-
x,t
haustive sequence of subspaces (Vk(M))k∈Z such that for any k,k′, we have
3
Vk(Dx,t)Vk′(M)⊂Vk+k′(M). As aboveletDxr,t[m]denotethe free moduleDxr,t
with basis e ,...,e endowed with the V-filtration
1 r
V (Dr [m])= V (D )e .
k x,t k−mi x,t i
X
IfM isendowedwithaV-filtrationdefinedfromasurjectivemapDr [m]→M,
x,t
then M admits a V-filtered free resolution: that is an exact sequence
···→Dr1 [m(1)]→Dr0[m(0)]→M →0
x,t x,t
which induces for each k an exact sequence of vector spaces
···→V (Dr1 [m(1)])→V (Dr0[m(0)])→V (M)→0.
k x,t k x,t k
A V-filtered free resolution is the object needed in the computation of the re-
striction, as we shall see in the next section.
Let us describe now how to define the notion of a minimal V-filtered free
resolution of M. To that end we introduce the bifiltration
F (D )=F (D )∩V (D )
d,k x,t d x,t k x,t
for each d,k ∈ Z. A bifiltration of a D -module M is an exhaustive se-
x,t
quence of subspaces (F (M)) satisfying F (M) ⊂ F (M)∩F (M)
d,k d,k d+1,k d,k+1
and Fd,k(Dx,t)Fd′,k′(M) ⊂ Fd+d′,k+k′(M) for any d,d′,k,k′. It is called a
good bifiltration if moreover there exist f ,...,f ∈ M and two vector shifts
1 r
n=(n ,...,n ) and m=(m ,...,m ) such that for any d,k,
1 r 1 r
F (M)= F (D )f .
d,k d−ni,k−mi x,t i
X
For example, we denote by Dr[n][m] the free module Dr with basis e ,...,e
x,t 1 r
endowed with the good bifiltration
F (Dr [n][m])= F (D )e .
d,k x,t d−ni,k−mi x,t i
X
Let us point out that if M is endowed with a good bifiltration (F (M)), then
d,k
it is endowed with a good F-filtration F (M) = ∪ F (M) and a V-filtration
d k d,k
V (M)=∪ F (M). A bifiltered free resolution of M is an exact sequence
k d d,k
···→Dr1[n(1)][m(1)]→Dr0[n(0)][m(0)]→M →0
x,t x,t
which induces for each d,k an exact sequence of vector spaces
···→F (Dr1 [n(1)][m(1)])→F (Dr0 [n(0)][m(0)])→F (M)→0.
d,k x,t d,k x,t d,k
TheringD(h) =RD isendowedbyaV-filtrationV (D(h))similarlyasabove,
x,t x,t k x,t
by giving the weight −1 to each t and the weight 1 to each ∂ . Equivalently,
i ti
we have V (D(h)) = V (D )Td. In the same way we endow RM with the
k x,t d k x,t
L
4
V-filtration V (RM)= F (M)Td. For example, we have an isomorphism
k d d,k
of V-filtered graded D(hL)-modules
x,t
R(D(r)[n][m])≃(D(h))r[n][m]
x,t x,t
where the vector shift [n] (resp. [m]) refers to the grading (resp. V-filtration).
Let us take a bifiltered free resolution
···→L →L →M →0. (4)
1 0
It induces a V-filtered graded free resolution
···→RL →RL →RM →0,
1 0
thus a bigraded free resolution
···→grV(RL )→grV(RL )→grV(RM)→0. (5)
1 0
Since it makes sense to talk about a minimal bigraded free resolution as (4),
the authors of [6] adopt the following definition: a bifiltered free resolution (4)
is said to be minimal if so is the bigraded free resolution (5). They prove the
existence and uniqueness (up to bifiltered isomorphism) of such a resolution.
2 Free resolutions of complexes of D-modules
Inthatsectionweintendtodefinethenotionofaminimalfilteredfreeresolution
ofacomplexofD-modulesinthecorrespondingderivedcategory. Wewillreview
some facts, perhaps well-known to specialists, for the clarity of our text.
Let us denote by CF(D) the category of bounded complexes ··· → M →
i
M →··· where for each i, M is a F-filtered D-module and the differentials
i−1 i
are F-adapted.
Amapα:M →N issaidtobeafilteredquasi-isomorphismiftheinduced
• •
map gr(α):grFM →grFN is a quasi-isomorphism.
• •
If α : M → N is a morphism in CF(D), we denote by C(α) the map-
• •
ping cone of α, defined by C(α) = M ⊕ N with differential (x,y) 7→
n i−1 i
(−φ(x),ψ(y)+α(x)). ItisendowedwiththefiltrationF (C(α) )=F (M )⊕
d i d i−1
F (N ). In that way we have natural isomorphisms F (C(α)) ≃ C(F (α)) and
d i d d
grF(C(α))≃C(grF(α)).
Lemma 2.1. Let α : M → N as above. If for any d, F (α) is a quasi-
• • d
isomorphism, then grF(α) is a quasi-isomorphism.
Proof. F (α) is a quasi-isomorphism is equivalent to saying that C(F (α)) is
d d
exact. ThusF (C(α))isexactanditeasilyimpliesthatgrF(C(α))≃C(grF(α))
d
is exact. That gives the result.
We have a converse statement:
5
Proposition 2.1. Let α : M → N be a morphism in CF(D), such that for
• •
each i, M and N are endowed with good filtrations. If grF(α) is a quasi-
i i
isomorphism, then α is a quasi-isomorphism and induces for each d a quasi-
isomorphism F (α).
d
Proof. We have that grF(C(α)) ≃ C(grF(α)) is exact. Since each C(α) is
i
endowedwithagoodfiltration,thenbyastatementanalogousto[6],Proposition
2.5,itfollowsthatC(α) is exact,moreoverfor eachd, F (C(α)) is exact,which
d
establishes the statement.
Corollary 2.1. Let α : M → N be a morphism in CF(D), such that for
• •
each i, M and N are endowed with good filtrations. Then α is a filtered quasi-
i i
isomorphism if and only if for each d, F (α) is a quasi-isomorphism. Moreover,
d
if α is a filtered quasi-isomorphism, then it is a quasi-isomorphism.
A complex
···→M →φi M →···
i i−1
in CF(D) is said to be strict if for each i and d we have Imφ ∩F (M ) =
i d i−1
φ (F (M )).
i d i
Lemma 2.2. Let
M φ2 //M φ1 // M
2 1 0
α1
N(cid:15)(cid:15) ψ2 // N(cid:15)(cid:15) ψ1 // N(cid:15)(cid:15)
2 1 0
be a commutative diagram of filtered D-modules, such that the rows are com-
plexes and α induces isomorphisms H (M )) ≃ H (N )) and H (F (M ))) ≃
1 1 • 1 • 1 d •
H (F (N ))). Then α induces an isomorphism
1 d • 1
F (φ (M )) F (ψ (N ))
d 2 2 d 2 2
≃ .
φ (F (M )) ψ (F (N ))
2 d 2 2 d 2
Proof. We have a commutative diagram with exact rows
Fd(Kerφ1) // Kerφ1 // Kerφ1 // 0
φ2(Fd(M2)) Imφ2 Fd(Kerφ1)+Imφ2
iso iso
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Fd(Kerψ1) // Kerψ1 // Kerψ1 //0,
ψ2(Fd(N2)) Imψ2 Fd(Kerψ1)+Imψ2
then by the five lemma, α induces an isomorphism
1
Kerφ Kerψ
1 1
≃ .
F (Kerφ )+Imφ F (Kerψ )+Imψ
d 1 2 d 1 2
6
Similarly, from the diagram with exact rows
0 // Fd(Kerφ1) // Kerφ1 // Kerφ1 //0
Fd(Imφ2) Imφ2 Fd(Kerφ1)+Imφ2
iso iso
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // Fd(Kerψ1) // Kerψ1 // Kerψ1 // 0,
Fd(Imψ2) Imψ2 Fd(Kerψ1)+Imψ2
we deduce that α induces an isomorphism
1
F (Kerφ ) F (Kerψ )
d 1 d 1
≃ .
F (Imφ ) F (Imψ )
d 2 d 2
Similarly, from the diagram with exact rows
0 // Fd(φ2(M2)) // Fd(Kerφ1) // Fd(Kerφ1) //0
φ2(Fd(M2)) φ2(Fd(M2)) Fd(φ2(M2))
iso iso
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // Fd(ψ2(N2)) // Fd(Kerψ1) // Fd(Kerψ1) // 0,
ψ2(Fd(N2)) ψ2(Fd(N2)) Fd(ψ2(N2))
the result follows.
Corollary 2.2. Let α : M → N be a morphism in CF(D) which is a quasi-
• •
isomorphism and which induces quasi-isomorphisms F (M ) → F (N ). Then
d • d •
C is strict if and only if D is strict.
• •
LetKF(D)denotethecategorywhoseobjectsaretheobjectsofCF(D)and
themapsaretakenmodulo(F-adapted)homotopies. ThenDF(D)denotesthe
localisation of KF(D) with respect to filtered quasi-isomorphisms, as done in
[12], 2.1.8 (see also [8]). In other terms, DF(D) is the localization of KF(D)
withrespecttothenullsystemcomposedofthecomplexesM suchthatgr(M )
• •
is acyclic (see in [7] an introduction to localization of categories). D(D) will
denote the derived category of D-modules.
Because of Corollary 2.2, the strictness of a complex M , where each M is
• i
endowed with a good filtration, makes sense in DF(D).
As pointedoutin[13],forastrictcomplexM ,foranyi,d,H (F (M ))isa
• i d •
subspace of H (M ), more precisely we can define a goodfiltrationon H (M )
i • i •
by F (H (M))=H (F (M )).
d i i d •
Let m denotes the maximal graded ideal of grF(D) generated by
x ,...,x ,ξ ,...,ξ
1 n 1 n
and m(h) the maximal graded two-sided ideal of D(h) generated by
x ,...,x ,∂ ,...,∂ ,h.
1 n 1 n
We have grF(D)/m≃C and D(h)/m(h) ≃C.
7
Definition 2.1. Let L =(···→L →ψi L →···) a complex of CF(D) with
• i i−1
for each i, Li = Dri[n(i)]. The complex L• is said to be minimal if for any i,
the matrix representing Rψ as its entries in m(h) (equivalently, for any i, the
i
matrix representing grψ as its entries in m).
i
Definition 2.2. Let M be a complex of DF(D). A minimal resolution of M
• •
is a minimal complex L isomorphic to M in DF(D).
• •
The numbers r(i) and the shifts n(i) arising in a minimal resolution of M
•
make sense. In fact, denoting
βi,j = dimCTorgirF(D)(grF(M),C)j
= dimCTorDi (h)(RF(M),C)j,
we have β =card{k,n(i) =j}. In particular r = β .
i,j k i j i,j
Let us see finally the link between resolutions ofPmodules and resolutions of
complexes. Thenotionofastrictfilteredfreeresolutionofacomplexgeneralizes
thenotionofafilteredfreeresolutionofamodule: regardingafilteredmoduleM
asacomplexinDF(D)concentratedindegree0,astrictfilteredfreeresolution
···→L →L →0ofM (thus H (L )≃M)providesafilteredfreeresolution
1 0 0 •
···→L →L →H (L )→0 in the sense of [11].
1 0 0 •
3 Filtered restriction
LetM be a D -module endowedwith a goodbibiltration(F (M)) suchthat
x,t d,k
M =D V (M). Weassumethatthereexistsanon-zeropolynomialb(s)∈C[s]
x,t 0
suchthatb(t ∂ +···+t ∂ )grV(M)=0,whichisthe caseif M is holonomic.
1 t1 p tp 0
An algorithmto compute such a polynomial can be found in [10]. Let k be an
1
integer such that b(k)6=0 if k>k .
1
Let i : (Cn,0) → (Cn ×Cp,0) denote the embedding x → (x,0) = (x,t).
The restriction i∗M of M along Y = {t = ··· = t = 0} is by definition the
1 p
complex (D / T D )⊗L M in D(D)-modules. Let Λi = ΛiCp. Then
i∗M is reprexs,etntPedi biy txh,te KoDsxz,utl complex over the sequence t1,··· ,tp:
0→M ⊗Λp →δ ···→M ⊗Λ1 →δ M ⊗Λ0 →0.
This complex is made of non finitely generated D-modules. By [9], Proposition
5.2 and [10], section 5, the truncation
0→V (M)⊗Λp →δ ···→V (M)⊗Λ1 →δ V (M)⊗Λ0 →0,
k1+p k1+1 k1
denoted by V ⊗Λ•, is quasi-isomorphic to the above Koszul complex, thus
k1+•
still represents i∗M. The bifiltration allows us to endow V ⊗ Λ• with a
k1+•
filtration F (V ⊗Λ•):
d k1+•
0→F (M)⊗Λp →δ ···→F (M)⊗Λ1 →δ F (M)⊗Λ0 →0.
d,k1+p d,k1+1 d,k1
8
LetusremarkthatsincethebifiltrationofM isgood,thenthereexistsf ,...,f
1 r
such that F (M) = F (D)f . If k is chosen so that for any i,
d,k i d−ni,k−mi i 1
k1 ≥mi, then each fi bPelongs to Vk1(M) and for each i=0,...,p, Fd,k1+i(M)
is a good filtration of V (M).
k1+i
Let
···→L →ψ1 L →ψ0 M →0
1 0
be any bifiltered free resolution of M, with L =Dr(i)[n(i)][m(i)]. In particular
i x,t
it is a V-filtered free resolution. By [10], Theorem 5.3, the complex
V (L ) V (L )
···→ k1 1 → k1 0 →0,
t V (L ) t V (L )
i i k1+1 1 i i k1+1 0
P P
denoted by V (L )/ t V (L ) is a free complex isomorphic to i∗M in
k1 • i i k1+1 •
D(D). We endow it wPith a F-filtration by setting
V (L ) F (L )
F k1 i = d,k1 i
d
(cid:18)tV (L )(cid:19) F (L )∩tV (L )
k1+1 i d,k1 i k1+1 i
F (L )
≃ d,k1 i .
tF (L )
d,k1+1 i
NotethatthefilteredD-moduleV (L )/ t V (L )isnaturallyisomorphic
k1 i i i k1+1 i
to some Dr[n]. P
Proposition 3.1. The complexes V (M)⊗Λ• and V (L )/ t V (L )
k1+• k1 • i i k1+1 •
are isomorphic in DF(D). Thus Vk1(L•)/ itiVk1+1(L•) is a fiPltered free res-
olution of i∗M. P
Proof. Since t ,...,t is a regular sequence in O , we have that the Koszul
1 p x,t
complex
R (D )
0→R (D )⊗Λp →···→R (D )⊗Λ0 → F,V x,t →0
F,V x,t F,V x,t t R (D )
i i F,V x,t
P
isexact. WemayreplaceD bythebifilteredfreemoduleL ,thenthecomplex
x,t i
R (L )
0→R (L )⊗Λp →δ ···→R (L )⊗Λ0 → F,V i →0
F,V i F,V i t R (L )
i i F,V i
P
is exact. Moreoverit is bigraded, thus the complex
F (L )
0→F (L )⊗Λp →δ ···→F (L )⊗Λ0 → d,k1 i →0
d,k1+p i d,k1 i t F (L )
i i d,k1+1 i
P
9
is exact. We have a commutative diagram:
0 0
OO OO
··· //F (M)⊗Λ1 // F (M)⊗Λ0 // 0
d,k1+1 OO d,k1 OO OO
··· // Fd,k1+1(LOO 0)⊗Λ1 //Fd,k1(LOO0)⊗Λ0 // PitFidF,dk,1kOO(1L+01)(L0) // 0
··· // Fd,k1+1(LOO 1)⊗Λ1 //Fd,k1(LOO1)⊗Λ0 // PitFidF,dk,1kOO(1L+11)(L1) // 0
. . .
. . .
. . .
Wededucethatthecomplexes(F (M)⊗Λi) and(V (L )/ t V (L ))
d,k1+i i k1 i i i k1+1 i i
are both isomorphic to the complex associated with the doublePcomplex
(F (L )⊗Λj) .
d,k1+j i i,j
Corollary 3.1. The complex V (L )/ t V (L ) is strict if so is the com-
k1 • i i k1+1 •
plex Vk1+•(M)⊗Λ•. P
Letusconsiderfromnowonaspecialcase: p=1. ThecomplexV (M)⊗
k1+•
Λ• is V (M) →t V (M). Let us assume furthermore that t : M → M is
k1+1 k1
injective. Therestrictioni∗M isconcentratedindegree0withH i∗M ≃M/tM.
0
Thus the complex
V (L ) V (L ) M
···→ k1 1 → k1 0 → →0 (6)
tV (L ) tV (L ) tM
k1+1 1 k1+1 0
is a free resolution of the module M/tM = (V (M)+tM)/tM. That module
k1
is naturally endowed with the filtration
F (M)
F (M/tM):= d,k1 .
d
F (M)∩tM
d,k1
and from Proposition 3.1 and Corollary 3.1 we obtain:
Proposition 3.2. Assume that t : M → M is injective and that for any d,
F (M)∩tV (M) = tF (M). Then the complex (6) is an F-filtered
d,k1 k1+1 d,k1+1
free resolution of M/tM, i.e. for any d the complex
F (L ) F (L ) M
···→ d,k1 1 → d,k1 0 →F →0 (7)
d
tF (L ) tF (L ) (cid:18)tM(cid:19)
d,k1+1 1 d,k1+1 0
10