Kähler differential algebras for 0-dimensional schemes

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Kähler differential algebras for 0-dimensional schemes

Martin Kreuzer^a,∗, TranN.K. Linh^b, Le Ngoc Long^a,b

aFakultätfürInformatikundMathematik,UniversitätPassau,D-94030Passau, Germany

bDepartmentofMathematics,HueUniversity’sCollegeofEducation,34LeLoi, Hue,VietNam

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received27January2017 Availableonline11January2018 CommunicatedbyBerndUlrich

MSC:

primary13N05

secondary13D40,14N05

Keywords:

Kählerdifferentialalgebra 0-Dimensionalscheme Fatpointscheme Regularityindex Hilbertfunction

Givena0-dimensional scheme Xina projectiven-spacePⁿ over a field K, we study the Kähler differential algebra Ω_RX/KofitshomogeneouscoordinateringR_X.Usingexplicit presentations of the modules Ω^m_R

X/K of Kähler differential m-forms,wedeterminemanyvaluesoftheirHilbertfunctions explicitlyandboundtheirHilbertpolynomialsandregularity indices.DetailedresultsareobtainedforsubschemesofP¹,fat pointschemes,andsubschemesofP²supportedonaconic.

©2018ElsevierInc.Allrightsreserved.

1. Introduction

Inthe paper[5], G. deDominicis andthe firstauthor introducedthe applicationof Kählerdifferential modulesto thestudyof 0-dimensionalsubschemesXof aprojective

* Correspondingauthor.

E-mailaddresses:martin.kreuzer@uni-passau.de(M. Kreuzer),tnkhanhlinh141@gmail.com (T.N.K. Linh),nglong16633@gmail.com(L.N. Long).

https://doi.org/10.1016/j.jalgebra.2017.12.023 0021-8693/©2018ElsevierInc.Allrightsreserved.

space Pⁿ over a field K of characteristic zero. They showed that this graded module overthehomogeneouscoordinateringR_Xcontainsnumericalandalgebraicinformation whichis notreadilyavailablefromthehomogeneousvanishingidealor fromR_X.Later, in [10], theauthors extended andrefined these techniques for fatpoint schemes inPⁿ. Following theclassicalconstructiondescribed byE.Kunzinhis book[15], itisnatural to define the Kähler differential algebra Ω_RX/K =

m∈NΩ^m_R

X/K of X as the exterior algebra ofits Kähler differential module Ω¹_R

X/K. Thisinvites thequestionwhether the Kähler differentialalgebracontainsnumericaland algebraicinformationaboutXwhich is not readily available in R_X or in Ω¹_R

X/K. Thus the following example provided the initial sparkto ignitethecuriosityoftheauthors.

Example 1.1.LetXandYbe twosets ofsixreducedK-rationalpointsinP² suchthat Xiscontainedinanon-singularconic,andsuchthatYconsistsofthreepointsonaline and threepointsonanotherline.ThentheHilbertfunctionsofR_XandR_Yagree,as do theHilbertfunctionsof Ω¹_R

X/K andΩ¹_R

Y/K, namely HF_X= HF_Y : 1 3 5 6 6 · · · HF_Ω¹

RX/K = HF_Ω¹

RY/K : 0 3 8 11 10 7 6 6· · ·. However,theHilbertfunctionsof Ω²_R

X/K andΩ²_R

Y/K aredifferent, andalso theHilbert functionsof Ω³_RX/K andΩ³_RY/K disagree:

HF_Ω²

RX/K : 0 0 3 6 41 0 0· · ·, HF_Ω²

RY/K : 0 0 3 651 0 0· · ·, HF_Ω³

RX/K : 0 0 0 1 00 · · ·, HF_Ω³

RY/K : 0 0 0 110 · · ·. So,theHilbertfunctionsoftheexteriorpowersof Ω¹_R

X/K “know”whetherXiscontained inanirreducibleorareducibleconic.

This observation motivated the studies underlying this paper. Let us now outline its contents in more detail. In the second section we start by recalling the definitions of the Kähler differential module Ω¹_R

X/K and the Kähler differential algebra Ω_RX/K =

R_X(Ω¹_RX/K) ofa0-dimensionalsubschemeXofPⁿ.Asexplainedin[15],wecancalculate an explicitpresentationof Ω^m_R

X/K =m R_X(Ω¹_R

X/K) for everym≥1.Moreover, weshow thatΩ^m_R

X/K =0form> n+ 1,provide asimplifiedpresentationforΩⁿ⁺¹_R

X/K,andshow thattheKoszulcomplexyields anexactsequence

0−→Ωⁿ⁺¹_R

X/K −→Ωⁿ_R

X/K −→ · · · −→Ω²_R

X/K−→Ω¹_R

X/K −→m_X−→0 where m_X isthehomogeneousmaximal idealm_X=x₀,. . . ,x_nofR_X.

In Section 3 we first have a brief glance at the case n = 1, i.e., at 0-dimensional subschemes of aprojective line.Unsurprisingly, inthis case the Hilbert functionsand regularity indices of Ω¹_RX/K and Ω²_RX/K can be writtendown explicitly. Then we look

at the Hilbert functionof Ω^m_RX/K in special degrees.We provide explicitvalues in low degrees,show that theHilbert polynomial(i.e., thevalue inhigh degrees) is constant, andexaminemonotonicityinintermediatedegrees.Theseinsightsarecomplementedby abound fortheregularity indexof Ω^m_R

X/K intermsoftheregularity indexofΩ¹_R

X/K in thelastpartofthesection.

Then,inthenextthreesections,welookatthemodulesofKählerdifferentialm-forms for fat point schemes W. Such schemes are defined by idealsof theform I_W = ℘^m₁¹ ∩

· · ·∩℘^m_s^s,wheretheideals℘i arethevanishingidealsofdistinctK-rationalpointsinPⁿ. InSection4weprovearegularity boundfor Ω^m_R

W/K which usestheregularity index of the fattening of W, i.e., the scheme V defined by I_V = ℘^m₁¹⁺¹ ∩ · · · ∩℘^m_s^s+1. For fat point schemes, we also give bounds on some specific values of the Hilbert polynomial of Ω^m_R

W/K.Inthereducedcase(i.e.,whenm1=· · ·=ms= 1),these valuesarezerofor m≥2,butas soonas oneoftheexponentsmi satisfiesmi ≥2,notallofthese values are zero anymore. Thus the property of W to be reduced is reflected in the values of the Hilbertpolynomials of Ω^m_R

W/K (see Corollary 4.8). Moregenerally, Proposition 4.7 providesupperandlower boundsfortheseHilbertpolynomials.

For the highest non-zero module of Kähler differentials Ωⁿ⁺¹_R

W/K, we can sometimes determineitsHilbertpolynomialexplicitly.Moreprecisely,wehaveformulasforschemes W contained in a hyperplane (see Proposition 5.1) and for uniform schemes W (see Theorem 5.2).Anothercaseinwhichwe havemoredetailedinformationis themodule Ω²_R

W/K forauniform fatpoint schemeW (i.e.,ascheme satisfyingm₁ =· · ·=m_s).In thiscase wecanextract thevalueofthe HilbertpolynomialofΩ²_R

W/K from acomplex connecting it to its fattening and second fattening (see Proposition 5.4). We end the discussionofHilbertpolynomialswithaconjecturefortheirvalueforΩⁿ⁺¹_R

W/K. InSection6,arichanddetailedsetofresultsdescribestheHilbertfunctionsofΩ^m_R

W/K, wherem= 1,2,3,inthecaseofafatpointschemeWinP² supportedonanon-singular conic.Inthiscase, theHilbertfunctionofΩ¹_R

W/K canbe computed explicitlyfrom the Hilbert functions of suitable fat point schemes (see Theorem 6.2). If W is a uniform fat point scheme, we construct a special homogeneous system of generators of I_W in Proposition 6.4and useittocompute theHilbertfunctionof Ω³_R

W/K (seeTheorem 6.6 andProposition 6.7).ThustheexactsequencegivenbytheKoszulcomplexallowsusto determinetheHilbertfunctionof Ω²_R

W/K (seeProposition 6.8).

Finally,inthe last section we pointout therelation betweenthe Kähler differential algebraΩR_X/K andtherelativeKählerdifferentialalgebraΩR_X/K[x0]anduseittodeduce manypropertiesoftheHilbertfunction,theHilbertpolynomial,andtheregularityindex of Ω_RX/K[x0] (seePropositions 7.1,7.2 and 7.3).

Throughoutthepaperweillustrateallresultswithexplicitlycomputedexamples.The necessary calculations were performedusing the second author’s package for thecom- puteralgebrasystemApCoCoA(see [1]).Unless explicitlystated otherwise,weadhere tothedefinitionsandnotationintroducedin[13,14]and[15].

2. Definitionandbasicproperties

Throughout this paper we work over a field K of characteristic zero. By Pⁿ we denote the projective n-space over K. The homogeneous coordinate ring of Pⁿ is S = K[X0,. . . ,Xn]. It is equipped with the standard grading deg(Xi) = 1 for i = 0,. . . ,n. Let X be a 0-dimensional scheme in Pⁿ, and let I_X be the (saturated) homogeneous vanishing ideal of X. Then the homogeneous coordinate ring of X is R_X = S/I_X. The ring R_X =

i≥0(R_X)i is a standard graded K-algebra. Its envelop- ing algebrais R_X⊗K R_X =

i≥0(

j+k=i(R_X)j⊗(R_X)k). ByJ we denote thekernel of the homogeneous R_X-linear map of degree zero μ : R_X⊗K R_X → R_X given by μ(f⊗g)=f g.ItiswellknownthatJ isthehomogeneousidealofR_X⊗KR_Xgenerated by {xi⊗1−1⊗xi |0≤i≤n},where xi istheimage ofXi inR_X fori= 0,. . . ,n. In thispaperweareinterestedinlookingatthealgebraicstructureandHilbertfunctionof thefollowing objects.

Definition 2.1.

(a) ThegradedR_X-moduleΩ¹_R

X/K = J/J² iscalled themoduleofKählerdifferential 1-forms ofR_X/K,orsimplythemoduleof Kählerdifferentials.

(b) The homogeneousK-linearmap d:R_X→Ω¹_R

X/K given byf →f⊗1−1⊗f +J² iscalled theuniversalderivationofR_X/K.

(c) The exteriorpowerΛ^m_RXΩ¹_R

X/K is called themodule of Kählerdifferential m-forms ofR_X/K and isdenotedby Ω^m_R

X/K. (d) The direct sum Ω_RX/K :=

m∈NΩ^m_R

X/K is an R_X-algebra. It is called the Kähler differentialalgebra ofR_X/K.Herewe useΩ⁰_R

X/K =R_X.

More generally,for any graded K-algebraT /S, wecan define themodule ofKähler differentialm-formsΩ^m_{T /S} andtheKähler differentialalgebraΩ_{T /S} analogously(cf.[15, Sect. 2]).TheKählerdifferentialalgebraofΩ_RX/K isinfactabigradedK-algebrawhose homogeneous component in degree (m,d) is given by (Ω^m_R

X/K)_d. Notice that we have deg(dxi) = deg(xi)= 1 for i = 0,. . . ,n. For m ≥0, the graded R_X-module Ω^m_RX/K is finitely generatedand itsHilbertfunctionisdefinedby

HFΩ^m_R

X/K(i) = dimK(Ω^m_RX/K)i for alli∈Z. NotethatΩ⁰_R

X/K =R_XandΩ¹_R

X/K =R_Xdx₀+· · ·+R_Xdx_n.HenceweobtainΩ^m_R

X/K =0 form> n+1 andΩ_RX/K =n+1

m=0Ω^m_RX/K.Furthermore,thereisapresentationofΩ_RX/K asΩ_RX/K∼= Ω_S/K/I_X,dI_XΩ_S/K(cf.[15,Prop. 4.12]).Fromthiswededucethefollowing presentationofΩ^m_R

X/K.

Proposition 2.2. Form≥1,thegradedR_X-moduleΩ^m_R

X/K has apresentation Ω^m_RX/K ∼= Ω^m_S/K/(I_XΩ^m_S/K+dI_XΩ^m_S/K⁻¹).

Inthecasem=n+ 1,thepresentationofΩⁿ⁺¹_R

X/K canbedescribed explicitly.

Corollary 2.3.Let {F1,. . . ,Fr}be a homogeneous system of generatorsof I_X. There is anisomorphism ofgradedR_X-modules

Ωⁿ⁺¹_R

X/K ∼= S/∂Fj

∂Xi |0≤i≤n,1≤j≤r

(−n−1).

Proof. Note thatΩⁿ⁺¹_S/K =SdX0∧ · · · ∧dXn ∼=S(−n−1).ForF ∈I_X and G∈S, we haveF dG =d(F G)−GdF ∈ dI_X. It follows thatI_XΩ^m_S/K ⊆ dI_XΩ^m−1_S/K for all m ≥1.

LettingI=_∂X^∂Fj_i |0≤i≤n,1≤j≤r,Euler’srelationyieldsI_X⊆I.So,itiseasyto checkthatdI_XΩⁿ_S/K=IdX₀∧ · · · ∧dX_n,and hencetheclaimfollowsreadily. 2

Nowletm_X=x₀,. . . ,x_nbethehomogeneousmaximalidealofR_X,andlete:R_X→ R_X be theEulerderivation ofR_X/K which isgivenby f →i·f forf ∈(R_X)i.Bythe universalpropertyof Ω¹_R

X/K (cf.[15,Sect. 1]),thereisauniquehomogeneousR_X-linear map γ : Ω¹_R

X/K → R_X such that e= γ◦d. In particular, we haveγ(dx_i) =x_i for all i= 0,. . . ,n and γ(df)= deg(f)·f for everyhomogeneouselement f ∈R_X\ {0}. The Koszulcomplexofγ isthecomplex

· · ·−→^γ Ω²_RX/K−→^γ Ω¹_RX/K −→^γ m_X−→0 whereγ: Ω^m_R

X/K →Ω^m−1_R

X/K isahomogeneousR_X-linearmapdefinedby γ(ω1∧ · · · ∧ωm) =

m j=1

(−1)^j+1γ(ωj)·ω1∧ · · · ∧ωj∧ · · · ∧ωm

for all ω₁,. . . ,ω_m ∈ Ω¹_R

X/K, and where γ(ω∧ω) = γ(ω)∧ω + (−1)^mω∧γ(ω) for ω ∈Ω^m_R

X/K and ω ∈ Ω^k_R

X/K (cf. [2, 1.6.1-2]). In our setting, this complex is an exact sequence,as thefollowing propositionshows.

Proposition2.4. The Koszulcomplex 0−→Ωⁿ⁺¹_R

X/K

−→γ Ωⁿ_RX/K−→ · · · −→Ω²_RX/K −→^γ Ω¹_RX/K −→^γ m_X−→0 (K) isanexact sequenceof gradedR_X-modules.

Proof. Let 1 ≤ m ≤ n+ 1,let i ≥ 0, and let ω = f dxi1 ∧ · · · ∧dxim ∈ Ω^m_R

X/K with f ∈(R_X)iand 0≤i1<· · ·< im≤n.Thenwe have

(γ◦d)(ω) =if dxi1∧ · · · ∧dxim−df∧γ(dxi1∧ · · · ∧dxim) and

(d◦γ)(ω) =d(f m j=1

(−1)^j+1xijdxi1∧ · · · ∧dxij∧ · · · ∧dxim)

=df∧γ(dx_i1∧ · · · ∧dx_im) +mf dx_i1∧ · · · ∧dx_im.

This implies(γ◦d+d◦γ)(ω)= (m+i)ω.Hence(γ◦d+d◦γ)(ω)= deg(ω)ω forevery homogeneouselementω∈Ω^m_R

X/K.Nowsupposethatω∈Ω^m_R

X/K\ {0}isahomogeneous element with γ(ω)= 0.Set ω= _deg(ω)¹ dω∈Ω^m+1_R

X/K.We getγ(ω) =ω, andthe proofis complete. 2

Obviously, the ring R_X is Noetherian and the graded R_X-module Ω^m_R

X/K is finitely generated, and so the Hilbert polynomial of Ω^m_R

X/K exists (cf. [14, 5.1.21]) and is denoted by HP_Ω^m_R

X/K(z). The number ri(Ω^m_R

X/K) = min{i ∈ Z | HF_Ω^m_R

X/K(j) = HP_Ω^m_R

X/K(j) for allj ≥ i} is called the regularity index of Ω^m_R

X/K. In the following, we denote the Hilbert function of R_X by HF_X and its regularity index by r_X. As a consequenceoftheexactsequence(K),wehavethefollowingbound forri(Ωⁿ⁺¹_R

X/K).

Corollary 2.5.Wehave ri(Ωⁿ⁺¹_R

X/K)≤max{r_X,ri(Ω¹_RX/K),. . . ,ri(Ωⁿ_RX/K)}. 3. GeneralpropertiesoftheHilbertfunctionofΩ^m_RX/K

InthissectionwedescribethevaluesoftheHilbertfunctionofthemoduleofKähler differential m-forms for a0-dimensional scheme Xin somespecial situations. Firstwe consider the easiest case, namely 0-dimensional subschemes X of P¹. It is well known that Hilbertfunctionsdo notchangeunder base fieldextensions (forinstance,see [14, 5.1.20]).Thus,inordertocomputetheHilbertfunctionoftheKählerdifferentialalgebra for the0-dimensional scheme Xof P¹, we mayassumethatthefieldK isalgebraically closed. In this case the homogeneous vanishing ideal I_X is a principal ideal generated byahomogeneouspolynomialF ∈S=K[X₀,X₁].Moreover,after asuitablechangeof coordinates, wemayalsoassumethatF isoftheform F =s

i=1(X₁−a_iX₀)^mi where s≥1,m₁,. . . ,m_s≥1 anda₁,. . . ,a_s∈K suchthata_i =a_j fori=j.

In[17,Sect. 4],L.G.RobertsgaveaformulafortheHilbertfunctionof Ω_RX/K when m₁=m₂=· · ·=m_s= 1.Nowweextendhisresulttoarbitraryexponentsm₁,. . . ,m_s≥ 1 asfollows.

Proposition 3.1.Let X⊆P¹ be a 0-dimensional scheme, and letI_X =F, where F = s

i=1(X₁−a_iX₀)^mi forsomes,m₁,. . . ,m_s≥1,anda_i∈Kwitha_i=a_j fori=j,and letμ=s

i=1mi.ThentheHilbertfunctionsoftheKählerdifferentialmodulesofR_X/K are givenby

HF_Ω¹

RX/K : 0 2 4 6 · · · 2(μ−2) 2(μ−1) 2μ−1 2μ−2 · · · 2μ−s2μ−s· · · HF_Ω²

RX/K : 0 0 1 2 · · · μ−2μ−1μ−2μ−3 · · · μ−s μ−s· · · In particular,we haveri(Ω¹_RX/K)= ri(Ω²_RX/K)=μ+s−1.

Proof. LetG=s

i=1(X1−aiX0)^mi−1, letH1 =s

i=1miai

j=i(X1−ajX0),and let H2 = s

i=1mi

j=i(X1−ajX0). Then we havedeg(G)= s

i=1(mi−1),deg(H1) = deg(H2)=s−1 andgcd(H1,H2) = 1. Also,{H1,H2}is an S-regular sequence. Con- sequently, this sequence is also aregular sequence for the principal idealG which is regardedasagradedS-module.So,fori∈Z,we have

HF_Ω²

RX/K(i) = HF_S/∂F

∂X0,_∂X^∂F₁(i−2)

= HF_S/G(i−2) + HF_G/GH1,GH2(i−2)

= HF_S/G(i−2) + HF_G(i−2)−2 HF_G(i−1−s) + HF_G(i−2s)

= HFS(i−2)−2 HFS(i−1−μ) + HFS(i−s−μ)

=_i−1

1

−2_i−μ

1

+_i−s−μ+1

1

.

ThustheHilbertfunctionofΩ²_RX/K is HF_Ω²

RX/K : 0 0 1 2 · · · μ−2μ−1μ−2μ−3 · · · μ−s μ−s· · ·.

Moreover,itisclearthatHFm_X : 0234 · · · μ−1μμ· · ·.ByProposition 2.4,wehave theexactsequenceof gradedR_X-modules

0−→Ω²_RX/K−→Ω¹_RX/K −→m_X−→0.

HencetheHilbertfunctionof Ω¹_R

X/K satisfies HF_Ω¹

RX/K(i)= HFm_X(i)+ HF_Ω²

RX/K(i) for alli∈Z,andmoreprecisely,itisoftheclaimedform. 2

More generally, we collect some properties of the Hilbert function of Ω^m_R

X/K in the nextproposition. From now on,the coordinates{X0,. . . ,Xn}ofPⁿ are always chosen suchthatnopointofXliesonthehyperplaneZ⁺(X0).Bythechoiceofthecoordinates, x0 is anon-zerodivisor of R_X. Clearly, x0 is also a non-zerodivisor for any non-trivial graded submoduleof agraded free R_X-module. Recall thatthe numberα_X= min{i ∈ N|(I_X)_i = 0}iscalled theinitialdegreeofI_X.

Proposition3.2. LetX⊆PⁿK bea0-dimensionalscheme,and let1≤m≤n+ 1.

(a) Fori< m,wehave HF_Ω^m_R

X/K(i)= 0.

(b) Form≤i< α_X+m−1,wehave HFΩ^m_R

X/K(i)=_n+1

m

·_n+i−m

n

. (c) TheHilbert polynomialof Ω^m_R

X/K isconstant.

(d) WehaveHF_Ω^m_R

X/K(r_X+m)≥HF_Ω^m_R

X/K(r_X+m+1)≥ · · ·,andifri(Ω^m_R

X/K)≥r_X+m then

HF_Ω^m_R

X/K(r_X+m)>HF_Ω^m_R

X/K(r_X+m+ 1)>· · ·>HF_Ω^m_R

X/K(ri(Ω^m_RX/K)).

Proof. (a) Obviously, every non-zero homogeneous element ω of Ω^m_RX/K has degree deg(ω)≥m,andhenceHFΩ^m_R

X/K(i)= 0 for alli< m.

(b) Let m ≤ i < α_X+m−1. Notice that I_XΩ^m_S/K ⊆ dI_XΩ^m−1_S/K. Also, we have (dI_XΩ^m−1_S/K)_i = 0 for all i < α_X+m−1, since a non-zero homogeneous element of dI_XΩ^m_S/K⁻¹isalwaysoftheform

kdFk∧ωk,whereFk ∈(I_X)_≥αX andωk ∈(Ω^m_S/K⁻¹)_≥m−1. ByProposition 2.2,foralli< α_X+m−1,we obtain

HFΩ^m_R

X/K(i) = HFΩ^m_S/K(i) =_n+1

m

·_n+i−m

n

.

(c) Itfollowsfrom Proposition 2.2that HF_Ω^m

RX/K(i)≤HF_Ω^m

S/K/I_XΩ^m_S/K(i) =_n+1

m

HF_X(i)≤_n+1

m

deg(X) foralli∈Z.HencetheHilbertpolynomialof Ω^m_R

X/K isaconstantpolynomial.

(d)Thegraded R_X-moduleΩ^m_RX/K hasthefollowing form:

(Ω^m_RX/K)_i+m= (R_X)_idx₀∧ · · · ∧dx_m−1+· · ·+ (R_X)_idx_n−m+1∧ · · · ∧dx_n. Observethat(R_X)_i=x₀(R_X)_i−1 ifi> r_X.Thus(Ω^m_R

X/K)_i+m=x₀(Ω^m_R

X/K)_i+m−1 forall i> r_X.So,foralli> r_X,wehavetheinequality

HF_Ω^m

RX/K(i+m−1)≥HF_Ω^m

RX/K(i+m).

Now letG=(_∂x^∂F₀,. . . ,_∂x^∂F_n)|F ∈I_XR_X.By[5,Prop. 1.3]and[18, X.83],there isan exactsequenceofgraded R_X-modules

0−→ G ∧R_Xm−1

R_X (R_Xⁿ⁺¹)−→m

R_X(R_Xⁿ⁺¹)−→Ω^m_R

X/K(m)−→0.

Suppose i≥r_X satisfiesHF_Ω^m

RX/K(i+m)= HF_Ω^m

RX/K(i+m+ 1). Then itfollows from theaboveexactsequence that

HF^m

RX(Rⁿ⁺¹_X )(i)−HF_G∧R

X

_m−1

RX (R_Xⁿ⁺¹)(i)

= HFm

RX(Rⁿ⁺¹_X )(i+ 1)−HF_G∧

RX

_m−1

RX (Rⁿ⁺¹_X )(i+ 1).

For every j ≥ r_X, HF_X(j) = deg(X), and so HFm

RX(Rⁿ⁺¹_X )(j) = HFm

RX(Rⁿ⁺¹_X )(j + 1).

Consequently, wehaveHF_G∧

RX

m−1

RX (Rⁿ⁺¹_X )(i)= HF_G∧

RX

m−1

RX (Rⁿ⁺¹_X )(i+ 1).Since x₀ is a non-zerodivisor forthegraded R_X-moduleG ∧R_X m−1

R_X (Rⁿ⁺¹_X ),thisimplies (G ∧R_X m−1

R_X (Rⁿ⁺¹_X ))_i+1=x₀(G ∧R_Xm−1

R_X (Rⁿ⁺¹_X ))_i.

Inviewof [7,Prop. 1.1], theidealI_X canbe generated byhomogeneouspolynomialsof degrees≤r_X+ 1.So,thegradedR_X-moduleG ∧R_Xm−1

R_X (Rⁿ⁺¹_X ) isgenerated indegrees

≤r_X. Thusweobtain

(G∧R_Xm−1

R_X (Rⁿ⁺¹_X ))i+2

=x0(G ∧R_Xm−1

R_X (Rⁿ⁺¹_X ))i+1+· · ·+xn(G ∧R_Xm−1

R_X (Rⁿ⁺¹_X ))i+1

=x₀

x₀(G ∧R_Xm−1

R_X (Rⁿ⁺¹_X ))_i+· · ·+x_n(G ∧R_Xm−1

R_X (Rⁿ⁺¹_X ))_i

=x₀(G ∧R_X

_m−1

R_X (Rⁿ⁺¹_X ))_i+1. Altogether,wehaveHFΩ^m_R

X/K(i+m+ 1)= HFΩ^m_R

X/K(i+m+ 2),andtheclaimfollows byinduction. 2

Thefollowing exampleshowsthatHF_Ω^m

RX/K(i) mayor maynotbemonotonic inthe rangeα_X+m≤i≤r_X+m.

Example 3.3. Let K = Q, and let X ⊆ P² be a set of six points on a line and three non-collinearpointsoffthatline,e.g.X={(1: 1: 0),(1: 1: 1),(1: 1: 2),(1: 1: 3),(1: 1: 4),(1: 1: 5),(1: 0: 1),(1: 2: 1),(1: 2: 2)}.It isclearthatHF_X: 1367899· · ·, α_X = 3,and r_X= 5. TheHilbertfunctionsof theKähler differentialmodulesof R_X/K aregivenby

HF_Ω¹

RX/K : 0 3 9 15 14 13 14 13 12 11 10 9 9· · · HF_Ω²

RX/K : 0 0 3 9 9 4 5 4 3 2 1 0 0· · · HF_Ω³

RX/K : 0 0 0 1 3 0 0· · ·. Wesee thatHF_Ω¹

RX/K(α_X+ 1)= 14>13= HF_Ω¹

RX/K(α_X+ 2) andHF_Ω¹

RX/K(α_X+ 2)= 13<14= HF_Ω¹

RX/K(r_X+ 1).So,HF_Ω¹

RX/K(i) isnotmonotonicintherangeα_X+ 1≤i≤ r_X+ 1.Similarly,HF_Ω²

RX/K(i) isnotmonotonicintherangeα_X+ 2≤i≤r_X+ 2.

Nextwe consider theset Y=X∪ {(1 : 0: 2)}.We haveHF_Y: 1 3689 1010· · ·, α_Y = 3,and r_Y= 5. TheHilbertfunctionsof theKähler differentialmodulesof R_Y/K are

HF_Ω¹

RY/K : 0 3 9 16 18 16 15 14 13 12 11 10 10· · · HF_Ω²

RY/K : 0 0 3 9 12 8 5 4 3 2 1 0 0· · · HF_Ω³

RY/K : 0 0 0 1 3 2 0· · ·. HenceHF_Ω¹

RY/K(i) ismonotonic in therange α_Y+ 1≤ i≤r_Y+ 1, and HF_Ω²

RY/K(i) is alsomonotonicintherangeα_Y+ 2≤i≤r_Y+ 2.

Inthelastpartofthissectionwegiveanupperboundfortheregularityindexofthe moduleofKählerdifferentialm-formsΩ^m_R

X/K fora0-dimensionalschemeXinPⁿ.Todo this,weneedthefollowing lemmas.

Lemma3.4. Letd≥1,letδ1,. . . ,δd∈Zsuchthatδ1≤ · · · ≤δd,letW =d

j=1R_X(−δj) be thegradedfreeR_X-module,andletV beanon-trivialgradedsubmoduleof W.Then, for1≤m≤d,wehave

ri(V ∧R_Xm

RX(W))≤ri(V) +δd−m+1+· · ·+δd.

Proof. First we note that the Hilbert polynomial of W is HPW(z) = d·deg(X) and that ri(W)=r_X+δd. Thisshows thattheHilbertpolynomialofV isaconstantpoly- nomial HP_V(z)=u≤d·deg(X). Letr= ri(V),and letv₁,. . . ,v_u be aK-basisof V_r. Then theelements {xⁱ₀v₁,. . . ,xⁱ₀v_u}form aK-basis of theK-vector space V_r+i for all i ∈N. We let{e1,. . . ,ed} be thecanonical R_X-basis of W, we lett =_d

m

, and we let {ε1,. . . ,εt} be a basis of the graded free R_X-module m

R_X(W) w.r.t. {e1,. . . ,ed}. We set δ=δd−m+1+· · ·+δd,andlet

N =x^δ₀^−deg(εk)vj∧εk∈V ∧R_Xm

RX(W)|1≤j≤u,1≤k≤tK.

Let = dimKN, and let w1,. . . ,w be a K-basis of N. It is not difficult to verify that N = w1,. . . ,wK = (V ∧R_X m

R_X(W))δ+r. Moreover, for any i ≥ 0, the set {xⁱ₀w₁,. . . ,xⁱ₀w} is K-linearly independent. Indeed, assume that there are elements a₁,. . . ,a ∈ K such that

j=1xⁱ₀a_jw_j = 0. Since x₀ is a non-zerodivisor for V ∧R_X

m

R_X(W),weget

j=1ajwj= 0,andhencea1=· · ·=a= 0.

Now it sufficesto provethattheset {xⁱ₀w1,. . . ,xⁱ₀w}generates theK-vectorspace (V ∧R_X m

R_X(W))δ+r+i for all i ≥ 0. Let w ∈ (V ∧R_X m

R_X(W))δ+r+i be a non-zero homogeneouselement.Thenw=

j,kv_j∧h_kε_k=

j,kh_kv_j∧ε_kforsomehomogeneous elements v_j ∈ V and h_k ∈ R_X such thatdeg(v_j)+ deg(h_k)= δ+r+i−deg(ε_k) for all j,k. Note that deg(hkvj) = δ+r+i−deg(εk) ≥ r+i. Also, we have hkvj ∈ Vδ+r+i−deg(εk)=x^δ+i₀ ^−deg(εk)v1,. . . ,x^δ+i₀ ^−deg(εk)vuK.So, there arebjk1,. . . ,bjku ∈K suchthath_kv_j =u

l=1b_jklx^{δ+i−deg(ε}₀ ^k)v_l.This implies w=

j,k

h_kv_j∧ε_k =

j,k

u l=1

b_jklx^{δ+i−deg(ε}₀ ^k)v_l∧ε_k

=

j,k

u l=1

b_jklxⁱ₀(x^δ−deg(ε₀ ^k)v_l∧ε_k) =

j,k

u l=1

q=1

b_jklc_jklqxⁱ₀w_q

for some cjklq ∈ K. Thus we get w ∈ xⁱ₀w1,. . . ,xⁱ₀wK, and consequently HF_V∧R

X

_m

RX(W)(i) = for all i ≥ δ+r. Therefore ri(V ∧R_X m

R_X(W)) ≤ ri(V)+δ, as wewantedtoshow. 2

Lemma3.5. LetV beagradedR_X-modulegeneratedbythesetofhomogeneouselements {v₁,. . . ,v_d}forsome d≥1. Letδ_j = deg(v_j)for j = 1,. . . ,d, andlet m≥1.Assume that δ₁ ≤ · · · ≤ δ_d,and set δ=δ_d−m+1+· · ·+δ_d if m≤d.Then the regularity index of m

R_X(V)satisfiesri(m

R_X(V))=−∞if m> dand