On the Torsion of Brieskorn Modules of Homogeneous Polynomials

On the Torsion of Brieskorn Modules of Homogeneous Polynomials.

KHURRAMSHABBIR(*)

ABSTRACT- Letf 2C[X1; :::;Xn] be a homogeneous polynomial andB(f) be t he corresponding Brieskorn module. We describe the torsion of the Brieskorn moduleB(f) fornˆ2and show that any torsion element has order 1. Forn>2, we find some examples in which the torsion order is strictly greater than 1.

1. The Milnor algebra and the Brieskorn module.

Letf 2RˆC[x₁; :::;x_n] be a homogeneous polynomial of degreed>1.

Then the Koszul complex of the partial derivativesf_jˆ @f

@xj;jˆ1; :::;ninR can be identified to the complex

0 !V^{0 df}!^{^} V^{1 df}!^{^} ::: ^df!^{^} V^{n 1 df^}!Vⁿ !0;

(1:1)

whereV^jdenotes the regular differential forms of degreejonCⁿ. Let Jf be the Jacobian ideal spanned by the partial derivatives fj, jˆ1; :::;n, inRandM(f)ˆR=Jf be the Milnor algebra off. One has the following obvious isomorphism of graded vector spaces

M(f)( n)ˆ Vⁿ df^V^{n 1}: (1:2)

Here, for any gradedC[t]-moduleM, the shifted moduleM(m) is defined by settingM(m)_sˆMm‡sfor alls2Z:

We define the (algebraic) Brieskorn module as the quotient B(f)ˆ Vⁿ

df ^d(V^{n 2}) (1:3)

(*) Indirizzo dell'A.: 68-B, New Muslim Town, School of Mathematical Sciences, GCU Lahore, Pakistan.

E-mail: khurramsms@gmail.com

in analogy with the (analytic) local situation considered in [3], see also [11], as well as the submodule

C(f)ˆ df^V^{n 1} df^d(V^{n 2}): (1:4)

These modules are modules over the ringC[t] and the multiplication bytis given by multiplying by the polynomialf. SometimesB(f) is denoted byG⁽⁰⁾_f andC(f) byG^{( 1)}_f , see [2], [6].

One has the following basic relation between the Milnor algebra and the Brieskorn module, see [7], Prop. 1.6.

PROPOSITION1.1.

df^V^{n 1}ˆdf^d(V^{n 2})‡fVⁿ: In particular

B(f)=f:B(f)'M(f)( n):

Let B(f)_tors be the submodule of C[t]-torsion elements in B(f) and define the reduced Brieskorn moduleB(f)ˆB(f)=B(f)_tors.

REMARK 1.2. The reduced Brieskorn module B(f) is known to be a free C[t]-module of rank bn 1(F), where Fˆ fx2Cⁿjf(x)ˆ1g is the affine Milnor fiber off. Indeed, it follows from the section (1.8) in [7] that one has a canonical isomorphism _{jˆ0;d 1}B(f)_qd‡j ˆH^{n 1}(F;C) for any qn. If we assume theC[t]-basis ofB(f) to be formed by homogeneous elements (which is always possible), each basis element will contribute by 1 to the dimension ofjˆ0;d 1B(f)_qd‡j forqlarge enough.

Moreover, it exists an integerN>0 such thatt^NB(f)_torsˆ0, see [7], Remark 1.7. The leastNsatisfying this condition is called thetorsion order off and is denoted byN(f).

REMARK1.3. As noted in [7], Remark 1.7, there is a slight difference between the Brieskorn module defined above and the Brieskorn module considered by Barlet and Saito in [1]. In fact, the latter one is defined to be then-th cohomology groupHⁿ(A_f), whereA^j_f ˆker (df^:V^j !V^j‡1) and the differential d_f :A_f^j ! A_f^j‡1 is induced by the exterior differential d:V^j!V^j‡1. Sincedf ^d(V^{n 2})d_f(Aⁿ_f ¹), one has an epimorphism

B(f)!Hⁿ(A_f);

(note that the direction of this arrow is misstated in [7], Remark 1.7).

Moreover, whennˆ2, it follows from Proposition 3.7 in [1] thatH²(A_f) is torsion-free. Our results in section 2 show that the torsion module B(f)_torscan even be not of finite type in this setting, and hence the above epimorphism is far away from an isomorphism in general.

DEFINITION1.4. Forb2B(f), we say thatbist-torsion of orderkb1 if t^kbbˆ0 andt^{kb 1}b6ˆ0:

It is clear that such a torsion orderk_bdivides the torsion orderN(f) of f, sincet^N(f)bˆ0 and thatN(f) is the G.C.D. of all thet-torsion ordersk_b forb2B(f) a torsion element.

1.5. The case of an isolated singularity.

Assume that f 2C[x1; :::;xn] is a homogeneous polynomial having an isolated singularity at the origin ofCⁿ. Then the dimension of the Milnor algebraM(f) (as aC-vector space) is the Milnor number off atthe origin, denoted bym(f). One has in this casem(f)ˆb_{n 1}(F), see for instance [5]. In this case, the structure of the moduleB(f) is as simple as possible.

PROPOSITION1.6. TheC[t]-moduleB(f)is free of rankm(f):

PROOF. Indeed, a homogeneous polynomial f having an isolated sin- gularity at the origin induces a tame mapping f :Cⁿ!C, to which the results in [6], [12] and [13] apply. Indeed, the Gauss-Manin systemG_f off, which is anA₁ˆC[t]h@_ti-module, contains aC[t]-submoduleG⁽⁰⁾_f , which is known to be free of finite type for a tame polynomial, see for instance Re- mark 3.3 in [6]. As we have already mentionned above,G⁽⁰⁾_f ˆB(f). p

COROLLARY1.7. There is an isomorphism B(f)(n)ˆM(f)_CC[t]

of graded modules over the graded ringC[t]. In particular, one has, at the level of the associated Poincare series, the following equality.

P_B(f)(t)ˆP_M(f)(t) tⁿ

1 tˆtⁿ(1 t^{d 1})ⁿ (1 t)^n‡1 :

EXAMPLE1.8. Fornˆ3;takef(x;y;z)ˆx³‡y³‡z³:Then aC-basis ofM(f) is given by 1;x;y;z;xy;yz;xz;xyz:The same 8 monomials form a

basis ofB(f) as a freeC[t]-module. In this case P_B(f)(t)ˆt³(1 t²)³

(1 t)⁴ ˆt³‡4t⁴‡7t⁵‡8t⁶‡8t⁷‡8t⁸‡ : COROLLARY1.9. TheC[t]-module B(f)is torsion free if and only if0 is an isolated singularity of the homogeneous polynomial f .

PROOF. If 0 is not an isolated singularity, then the Milnor algebra M(f) is an infinite dimensional C-vector space. The isomorphism in Proposition 1.1 implies that in this case B(f) is notfinitely generated overC[t]. By Remark 1.2, it follows that the canonical projection

B(f)!B(f)

is notan isomorphism, henceB(f)_tors6ˆ0: p Proposition 1.1 implies that f B(f)ˆC(f), which in turn yields the following.

COROLLARY1.10. Assume that0is not an isolated singularity of the homogeneous polynomial f . Then N(f)ˆ1if and only if theC[t]-module C(f)is torsion free.

PROOF. Let b2B(f)_tors be a non-zero element, which exists by our assumption. By Remark 1.2, it follows thatbist-torsion, say of orderk:If k>1;then

0ˆt^kbˆt^{k 1}(tb):

By Proposition 1.1, we know thattb2C(f):IfC(f) is torsion free, we get thattbˆ0 inC(f), i.e.tbˆ0 inB(f), a contradiction. Hencekˆ1 for anyb2B(f)_tors, in other wordsN(f)ˆ1. p

2. The casenˆ2.

In this section we suppose thatf 2C[x;y] is a homogeneous polynomial of degree d>1, which is notthe power g^r of some other polynomial g2C[x;y] for somer>1:This condition is equivalent to asking the Mil- nor fiberFoffto be connected and such polynomials are sometimes called primitive. For more on this, see [8], final Remark, part (I).

PROPOSITION2.1. The submoduleC(f)is a freeC[t]-module of rank b1(F).

PROOF. The facttheC(f) is torsion free follows from Proposition 7. (ii) in [2]. Indeed, it is shown there that the localization morphism

`:C(f)!C(f)_C[t]C[t;t ¹]

is injective, which is clearly equivalent to the fact that there are not-torsion elements.

For the second claim, note that the composition of the inclusion C(f)!B(f) and the canonical projection B(f)!B(f) gives rise to an embedding of C(f) into B(f) whose image is exactly tB(f) (use again Proposition 1.1 and the fact that C(f) is torsion free). Since C[t] is a principal ideal domain, the claim follows from the structure theorem of submodules of free modules of finite rank over such rings. p

Corollary 1.10 implies the following:

COROLLARY 2.2. If f 2C[x;y] is a homogeneous polynomial with a non-isolated singularity at the origin, then N(f)ˆ1.

The above Corollary can be restated by saying that 0!B(f)_tors!B(f)!^t C(f)!0

is an exactsequence of gradedC[t]-modules. We getthus an isomorphism of gradedC[t]-modules

B(f)( d)'C(f):

EXAMPLE 2.3. Let f ˆx^py^q with (p;q)ˆ1. In order to compute the torsion of the Brieskorn moduleB(f);we start by findingC-vector space monomial bases forB(f)

C(f)'M(f) (up-to a shift in gradings) andC(f). Note that the Jacobian ideal is given by

J_f ˆ hx^{p 1}y^q;x^py^{q 1}i ˆx^{p 1}y^{q 1}hx;yi:

Itfollows thatB(f) C(f)' S

J_f is an infinite dimensionalC-vector space with a monomial basis given by x^ay^b with ap 2 or bq 2 or (a;b)ˆ

ˆ(p 1;q 1):

We have to compute df ^dV⁰ˆ fdf^dgg; where g is a polynomial function onC². Since we are working with homogeneous polynomials, it is enough to work with one monomial, say (x^ay^b);ata time. We have

df^d(x^ay^b)ˆ(pb qa)x^{a‡p 1}y^{b‡q 1}dx^dy:

Since

C(f)ˆ J_f V²

df^dV⁰ˆx^{p 1}y^{q 1}hx;yiV² df ^dV⁰ ;

a system of generators of theC-vector spaceC(f) is given by the classes of the elementsw2x^{p 1}y^{q 1}hx;yiV²;which do notbelong todf^dV⁰:To find them, it is enough to look at monomial differential forms i.e.x^ay^bdx^dy withap 1,bq 1 anda‡bp‡q 1:

So, ifa‡p 1ˆa;b‡q 1ˆb andpb qa6ˆ0;then df^d x^ay^b

pb aq

ˆx^ay^bdx^dy:

Hence, the only elements x^ay^bdx^dy which are notin df^dV⁰ are x^{a‡p 1}y^{b‡q 1}dx^dy;wherepbˆqa, i.e.aˆkp;bˆkqfor somek0:It follows thatC(f), as aC-vector space, has a basis given byx^{(k‡1)p 1}y^{(k‡1)q 1} wherek0, which can be written as

C(f)ˆC[t]x^{2p 1}y^{2q 1}dx^dy i.e.C(f) is a freeC[t]-module of rank 1.

Moreover, the corresponding monomial basis as aC-vector space for the Brieskorn module B(f) is given by x^ay^bdx^dy with ap 2 or bq 2 oraˆ(k‡1)p 1;bˆ(k‡1)q 1 for somek0:

With respect to this basis,B(f)_torsis the linear span ofx^ay^bdx^dywith ap 2 orbq 2. Indeed our computation above shows that

df^d(x^a‡1y^b‡1)ˆ(p(b‡1) q(a‡1))fw i.e.twˆ0 if the coefficientp(b‡1) q(a‡1) is non-zero.

Itfollows that

B(f)ˆC[t][x^{p 1}y^{q 1}dx^dy];

a freeC[t]-module of rank 1.

Itremains to explain whyb₁(F)ˆ1, whereFˆ f(x;y)2C²;x^py^qˆ1g:

Consider the covering

Z_q,!F!^f C wherefmaps (x;y) t ox:

Now a covering yields an exactsequence (see for instance [10], page 376), 0,!p1(F;(x0;y0))!^f# p1(C;x0)!p0(Zq;(x0;y0))ˆZq!0:

Hencep1(F)'Z, which shows that the first Betti numberb1(F) is 1:

3. Eigenvalues of the monodromy and torsion of Brieskorn modules.

Any homogeneous polynomialf 2C[x1; :::;xn] induces a locally trivial fibration f :Cⁿnf ¹(0)!C, with fiber F and semisimple monodromy operators

T_f^k:H^k(F;C)!H^k(F;C)

forkˆ0; :::;n 1:The eigenvalues ofT_f ared-roots of unity, wheredis the degree of f, and for each such eigenvaluel we denote byH(F;C)_l the corresponding eigenspace.

According to [7], see the discussion just before Remark 1.9, one has for q<nan inclusion

t^{n q}:B(f)_{qd j}!B(f)_{nd j} and, forqn, an identification

B(f)_{qd j}ˆH^{n 1}(F;C)_l;

wherelˆexp 2pj 

p 1 d

!

;withjˆ0;1; :::;d 1:Letvnˆdx1^:::^dxn

and note that [vn]2B(f)_{qd j}for someqif and only ifn jis divisible byd.

This yields the following, see [7], Corollary 1.10.

COROLLARY3.1. Assume thatlˆexp 2pn 

p 1 d

!

is not an eigenva- lue of the monodromy operator T_f acting on H^{n 1}(F;C). Then[vn]is a non-zero torsion element in B(f):

Here are some examples of torsion elements inB(f) obtained using this approach, and having torsion orderN(f)>1.

EXAMPLE3.2. Let f ˆx³‡y²z; nˆ3; dˆ3; jˆ0 (this defines a cuspidal cubic curve C in P²). In this case lˆ1 is notan eigenvalue of the monodromy acting on H²(F;C). Indeed, one has H²(F;C)₁ ˆ

ˆH²(U;C)ˆ0. Here UˆP²n C and the last vanishing comes from

the following obvious equalities: b0(U)ˆ1, b1(U)ˆ0 and x(U)ˆ

ˆx(P²) x(C)ˆ3 2ˆ1 as C is homeomorphic to a sphere S². Forwˆdx^dy^dz;to find the value ofksuch thatt^k[w]ˆ0 inB(f) is equivalent to saying thatf^kw2df^dV¹. We have to check for which value ofk;we have solutions of the equation:

f^kwˆ 3x² @R

@y

@Q

@z

‡2yz @P

@z

@R

@x

‡y² @Q

@x

@P

@y

dx^dy^dz;

whereP;Q;R2C[x;y;z] are homogeneous polynomials of degree 3k 1.

Forkˆ1;we get a system of non-homogeneous linear equations in which we have 15 unknowns and 9 equations. Then using the software Mat Lab we compute the rank of the matrix corresponding to the homogeneous system (containing 9 rows and 15 columns) and get 8:On other hand, the rank of the matrix associated to the non-homogeneous system (containing 9 rows and 16 columns) is 9; which show that this system has no solution.

Forkˆ2;we get another system of non-homogeneous linear equations containing 60 unknowns and 26 equations. Then using Mat Lab we com- pute the rank of the matrix corresponding to the homogeneous system (containing 26 rows and 60 columns) and get 26: This time the corre- sponding non-homogeneous system matrix (containing 26 rows and 61 columns) has also rank 26;which shows that this system of equations has a solution. An explicit solution for kˆ2; is Pˆ32

3 (x³yz); Qˆxy²z² and Rˆ1

3(x⁴y):Hence [w] hast-torsion order 2 inB(f):

The nextexample shows somehow whathappens whenH^{n 1}(F;C)_l6ˆ0.

EXAMPLE 3.3. Let f ˆx²y²‡xz³‡yz³, where nˆ3; dˆ4 and jˆ3. The corresponding curveC:f ˆ0 in P² has two cusps as singula- rities. It follows from the study of the plane quartic curves, see [5], p. 130, thatp₁(U)ˆZ₄and henceH¹(F;C)ˆ0.

Nextx(U)ˆx(P²) x(C)ˆ3 (2 23‡22)ˆ3. The zeta-function of the monodromy operatorTf looks like

det(tId T⁰_f)det(tId T_f²)ˆ(t⁴ 1)³

see for instance [5], p. 108. Hence we can not apply Corollary 3.1 to this polynomial to infer that [w]ˆ[v3] is a torsion element inB(f).

However, we can try to find values ofk;for which we have solutions of the equation:

f^kwˆ

"

(2xy²‡z³) @R

@y

@Q

@z

‡(2x²y‡z³) @P

@z

@R

@x

‡(3xz²‡3yz²) @Q

@x

@P

@y

#

w

where P;Q;R2C[x;y;z] are homogeneous polynomials of degree 4k 2.

Forkˆ1 andkˆ2 a similar computation of matrix ranks as above shows that the corresponding systems have no solution. Hence [w] is either a non-torsion element, or it has t-torsion order greater or equal t o 3 in B(f):

REMARK3.4. Note that in both examples above the elementt[w] is a non-zero torsion element inC(f). Hence Proposition 2.1 fails forn>2. It also shows that in these cases the moduleC(f) is nottorsion free, compare to Corollary 1.10.

The last example shows that even for rather complicated examples (here the zero set off is a surfaceS with non-isolated singularities) one may still have 1 as thet-torsion order of [vn].

REMARK 3.5. Let f ˆx²z‡y³‡xyt; nˆ4; dˆ3; jˆ1 be t he equation of a cubic surfaceSinP³. Itfollows from [4], Example 4.3, that H³(F;C)ˆ0. Hence we can apply Corollary 3.1 to this polynomial to infer that [w]ˆ[v4] is a torsion element inB(f).

We have to check for which values of k; we have solutions of the equation:

f^kwˆ

"

(2xz‡yt) @S

@t

@T

@z‡@U

@y

‡(3y²‡xt) @Q

@t ‡@R

@z

@U

@x

‡x² @P

@t

@R

@y‡@T

@x

‡xy @P

@z

@Q

@y

@S

@x

#

dx^dy^dz^dt;

where P;Q;R;S;T;U2C[x;y;z;t] are homogeneous polynomials of de- gree 3k 1.

For kˆ1; we get a system of non-homogeneous linear equations in which we have 42 unknowns and 15 equations. Using MatLab we compute

the rank of the corresponding homogeneous system (containing 15 rows and 42 columns) and get14: And the rank of non-homogeneous system (containing 15 rows and 43 columns) has the same rank 14;which show that this system has a solution.

Hencet[w]ˆ0 inB(f):

REFERENCES

[1] D. BARLET- M. SAITO,Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities, preprint(math.CV/0411406).

[2] PH. BONNET- A.DIMCA,Relative differential forms and complex polynomials, Bull. Sci. Math.,124(2000), pp. 557-571.

[3] E. BRIESKORN, Die Monodromie der isolierten SingularitaÈten von Hyper- flaÈchen, Manuscripta Math.,2(1970), pp. 103-161.

[4] A. DIMCA,On the Milnor fibrations of weighted homogeneous polynomials, Composito Math.,76(1990), pp. 19-47.

[5] A. DIMCA, Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, 1992.

[6] A. DIMCA - M. SAITO, Algebraic Gauss-Manin systems and Brieskorn modukles, Amer. J. Math.,123(2001), pp. 163-184.

[7] A. DIMCA- M. SAITO, A generalization of Griffiths's Theorem on rational integrals, Duke Math. J.,135(2006), pp. 303-326.

[8] A. DIMCA- L. PAUNESCU,On the connectivity of complex affine hypersurfaces, II, Topology,39(2000), pp. 1035-1043.

[9] D. EISENBUD, Commutative Algebra witha View Towards Algebraic Geo- metry, Springer-Verlag New York, 1995.

[10] A. HATCHER,Algebraic Topology, Cambridge Univ. Press, 2002.

[11] E. LOOIJENGA, Isolated singular points on Complete Intersections, Cam- bridge Univ. Press 1984.

[12] A. NEÂMETHI- C. SABBAH,Semicontinuity of the spectrum at infinity, Abh.

Math. Sem. Univ. Hamburg,69(1999), pp. 25-35.

[13] C. SABBAH,Hypergeometric periods for a tame polynomial, C.R. Acad. Sci.

Paris,328(1999), pp. 603-608.

Manoscritto pervenuto in redazione il 5 maggio 2007.