Essentially normal Hilbert modules and K-homology

Math. Ann. (2008) 340:907–934DOI 10.1007/s00208-007-0175-2 Mathematische Annalen

Essentially normal Hilbert modules and K-homology

Kunyu Guo · Kai Wang

Received: 7 May 2007 / Revised: 24 August 2007 / Published online: 23 October 2007© Springer-Verlag 2007

Abstract This paper mainly concerns the essential normality of graded submod-ules. Essentially all of the basic Hilbert modules that have received attention overthe years are p-essentially normal—including the d-shift Hilbert module, the Hardyand Bergman modules of the unit ball. Arveson conjectured graded submodules overthe unit ball inherit this property and provided motivations to seek an affirmativeanswer. Some positive results have been obtained by Arveson and Douglas. However,the problem has been resistant. In dimensions d = 2, 3, this paper shows that theArveson’s conjecture is true. In any dimension, the paper also gives an affirmativeanswer in the case of the graded principal submodule. Finally, the paper is associ-ated with K -homology invariants arising from graded quotient modules, by whichgeometry of the quotient modules and geometry of algebraic varieties are connected.In dimensions d = 2, 3, it is shown that K -homology invariants determined bygraded quotients are nontrivial. The paper also establishes results on p-smoothness ofK -homology elements, and gives an explicit expression for K -homology invariant indimension d = 2.

Mathematics Subject Classification (2000) 47A13 · 47A20 · 46H25 · 46C99

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9082 Each graded principal submodule is p-essentially normal . . . . . . . . . . . . . . . . . . . . . 910

K. Guo (B) · K. WangSchool of Mathematical Sciences, Fudan University,Shanghai 200433, People’s Republic of Chinae-mail: [email protected]

K. Wange-mail: [email protected]

123

908 K. Guo, K. Wang

3 In dimensions d = 2, 3, each graded submodule is p-essentially normal . . . . . . . . . . . . . 9154 p-essentially normal graded quotient modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 9195 K -homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922

5.1 In dimension d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9265.2 In dimension d = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928

1 Introduction

In the study of multivariable operator theory, there is a natural approach via Hilbertmodule [21]. Let T = (T1, . . . , Td) be a tuple of commuting operators acting on a Hil-bert space H . Then, one naturally makes H into a Hilbert module over the polynomialring C[z1, . . . , zd ]. The C[z1, . . . , zd ]-module structure is define by

p · ξ = p(T1, . . . , Td)ξ, p ∈ C[z1, . . . , zd ], ξ ∈ H.

In Arveson’s language [5], a Hilbert module is said to be essentially normal if thecross-commutators T ∗

k Tj − Tj T ∗k of its ambient operators are all compact, and more

specifically, p-essentially normal if the cross-commutators belong to the Schattenclass Lp, where p ∈ [1,∞). For essentially normal Hilbert modules, Douglas calledsuch Hilbert modules as essentially reductive, see [16–19,21]. The study of essen-tial normality and p-essential normality facilitates the introduction of techniques andmethods drawn from algebraic geometry, homology theory and complex analysis, etc,and this study establishes important connections between operator theory and thesebranches. Essentially all of the basic Hilbert modules that have received attention overthe years are p-essentially normal–including the d-shift Hilbert module, the Hardyand Bergman modules of the unit ball. Arveson conjectures that graded submodulesover the unit ball inherit this property and seeks an affirmative answer [3–7].

Observing the following example will serve as a good illustration of motivations.Let I be a homogeneous ideal of the polynomial ring C[z1, . . . , zd ], and let [I ] be thegraded submodule of the Bergman module L2

a(Bd) generated by I . Then there corre-sponds a graded quotient submodule L2

a(Bd)/[I ] (naturally identical with L2a(Bd)�

[I ]) with the module action endowed by d-tuple {S1, . . . , Sd} obtained by compressing{Mz1, . . . ,Mzd } to the quotient. Let C∗〈I 〉 denote the C∗-algebra on L2

a(Bd)/[I ] gen-erated by the identity operator and d-tuple {S1, . . . , Sd}. If [I ] is essentially normal,then applying a result in [4,16] here shows that the quotient submodule L2

a(Bd)/[I ]also is, and hence one has a C∗-extension defined by

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0. (1.1)

This extension yields an odd K -homology element eI for the variety Z(I ) by BDF-theory[9], where Z(I ) denotes Z(I ) ∩ ∂Bd . As the first step, one would expect thisK -homology element eI to be nontrivial [7], which would establish connectionsbetween operator theory, index theory and algebraic geometry. A more challengingproblem is which element of the K1(Z(I )) is obtained. Douglas suspected that thisK -homology element in fact is identical with the fundamental class for the varietyZ(I ) [17].

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Essentially normal Hilbert modules and K -homology 909

Concerning p-essential normality of the graded quotient module L2a(Bd)/[I ], it

naturally arises from the study of p-smoothness and index theory of K -homologyelement eI [20,24]. Furthermore, a possible efficient connection to geometry is theConnes’ cyclic cohomology, a generalized de Rham cohomology which is essentiallyimportant in the noncommutative geometry [13]. Related to p-summable Fredholmmodules in the cyclic cohomology, p-essential normalty will play an role in the calcula-tion of Chern character. This requires a deep understanding on topology and geometryof algebraic varieties. In the case of the bounded strongly pseudo-convex domains,the reader might consult [17,18] for related discussions.

For the study of essential normality and p-essential normality, there are many othermotivations coming from topology and algebraic geometry. Considering an analyticmanifold M in the projective space P

n , by a theorem of Chow [34], M can be obtainedby the zero variety of a homogeneous ideal, with suitable criterion which ensures Msmooth. That fact implies that M contains plenty of subvariety. By an elementary resultof algebraic geometry, each subvariety of M determines a fundamental cohomologyelement in H∗(M). The famous Hodge’s conjecture asserts, roughly speaking, thatwhether there are enough subvarieties such that their fundamental cohomology classesrationally generate the cohomology group H (−,−)(M). The study of essentially nor-mal Hilbert modules may provide a connection between operator theory and Hodge’sconjecture. For a smooth homogeneous variety Z(J ), considering any homogeneouspolynomial ideal I ⊇ J and then Z(I ) ⊆ Z(J ), one wonders whether the funda-mental class of Z(I ) in cohomology group H∗(Z(J )) is identical with the image ofeI ∈ K1(Z(I )) �→ K1(Z(J )). Furthermore, one can analogously consider the sub-group of K1(Z(J )) generated by the images of eI for all I ⊇ J . There is even noguess about the possible result in that Hilbert module analogs and the connection withthe Hodge’s conjecture.

This paper is devoted to the Arveson’s conjecture and K -homology invariant prob-lem above mentioned. Some positive results were obtained by Arveson and Douglas.For the d-shift Hilbert module with finite multiplicity, Arveson established p-essentialnormality in the case the submodule is generated by monomials [4]. That result on“monomial” submodules was generalized by Douglas to cases in which the d-shift isreplaced by more general weighted shifts [16]. Furthermore, Arveson has developeda theory of “Standard Hilbert modules” in an effort to establish his conjecture [5].Recently, Douglas indicated and discussed a new kind of index theorem arising fromthe study of essential normality [17].

For convenience of discussion, we will mainly work on the d-shift Hilbert moduleH2

d over the unit ball Bd of Cd . It is worth noticing that the same arguments parallel

run through the most “natural” Hilbert modules over the unit ball, especially the Hardymodule and the Bergman module.

This paper is organized as follows. Section 2 shows that in any dimension d, eachgraded principal submodule is p-essentially normal for p > d, the proof of which isconsiderably technical, based on two interesting inequalities of multiplication oper-ators. Therefore, the submodule [zn

1 + zn2 + zn

3] of 3-shift Hilbert module on whichfocused by Arveson [5,7], is p-essentially normal for p > 3.

Section 3 is devoted to the Arveson’s conjecture in dimension d = 2, 3. Usinga method on trace estimation, it is shown that in dimension d = 2, each graded

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910 K. Guo, K. Wang

submodule of H2d ⊗ C

r is p-essentially normal for p > 2. Combining trace estima-tion and techniques from commutative algebra, a more technical analysis shows thatin the case of dimension d = 3, each graded submodule of H2

d is p-essentially normalfor p > 3.

Concerning the study of p-smoothness of extensions arising from quotient sub-modules, in scalar-valued version, Douglas [17] raised a refinement for Arveson’sconjecture as follows: Let I be an ideal of C[z1, . . . , zd ], then the quotient moduleL2

a(Bd)/[I ] is p-essentially normal for p > dimC Z(I ). In dimensions d = 2, 3,Sect. 4 shows that the assertion is true.

Section 5 is devoted to the study of K -homology invariants arising from gradedquotient modules. For each essentially normal graded submodule M = [I ], consid-ering the corresponding quotient module H2

d /[I ], and let C∗〈I 〉 be the C∗-algebra onH2

d /[I ] generated by the identity operator and d-tuple {S1, . . . , Sd} obtained by com-pressing {Mz1, . . . ,Mzd } to the quotient module. Then one obtains a C∗-extension asin (1.1). This extension yields an odd K -homology element eI for the variety Z(I ).It is shown that this element eI is nontrivial in dimensions d = 2, 3. The proofs arebased on methods of commutative algebra and the characteristic space theory [12,29].This result applies to the submodule [zn

1 + zn2 − zn

3] of 3-shift Hilbert module to obtaina nontrivial geometric invariant for noncommutative Fermat curve “ Xn + Y n = Zn ”on which focused by Arveson [5,7]. It is remarkable that the K -homology elementsdoesn’t depend on the kernel Hilbert module with which one begins. In [17], Douglassuspected the same thing is true for more general case, especially in the case of alge-braic variety. For general cases, a deep understanding on topology and geometry ofalgebraic variety is required. Douglas suggested, in a private communication, that oneneeds a generalization of the calculus of pseudo-differential operators to the contextof complex algebraic varieties.

2 Each graded principal submodule is p-essentially normal

We will work on the d-shift Hilbert module H2d over the unit ball Bd of the d-dimen-

sional complex space Cd , where Bd = {z ∈ C

d : |z| < 1}. Recall H2d is the Hilbert

space of analytic function determined by the reproducing kernel Kw(z) = 11−〈z,w〉 ,

where 〈z, w〉 = ∑di=1 ziwi . It is easy to verify that H2

d has a canonical orthonor-

mal basis {( |α|!α! )

12 zα}, where α = (α1, . . . , αd) run over multi-indices of nonnegative

integers, and α! = α1! . . . αd !, |α| = α1 + · · · + αd .The d-shift Hilbert module H2

d , known as the symmetric Fock space, plays a basicrole in the study of multivariable operator theory. The space was first considered byDrury in [23] to generalize Von Neumann’s inequality. From the point of view ofHilbert modules, the module has been comprehensively investigated by Arveson, werefer the reader to the references [1–5,7] and [26–28] for a far-reaching operator-theoretic and index-theoretic developments of the Hilbert module. Arveson raiseda conjecture about the essentially normality or p-essentially normality (p > d) ofgraded submodules of H2

d ⊗ Cr with finite multiplicity r [3–7].

Let M be a submodule of H2d ⊗ C

r . Write Ri = (Mzi ⊗ I )|M for the restriction ofMzi ⊗ I to M and Si = PM⊥(Mzi ⊗ I )|M⊥ for the compression of Mzi ⊗ I to M⊥

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for i = 1, . . . , d. In Arveson’s language [5], the submodule M (the quotient mod-ule M⊥) is said to be p-essentially normal if the cross-commutators Ri R∗

j − R∗j Ri

(Si S∗j − S∗

j Si ) are in Lp for i, j = 1, . . . , d, where p ∈ [1,∞], and where L∞ means

the ideal K of compact operators.Let Hn be the space of homogeneous polynomials with degree n. Then H2

d ⊗ Cr

has a canonical graded decomposition H2d ⊗ C

r = ⊕n(Hn ⊗ Cr ). A submodule M

is called graded if M = ⊕n M ∩ (Hn ⊗ Cr ). Equivalently, M is generated by finitely

many vector-valued homogeneous polynomials. When M is generated by one poly-nomial, we say that M is a principal submodule. In this section we will show that theArveson’s conjecture is true in the case of the graded principal submodule.

We begin with the following lemma that comes from [4,5]. The lemma also appearedin [16].

Lemma 2.1 Let M be a submodule of H2d ⊗ C

r . Then the followings are equivalent:for p > d,

1. M is p-essentially normal,2. M⊥ is p-essentially normal,3. [PM ,Mzi ] = PM Mzi − Mzi PM are in L2p for 1 ≤ i ≤ d.

The following is the main result in this section.

Theorem 2.2 Each graded principal submodule of H2d is p-essentially normal for

p > d, that is, if q be a homogeneous polynomial, then the submodule [q] generatedby q is p-essentially normal for p > d.

From the above theorem, the submodule [zn1 + zn

2 + zn3] of 3-shift Hilbert module

H23 is p-essentially normal for p > 3. This answers a problem in [5,7].The proof of Theorem 2.2 is considerably technical. We begin with some notations.

Let α = (α1, . . . , αd), β = (β1, . . . , βd) be multi-indices of nonnegative integers. Asusual, set α±β = (α1 ±β1, . . . , αd ±βd) and zα = zα1

1 , . . . , zαdd , ∂α = ∂

α11 , . . . , ∂

αdd ,

where ∂ j = ∂∂z j

. Given two functions g, h, using iteration of the formula ∂ j (gh) =(∂ j g) h + g (∂ j h), one has the following Newton–Leibnitz formula

∂α(gh) =∑

β+γ=α

α!β!γ ! (∂

βg)(∂γ h). (2.1)

Given a polynomial p(z1, . . . , zd) = ∑α aαzα , define the differential operator p(∂) =∑

α aα∂α. In general, p(∂) acting in H2d , is unbounded, but it is densely defined and

closed. For a polynomial q, applying the Newton-Leibnitz formula (2.1), we have

p(∂)Mq =∑

α

1

α! M∂αq (∂α p)(∂). (2.2)

Define the number operator N acting in H2d = ⊕

n Hn by Nh = nh if h ∈ Hn . Theoperator N is unbounded, and its n-th eigenspace is Hn . Given a function F : Z+ → R,

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912 K. Guo, K. Wang

the operator F(N ) acting in H2d is understood as F(N )h = F(n)h if h ∈ Hn . Let A

be an operator acting in H2d , A is called graded if there exists an integer m such that

AHn ⊆ Hn+m for n = 0, 1, . . . , where the integer m is called the degree of A. If Ais graded and deg A = m, then A∗ also is graded and deg A∗ = −m. Moreover, wehave

AF(N ) = F(N − m)A. (2.3)

Let p = ∑α aαzα be a homogeneous polynomial with the degree m, and write p =∑

α aαzα . We consider multiplication operator Mp acting on H2d . Then a straightfor-

ward computation shows that

p(∂)En = n!(n − m)! M∗

p En, n = 0, 1, . . . ,

where En denotes the projection from H2d onto the space Hn of homogeneous poly-

nomials with the degree n. For each nonnegative integer k, we define Fk : Z+ → R

by Fk(n) = (n+k)!n! . Then we have

p(∂) = Fm(N )M∗p. (2.4)

Let q be a homogeneous polynomial with the degree l. Combining (2.2), (2.3) and(2.4) we have

Fm(N )M∗p Mq = p(∂)Mq =

∑

α

1

α! M∂αq(∂α p)(∂)

=∑

α

1

α! M∂αq Fm−|α|(N )M∗∂α p

=∑

α

1

α! Fm−|α|(N − l + |α|)M∂αq M∗∂α p.

This implies that

M∗p Mq =

∑

α

1

α! Fα(N )M∂αq M∗∂α p, (2.5)

where Fα : Z+ → R is defined by

Fα(n) = Fm−|α|(n − l + |α|)Fm(n)

= n!(n + m − l)!(n + m)!(n − l + |α|)! .

Let p be a homogeneous polynomial with degree m, applying (2.5) gives that forj = 1, . . . , d,

M∗z j

Mp = (N − m + 1)(N + 1)−1 Mp M∗z j

+ (N + 1)−1 M∂ j p. (2.6)

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When p = q, the equality (2.5) gives that

M∗p Mp =

∑

α

1

α! Fα(N )M∂α p M∗∂α p, (2.7)

where the operator Fα(N ) is defined by the function

Fα(n) = Fm−|α|(n − m + |α|)Fm(n)

= n!2(n + m)!(n − m + |α|)! .

The following inequalities will play a basic role in the proof of Theorem 2.2.

Proposition 2.3 Let p be a homogeneous polynomial with degree m. Then there existpositive constants C1 and C2 such that

1. M∗p Mp ≥ C1 (N + m)−1 ∑d

j=1 M∗∂ j p M∂ j p ;

2. M∗p Mp ≥ C2 (N + 1)−1 . . . (N + m)−1.

Proof Let e j denote the multi-index with 1 in the j-th position and 0 for all others.By (2.7), we have

M∗∂ j p M∂ j p =

∑

β

1

β!Gβ(N )M∂β+e j p

M∗∂β+e j p

(2.8)

where the operator Gβ(N ) is defined by the function Gβ(n) = Fm−1−|β|(n−m+|β|+1)Fm−1(n)

.

Setting α = β + e j , a simple computation gives that Gβ(N ) = (N + m) Fα(N ),where Fα(N ) is that in (2.7). Therefore, from (2.8) and (2.7), there exists a positiveconstant C0 such that

M∗∂ j p M∂ j p = (N + m)

∑

α:α j ≥1

α j

α! Fα(N )M∂α p M∗∂α p

≤ C0 (N + m)∑

α

1

α! Fα(N )M∂α p M∗∂α p

= C0 (N + m)M∗p Mp.

The inequality (1) comes easily from the above. The inequality (2) is an immediateconsequence of the inequality (1), completing the proof. ��

Let p be a homogeneous polynomial with deg p = m. The submodule [p], gener-ated by p, is the closure of the algebraic submodule {ph : h ∈ C[z1, . . . , zd ]}. Definea densely defined operator S on H2

d as follows

S [p]⊥ = 0, S (ph) = (N + m)−m2 h, h ∈ C[z1, . . . , zd ]. (2.9)

Lemma 2.4 Both the densely defined operators S and (N+1)m−1

2 M∂ j p S are bounded,and hence they can be extended as the bounded operators on H2

d .

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914 K. Guo, K. Wang

Proof It is enough to show that the operators are bounded on the subspace {ph : h ∈C[z1, . . . , zd ]}. For any polynomial h, applying Proposition 2.3 (2) shows that thereis a positive constant C2 such that

‖(N + m)−m2 h‖2 = 〈(N + m)−mh, h〉 ≤ C2 〈M∗

p Mph, h〉 = C2 ‖p h‖2.

By the definition (2.9) of S, we see that S is bounded.Using the formula (2.3) and Proposition 2.3(1), there exists a positive constant C1

such that

‖(N + 1)m−1

2 M∂ j p S(ph)‖2 = ‖(N + 1)m−1

2 M∂ j p(N + m)−m2 h‖2

= ‖(N + 1)−12 M∂ j p h‖2 = 〈(N + m)−1 M∗

∂ j p M∂ j ph, h〉≤ C1 〈M∗

p Mph, h〉 = C1 ‖ph‖2.

Therefore, the operator (N + 1)m−1

2 M∂ j p S is bounded, completing the proof. ��

Let Q p be the orthogonal projection from H2d onto [p]. It is easy to check

Q p = Nm2 Mp S. (2.10)

The proof of Theorem 2.2 By Lemma 2.1 (3), it is enough to show that [Q p,Mz j ] ∈L2s for s > d and 1 ≤ j ≤ d. Setting Q⊥

p = I − Q p, and applying (2.3), (2.6) and(2.10) we have

M∗z j

Q p − Q p M∗z j

= M∗z j

Q p − Q p M∗z j

Q p = Q⊥p M∗

z jQ p

(2.10)= Q⊥p M∗

z jN

m2 Mp S

(2.3)= (N + 1)m2 Q⊥

p M∗z j

Mp S

(2.6)= (N + 1)m2 Q⊥

p

((N − m + 1)(N + 1)−1 Mp M∗

z j+ (N + 1)−1 M∂ j p

)S

= (N + 1)m2 (N − m + 1)(N + 1)−1 Q⊥

p Mp M∗z j

S + (N + 1)m2 −1 Q⊥

p M∂ j p S

= (N + 1)−12 Q⊥

p (N + 1)m−1

2 M∂ j p S,

where the last equation follows from Q⊥p Mp = 0.

Setting A j = Q⊥p (N +1)

m−12 M∂ j p S, Lemma 2.4 shows that A j is a bounded graded

operator with deg A j = −1. The above reasoning gives the equality [Q p,Mz j ] =A∗

j (N + 1)− 12 . Since (N + 1)− 1

2 ∈ L2s for s > d, we see that [Q p,Mz j ] ∈ L2s fors > d and 1 ≤ j ≤ d, completing the proof. ��

As a consequence of the above proof, the following corollary will be used in thenext section.

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Corollary 2.5 There exists a bounded graded operator B j with deg B j = 1 such that

C j = Q p Mz j Q⊥p = [Q p,Mz j ] = B j (N + 1)−

12 , j = 1, . . . , d.

In the essentially same way as done in the scalar-valued case, Theorem 2.2 can begeneralized to the vector-valued version, and described as follows. Let p = (p1, . . . ,

pr ) ∈ C[z1, . . . , zd ] ⊗ Cr , where each component pi is a homogeneous polynomial,

and not necessarily have the same degree. Considering the submodule [p] of H2d ⊗C

r

generated by p, then [p] is the closure of the set {ph : h ∈ C[z1, . . . , zd ]}. Writem1 = deg p1, . . . ,mr = deg pr , and set m = max{m1, . . . ,mr }. We define theoperator S : H2

d ⊗ Cr → H2

d by

S [p]⊥ = 0, S (ph) = (N + m)−m2 h, h ∈ C[z1, . . . , zd ].

Then the operator S is bounded. Let Mp : H2d → H2

d ⊗Cr be a multiplication operator

defined by Mph = ph for h ∈ H2d , and let Qp denote the orthogonal projection from

H2d ⊗ C

r onto [p], then

Qp = Mp(N + m)m2 S.

By a slight careful modification of the proof of Theorem 2.2, one can obtain thatQ⊥

p M∗z j

Qp ∈ L2p for p > d. Therefore, by Lemma 2.1(3), the principal submodule[p] is p-essentially normal for p > d.

3 In dimensions d = 2, 3, each graded submodule is p-essentially normal

The following theorem shows that Arveson’s conjecture is true in dimension d = 2.Essentially, this is a generalization of [31, Theorem 2.4].

Theorem 3.1 If d = 2 and M is a graded submodule of H2d ⊗C

r , then M is p-essen-tially normal for p > 2.

Proof Since M is graded, this means that it can be decomposed as

M = M0 ⊕ M1 ⊕ · · · ⊕ Mn ⊕ · · · ,

where Mn ⊆ Hn ⊗ Cr for n = 0, 1, . . . . Writing Ai for [PM ,Mzi ][PM ,Mzi ]∗ for

i = 1, 2, a simple verifying gives that

Ai = PM Mzi M∗zi

PM − Mzi PM M∗zi,

and hence the operator Ai maps Mn to Mn . Thus we have

Trace(Ai |Mn ) = Trace(PMn Mzi M∗zi

PMn )− Trace(PMn Mzi PMn−1 M∗zi

PMn )

= Trace(PMn Mzi M∗zi

PMn )− Trace(PMn−1 M∗zi

Mzi PMn−1).

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916 K. Guo, K. Wang

By [1, Lemma 2.8 and Propostion 5.3], it holds that

Mz1 M∗z1

+ Mz2 M∗z2

= I − 1 ⊗ 1, M∗z1

Mz1 + M∗z2

Mz2 = N + 2

N + 1.

Hence

Trace(A1 + A2)|Mn = dim Mn − n + 1

ndim Mn−1.

Applying [2, Theorem 4.2], there exist integers a, b and N such that dim Mn = a+bnfor all n ≥ N . Therefore, when n ≥ N + 1, we have

Trace(A1 + A2)|Mn = dim Mn − n + 1

ndim Mn−1 = b − a

n.

From Sect. IV.2 in [13], the operator A1 + A2 belongs to the weak trace class L(1,∞),

and hence A1 + A2 ∈ Lt for t > 1. This implies that all [PM ,Mzi ] belong to Schattenclass Lq for q > 2. The desired conclusion comes from Lemma 2.1, completing theproof. ��

In what follows we will show Arveson’s conjecture is true in the case of the dimen-sion d = 3.

Theorem 3.2 In dimension d = 3, each graded submodule of H2d is p-essentially

normal for p > 3.

In order to prove this theorem we require some notations and lemmas.The Hilbert polynomial is an invariant associated with the graded modules. We

refer the reader to references [2,25,38] for more details. For convenience, we remarka variation of [38, Vol(II), Chapter VII, Th.41 and Th.42’] as follows.

Lemma 3.3 Let L be a graded algebraic submodule of C[z1, . . . , zd ] ⊗ Cr over the

polynomial ring C[z1, . . . , zd ], and let

L =∑

n

Ln, C[z1, . . . , zd ] ⊗ Cr =

∑

n

Hn ⊗ Cr

be graded decompositions of L and C[z1, . . . , zd ] ⊗ Cr , respectively. Then for suf-

ficiently large n, the dimension dim Hn ⊗ Cr/Ln is a polynomial whose degree is

equal to the projective dimension of the zero variety determined by the annihilatorideal Ann(C[z1, . . . , zd ] ⊗ C

r/M).

Let M be a graded submodule of H2d , then there exists a unique homogeneous ideal

I of C[z1, . . . , zd ] such that M = [I ], where [I ] denotes the closure of I in H2d .

In fact, I = M ∩ C[z1, . . . , zd ]. For each homogeneous ideal I , the ideal I can beuniquely written as I = pL , which is called the Beurling form of I , where p is thegreatest common divisor of I , i.e., p = GCD(I ) = GCD{q : q ∈ I }. It is easy to

123


check that both p and L are homogeneous. In dimension d = 3, by the elementary factof algebraic geometry, one sees that dimC Z(L) ≤ 1. Let L = ⊕n Ln be the homo-geneous decomposition of L . Applying lemma 3.3 shows that there exists a positiveinteger N0 such that when n ≥ N0, dim Hn/Ln = l for some constant l.

Decompose the coordinate multiplication operators Mz j ( j = 1, 2, 3) with respectto [p] ⊕ [p]⊥ as follows

Mz j =[

R j C j

0 S j

]

.

Then we have

M∗z j

Mz j =[

R∗j R j R∗

j C j

C∗j R j C∗

j C j + S∗j S j

]

, Mz j M∗z j

=[

R j R∗j + C j C∗

j C j S∗j

S j C∗j S j S∗

j

]

This gives that

R∗j R j = Q p M∗

z jMz j Q p, R j R∗

j = Q p Mz j M∗z j

Q p − C j C∗j , (3.1)

where Q p is the orthogonal projection from H2d onto [p].

The proof of Theorem 3.2. Let QI denote the orthogonal projection from H2d onto

[I ]. From Lemma 2.1(3), it suffices to show that [QI ,Mz j ] ∈ L2s for s > 3, andj = 1, 2, 3. Considering the equality

[QI ,Mz j ] = [QI ,Mz j ]Q p + [QI ,Mz j ]Q⊥p = [QI ,Mz j ]Q p + QI Mz j Q⊥

p

= [QI ,Mz j ]Q p + QI Q p Mz j Q⊥p = [QI ,Mz j ]Q p + QI C j ,

Corollary 2.5 shows C j ∈ L2s for s > 3. It remains to show [QI ,Mz j ]Q p ∈ L2s fors > 3. Write Q = Q p − QI , that is, Q is the orthogonal projection from H2

d onto[p] � [pL]. By the equalities

[QI ,Mz j ]Q p = −[Q, R j ], j = 1, 2, 3, (3.2)

it suffices to show that the following assertion is true

3∑

j=1

[Q, R j ]∗[Q, R j ] ∈ Ls

for s > 3. Since Q R∗j Q = R∗

j Q, Q R j Q = Q R j , we have

[Q, R j ]∗[Q, R j ] = Q R∗j R j Q − R∗

j Q R j . (3.3)

123

918 K. Guo, K. Wang

Let L = ∑n Ln, C[z1, z2, z3] = ∑

n Hn be homogeneous decompositions of L andC[z1, z2, z3], respectively. Then [p] = ⊕

n pHn, [pL] = ⊕n p Ln are homoge-

neous decompositions of [p] and [pL], respectively, and hence

[p] � [pL] =⊕

n

(pHn � pLn).

Let Pn and Qn be the orthogonal projections from H2d onto pHn and pHn � pLn ,

respectively. Setting An = Pn

(∑3j=1[Q, R j ]∗[Q, R j ]

)Pn , then

3∑

j=1

[Q, R j ]∗[Q, R j ] =∑

n

An .

Noticing the fact∑d

i=1 Mzi M∗zi

= I − 1 ⊗ 1,∑d

i=1 M∗zi

Mzi = N+dN+1 and applying

(3.1), (3.3),

An = Pn

⎛

⎝3∑

j=1

[Q, R j ]∗[Q, R j ]⎞

⎠ Pn =3∑

j=1

Qn[Q, R j ]∗[Q, R j ]Qn

= Qn

⎛

⎝3∑

j=1

M∗z j

Mz j

⎞

⎠ Qn −3∑

j=1

Qn R∗j Q R j Qn

= m + n + d

m + n + 1Qn −

3∑

j=1

Qn R∗j Qn+1 R j Qn .

Combining Corollary 2.5 and (3.1), we see

Trace An = m + n + d

m + n + 1dim Qn − Trace

⎛

⎝3∑

j=1

Qn R∗j Qn+1 R j Qn

⎞

⎠

= m + n + d

m + n + 1dim Qn − Trace Qn+1

⎛

⎝3∑

j=1

R j R∗j

⎞

⎠ Qn+1

=(

m + n + d

m + n + 1dim Qn − dim Qn+1

)

+ Trace Qn+1

⎛

⎝3∑

j=1

C j C∗j

⎞

⎠ Qn+1

=(

m + n + d

m + n + 1dim Qn − dim Qn+1

)

+Trace Qn+1

(∑3j=1 B j B∗

j

)Qn+1

m + n + 1.

123


By Lemma 3.3 there exists a natural number N0 such that when n ≥ N0, dim Qn =dim(pHn � pLn) = l for some constant l. Therefore,

Trace Qn+1

⎛

⎝3∑

j=1

B j B∗j

⎞

⎠ Qn+1 ≤ C0 Trace Qn+1 = C0 l,

where C0 = ‖∑3j=1 B j B∗

j ‖. This means that there exists a positive constant C suchthat when n ≥ N0,

Trace An ≤ C

m + n + 1. (3.4)

Therefore, An ∈ L(1,∞) and it follows that

∑

n

An =3∑

j=1

[Q, R j ]∗[Q, R j ] ∈ Ls

for s > 1. This gives the desired assertion, completing the proof. ��Using the proof of Theorem 3.3, we have

Proposition 3.4 Assume that I is a homogeneous ideal of C[z1, . . . , zd ] and I = pLis its Beurling form. If the complex dimension of the zero variety Z(L) dimC Z(L) =0, or 1, then [I ] is p-essentially normal for p > d.

4 p-essentially normal graded quotient modules

To any submodule M of H2d ⊗C

r , there corresponds a quotient submodule H2d ⊗C

r/M.From Lemma 2.1, if the submodule is p-essentially normal, then the correspondingquotient submodule necessarily enjoys this property. However, for the most “natural”examples, the quotient submodules enjoy more strong property than this. To studyp-smoothness of extensions arising from quotient submodules, Douglas [17] raised arefinement for Arveson’s conjecture as follows:

Douglas’s conjecture Let I be an ideal of C[z1, . . . , zd ]. Then the quotient moduleL2

a(Bd)/[I ] is p-essentially normal for p > dimC Z(I ).For convenience of discussion, we will work on the d-shift Hilbert module H2

d . Thesame arguments parallel run through the most “natural” Hilbert modules over the unitball, especially the Bergman module over the unit ball. It is shown that in dimensiond = 2, 3, if I is homogeneous, then the quotient module H2

d /[I ] is p-essentiallynormal for p > dimC Z(I ).

The following result comes from the discussion with Douglas.

123

920 K. Guo, K. Wang

Proposition 4.1 Let M be a graded submodule of H2d ⊗ C

r . If the zero variety ofthe annihilator ideal Ann(H2

d ⊗ Cr/M) has the complex dimension ≤ 1, Then the

quotient module H2d ⊗ C

r/M is p-essentially normal for p > 1.

Proof Let {S1, . . . , Sd} be the tuple obtained by compressing {Mz1, . . . ,Mzd } to M⊥(here H2

d ⊗ Cr/M is naturally identified with M⊥). Writing Mi for Mzi for i =

1, . . . , d, then it is easy to check that

[S∗i , S j ] = P⊥[M∗

i ,M j ]P⊥ − [P⊥,Mi ]∗[P⊥,M j ], (4.1)

where P⊥ denotes the projection onto M⊥. From [1, Proposition 5.3]

M∗i M j − M j M∗

i = −(N + 1)−1 M j M∗i , if i �= j, and

(4.2)M∗

i Mi − Mi M∗i = (N + 1)−1 − (N + 1)−1 Mi M∗

i .

Since the zero variety of the annihilator ideal has the complex dimension ≤1, applyingLemma 3.3 shows that there exists some positive integer K0 such that when n > K0,the dimension dim P⊥En P⊥ is a constant, here En is the orthogonal projection fromH2

d ⊗ Cr onto Hn ⊗ C

r . Therefore, for p > 1, the operator P⊥(N + 1)−1 P⊥ ∈ Lp

and it follows from (4.2) that

P⊥[M∗i ,M j ]P⊥ ∈ Lp. (4.3)

Setting Bi = [P⊥,Mi ]∗[P⊥,Mi ], i = 1, . . . , d, then Bi ≥ 0 and a simple veri-fying shows that

Bi = P⊥M∗i Mi P⊥ − M∗

i P⊥Mi P⊥.

Let M⊥ = N0 ⊕ N1 ⊕ · · · ⊕ Nn ⊕ · · · be the homogeneous decomposition of M⊥,and P⊥

n denote the orthogonal projection onto Nn . Put

B = B1 + · · · + Bd , B(n) = P⊥n B P⊥

n .

Then as done in the proof of Theorem 3.2, when n > K0, there exists a positiveconstant C such that

Trace B(n) ≤ C

n + 1.

Therefore, B ∈ L(1,∞) ⊂ Lt for t > 1, and it follows that

[P⊥,Mi ] ∈ Lt

for t > 2 and i = 1, . . . , d. Applying this fact and (4.3) to (4.1) shows that [S∗i , S j ] ∈

Lp for p > 1, that is, the quotient module is p-essentially normal for p > 1. ��

123


The following results support Douglas’s conjecture.

Proposition 4.2 In dimensions d = 2, 3, if I is homogeneous, then the quotientmodule H2

d /[I ] is p-essentially normal for p > dimC Z(I ).

The proof of Proposition 4.2 In dimension d = 2, we apply Proposition 4.1 to obtainthe desired conclusion. It remains to prove the case of dimension d = 3. Since I �= 0,this means dimC Z(I ) ≤ 2. When dimC Z(I ) ≤ 1, the result comes from Proposi-tion 4.1. In the case dimC Z(I ) = 2, then necessarily, the ideal I has the nonconstantgreatest common divisor p, and let I = p L be its Beurling form. Applying Lemma 3.3shows that there exist some positive integers K0, a, b, such that when n > K0, thedimension dim Q⊥

I En Q⊥I = a + bn, where Q⊥

I = I − QI , and QI denotes theorthogonal projection from H2

d onto [I ], En is the orthogonal projection from H2d

onto Hn . Therefore, for p > 2 the operator Q⊥I (N + 1)−1 Q⊥

I ∈ Lp. From (4.2),

Q⊥I [M∗

i ,M j ]Q⊥I ∈ Lp, p > 2.

To complete the proof, it suffices to prove the following claim:

[QI ,M j ]∗[QI ,M j ] ∈ Lp for p > 2 and j = 1, 2, 3. (4.4)

Below, We use the same notations as the proof of Theorem 3.2 in Sect. 3. A simplecalculation gives

[QI ,M j ]∗[QI ,M j ] = [Q, R j ]∗[Q, R j ] − [Q, R j ]∗QI C j

− C∗j Q I [Q, R j ] + C∗

j Q I C j .

From the proof of Theorem 3.2, the commutators [Q, R j ] ∈ Lp for p > 2 andj = 1, 2, 3. Therefore it is enough to show that C∗

j Q I C j ∈ Lp for p > 2 andj = 1, 2, 3. By Corollary 2.5,

C∗j Q I C j = B∗

j Q I B j (N + 1)−12 (N + 2)−

12

= B∗j Q I B j (N + 1)−

12 (N + 2)−

12 Q⊥

p

= (N + 1)12

(N + 2)12

B∗j Q I B j Q⊥

p (N + 1)−1 Q⊥p .

It is easy to verify Q⊥p (N + 1)−1 Q⊥

p = Q⊥p Q⊥

I (N + 1)−1 Q⊥I Q⊥

p ∈ Lp for p > 2.The above reasoning shows that the claim (4.4) is true, and hence the proof is com-plete. ��

Applying the same proof of Proposition 4.2, we have

Proposition 4.3 Let I = p L be the Beurling form of a homogeneous ideal I ofC[z1, . . . , zd ]. If the complex dimension of the zero variety Z(L), dimC Z(L) =0, or 1, then the quotient module H2

d /[I ] is p-essentially normal for p > d − 1.

123

922 K. Guo, K. Wang

5 K-homology

To study K -homology yielded by graded quotient modules, one naturally is associ-ated with Taylor spectrum and Taylor essential spectrum for operator tuples. There is alarge literature concerning Taylor spectrum and Taylor essential spectrum for operatortuples on a Hilbert space. We have made no attempt to compile a comprehensive list ofreferences, but refer the reader’s attention to [14,15,35,36]. The spectrum and essen-tial spectrum of graded submodules and quotient modules is well known for experts.Because of the lack of references, we remark a short proof as follows.

Let M be a submodule of H2d , and write Ri , Si for the restriction of Mzi to M , and

the compression of Mzi to M⊥ for i = 1, . . . , d, respectively.

Theorem 5.1 Assume that I is a homogeneous ideal of C[z1, . . . , zd ]. For the gradedsubmodule M = [I ] of H2

d , if M is essentially normal, then we have

1. σ(R1, . . . , Rd) = Bd ,

2. σe(R1, . . . , Rd) = ∂Bd ,

3. σ(S1, . . . , Sd ) = Z(I ) ∩ Bd ,

4. σe(S1, . . . , Sd) = Z(I ) ∩ ∂Bd .

Proof It is easy to see that (1) and (2) come from [33, Sect. 3]. Applying [33] again,it holds that Z(I ) ∩ Bd ⊆ σ(S1, . . . , Sd) ⊆ Z(I ) ∩ Bd , and σe(S1, . . . , Sd) ⊆Z(I ) ∩ ∂Bd . Therefore, it suffices to show that the following inclusion is true

σe(S1, . . . , Sd) ⊇ Z(I ) ∩ ∂Bd .

Given anyw = (w1, . . . , wd) ∈ Z(I )∩∂Bd ,we claim that∑d

i=1(Si −wi )(Si −wi )∗

is not Fredholm. Otherwise, there is a positive finite rank operator F such that theoperator

A =d∑

i=1

(Si − wi )(Si − wi )∗ + F

is positive and invertible. But, by the fact that I is homogeneous, one sees that tw ∈Z(I )∩Bd and the reproducing kernel Ktw ∈ M⊥ for any t < 1. Setting ktw = Ktw‖Ktw‖ ,then

⟨d∑

i=1

(Si − wi )(Si − wi )∗ktw, kt w

⟩

=⟨

d∑

i=1

(Mzi − wi )∗ktw, (Mzi − wi )

∗kt w

⟩

= |t − 1|2|w|2 → 0

as t → 1. Since ktw weakly converges to 0 as t → 1, we have that ‖F ktw‖ → 0.Hence,

〈Aktw, ktw〉 → 0.

123


This leads to a contradiction since the operator A is positive and invertible. Thisshows that the operator

∑di=1(Si −wi )(Si −wi )

∗ is not Fredholm, and it follows thatσe(S1, . . . , Sd) = Z(I ) ∩ ∂Bd , completing the proof. ��

Let I be a homogeneous ideal of C[z1, . . . , zd ]. For the graded submodule M = [I ]of H2

d , we use C∗[I ] to denote the Toeplitz algebra on M , which is a C∗-algebra gen-erated by {I, R1, . . . , Rd}. An easy argument shows that C∗[I ] is irreducible. If M isessentially normal, then C∗[I ] contains all compact operators K. By Theorem 5.1(2),we have a C∗-extension defined by C∗[I ],

0 → K ↪→ C∗[I ] π→ C(∂Bd) → 0,

where π is the unital ∗-homomorphism given by π(Ri ) = zi for i = 1, . . . , d. By [9],this extension yields a canonical element in the odd K -homology group K1(∂Bd) ∼= Z.In fact, this K -homology element is identified with the index of the Koszul complexdefined by the tuple {R1, . . . , Rd} [8,9,33], and hence it corresponds to (−1)d .

This section will be devoted to K -homology invariants arising from graded quotientmodules, by which geometry of the quotient modules and geometry of algebraic vari-eties are connected. In dimensions d = 2, 3, it is shown that K -homology invariantsdetermined by graded quotients are nontrivial.

Let I be a homogeneous ideal of C[z1, . . . , zd ]. For the graded submodule M = [I ]of H2

d , we consider the corresponding quotient module H2d /M , and naturally iden-

tify the quotient module H2d /M with M⊥. Let C∗〈I 〉 denote the C∗-algebra on the

graded quotient [I ]⊥ generated by Id, S1, . . . , Sd . If [I ] is essentially normal, thenapplying Lemma 2.1 shows that [I ]⊥ also is. In this case, using the similar argumentas in [5, Proposition 2.5] shows that C∗〈I 〉 is irreducible and hence it contains allcompact operators. Indeed, suppose that there exists a projection P such that for anypolynomial q, P Sq = Sq P. Since M is graded, 1 ∈ M⊥ and we put e = P1. For anyf ∈ M⊥, f − f (0) ∈ M⊥. Taking a polynomial sequence {qn} with qn(0) = 0 suchthat qn converges to f − f (0) in the norm of H2

d , then we have

〈e, f − f (0)〉 = limn

〈e, qn〉 = limn

〈P1, Sqn 1〉= lim

n〈1, Sqn e〉 = lim

n〈1, qne〉

= 0.

Taking f = e shows that ‖e − e(0)‖2 = 0 and hence e = e(0). Since e = P1, itfollows that e(0)2 = e(0). This implies that e = 0 or e = 1. In the case e = 0, forany f ∈ M⊥, taking a polynomial sequence {qn} such that qn converges to f , thenSqn 1 = PM⊥qn → f , and hence

P Sqn 1 = Sqn P1 = Sqn e = 0 → P f.

This gives P = 0. In the case e = 1, the same reasoning shows P = I dM⊥ . Therefore,C∗〈I 〉 is irreducible.

123

924 K. Guo, K. Wang

Setting Z(I ) = Z(I ) ∩ ∂Bd , then from Theorem 5.1(4) we have a C∗-extensiondefined by C∗〈I 〉 as follows

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0. (5.1)

This extension yields an odd K -homology element eI for the variety Z(I ) by BDF-theory[9]. Moreover, in dimension d = 2, 3, Proposition 4.2 shows that this elementeI is p-smooth for p > dimC Z(I ). One naturally would expect this element eI tobe nontrivial [7], which would establish a connection between operator theory andalgebraic geometry. A more challenging problem is which element of the K1(Z(I ))is obtained. Douglas suspected that this K -homology element in fact is identical withthe fundamental class for the variety Z(I ) [17].

In what follows, we will shows that this element eI is nontrivial in dimensionsd = 2, 3. In particular, in dimension d = 2, we will give an explicit expression of thisinvariant, and it is identical with the fundamental class for the variety.

We begin first with a useful operation on extensions, which is called the disjointsum. Given two compact metric spaces X1 and X2 with X1 ∩ X2 = ∅, and two exten-sions (E1, π1), (E2, π2) of K by C(X1) on H1 and C(X2) on H2, respectively, anextension (E, π) of K by C(X1 � X2) on H1 ⊕ H2, which is called the disjoint sumof (E1, π1) and (E2, π2), is defined by

E = {T1 ⊕ T2 + K : Ti ∈ Ei , i = 1, 2, K is compact}π(T1 ⊕ T2 + K )|Xi = πi (Ti ), i = 1, 2.

where � denotes disjoint union of spaces. From [10, Theorem 4.8], the above operationgives the following isomorphism

K1(X1 � X2) ∼= K1(X1)⊕ K1(X2). (5.2)

Proposition 5.2 Let I and J be homogeneous ideals, and Z(I ) ∩ Z(J ) = {0}. If[I ], [J ] and [I ∩ J ] are all essentially normal, then one has

eI∩J = eI ⊕ eJ .

Proof For homogeneous ideals I, J, I ∩ J , we consider three extensions

0 → K ↪→ C∗〈I 〉 π1→ C(Z(I )) → 0,

0 → K ↪→ C∗〈J 〉 π2→ C(Z(J )) → 0,

0 → K ↪→ C∗〈I ∩ J 〉 π3→ C(Z(I ∩ J )) → 0.

Since Z(I ) ∩ Z(J ) = {0} and Z(I ∩ J ) = Z(I ) ∪ Z(J ), we have

Z(I ∩ J ) = Z(I )⊔

Z(J ).

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We will prove that the third extension is the disjoint sum of the first two. Consideringthe equality

[I ∩ J ]⊥ = [I ]⊥ ⊕ ([I ] � [I ∩ J ])

and for a polynomial p, setting

S(1)p = P[I ]⊥ Mp|[I ]⊥ , S(2)p = P[I∩J ]⊥ Mp|[I∩J ]⊥ , S(3)p = P[I ]�[I∩J ]Mp|[I ]�[I∩J ],

then we have

S(2)p =[

S(1)p 0K p S(3)p

]

. (5.3)

By assumption and Lemma 2.1, it is easy to check that K p is compact, and S(3)p isessentially normal. Hence

σe

(S(2)z1

, . . . , S(2)zd

)= σe

(S(1)z1

, . . . , S(1)zd

) ⋃σe

(S(3)z1

, . . . , S(3)zd

).

From Theorem 5.1, it holds that

σe

(S(2)z1

, . . . , S(2)zd

)= Z(I )

⊔Z(J )

and

σe

(S(1)z1

, . . . , S(1)zd

)= Z(I ).

Hence, we have

Z(J ) ⊆ σe

(S(3)z1

, . . . , S(3)zd

)⊆ ∂Bd .

For each polynomial Q ∈ J , it is easy to check that

Q(

S(3)z1, . . . , S(3)zd

)= S(3)Q = 0.

This means that

σe

(S(3)z1

, . . . , S(3)zd

)⊆ Z(J ),

and hence we get that

σe

(S(3)z1

, . . . , S(3)zd

)= Z(J ).

123

926 K. Guo, K. Wang

From the decomposition (5.3), we see that

eI∩J = eI ⊕J ,

whereJ ∈ K1(Z(J )) is the K -homology element coming from the extension definedby C∗(Id, S(3)z1 , . . . , S(3)zd )+ K. The same reasoning shows that

eI∩J = I ⊕ eJ ,

where I ∈ K1(Z(I )). By (5.2),

K1(Z(I ∩ J )) = K1(Z)⊕ K1(Z(J )).

This means that

J = eJ , I = eI ,

and hence eI∩J = eI ⊕ eJ . The proof is complete. ��Remark Using the argument in the proof of Proposition 5.2, one can prove the fol-lowing: If I and J are two homogeneous ideals satisfying Z(I ) ∩ Z(J ) = {0}, and[I ], [J ] and [I J ] are all essentially normal, then

eI J = eI ⊕ eJ .

5.1 In dimension d = 2

Let I be a homogeneous ideal in two variables, and let M = [I ] be a graded sub-module of 2-shift Hilbert module H2

2 . Then by Theorem 3.1, M is essentially normal,and hence gives an extension as in (5.1). This extension yields an odd K -homologyelement eI for Z(I ). We will prove eI �= 0, and give an explicit expression of thisinvariant.

Proposition 5.3 Let p(z1, z2) = αz1 + βz2, then we have K1(Z(p)) = Z andepn = −n, where epn denotes K -homology element determined by (5.1) when I =pn

C[z1, z2].Proof Firstly, consider the case α = 1, β = 0, that is, p(z1, z2) = z1. Since Z(z1) =Z(z1)∩ ∂B2 = {(0, z2) : |z2| = 1}, and this set is a unit circle, we have K1(Z(z1)) ∼=Z. With this identification, ezn

1= Ind S2 [9]. We claim that Ind S2 = −n. In fact,

letting h ∈ ker S2, then z2h ∈ [zn1]. Write h(z) = ∑

ai j zi1z j

2, then i ≥ n if ai j �= 0.This implies h ∈ [zn

1], and hence h = 0. Therefore, ker S2 = 0. Since

ker S∗2 = [zn

1]⊥ ∩ ker M∗z2

= span{1, z1, . . . , zn−11 },

123


we have

Ind S2 = dim ker S2 − dim ker S∗2 = −n.

Now let us consider the general case p(z1, z2) = αz1+βz2. Define a unitary transformµ : C

2 → C2 by

µ(z1, z2) =(αz1 + βz2

√|α|2 + |β|2 ,−βz1 + αz2√|α|2 + |β|2

)

.

It is easy to check µ(Z(p)) = Z(z1) and the transform µ induces a unitary operatorU : f → f ◦ µ on H2

2 . Then the operator U maps [zn1]⊥ onto [pn]⊥, and gives the

following commutative diagram,

0 → K → C∗〈zn1〉 → C(Z(z1)) → 0

∼=↓ U ∼=↓ U ∼=↓ µ0 → K → C∗〈pn〉 → C(Z(p)) → 0.

Hence K1(Z(p)) ∼= Z and epn = −n. ��We turn to any homogeneous polynomial q(z1, z2). Then q(z1, z2) can be decom-

posed as

q(z1, z2) = (α1z1 + β1z2)γ1 . . . (αnz1 + βnz2)

γn ,

such that

Z(αi z1 + βi z2) ∩ Z(α j z1 + β j z2) = {0}, if i �= j.

Combining Propositions 5.2 with 5.3 gives that

eq = e(α1z1+β1z2)γ1 ⊕ · · · ⊕ e(αn z1+βn z2)γn = (−γ1, . . . ,−γn).

For each homogeneous ideal I in C[z1, z2], the ideal I can be uniquely written asits Beurling form I = p L . By [37, Lemma 6.1], L is a finite codimensional idealof C[z1, z2]. This implies that [p] = [I ] ⊕ N for some finite dimensional space N ,and hence [I ]⊥ = [p]⊥ ⊕ N . Therefore, for quotient modules [I ]⊥ and [p]⊥, thecorresponding extensions (5.1) are weakly equivalent, and hence they are equivalent[9, Theorem 1.8]. This means ep = eI . We summarizes the above discussion to thefollowing theorem.

Theorem 5.4 In dimension d = 2, let I = p L be the Beurling form of a homogeneousideal I . Then the extension (5.1) determines one nontrivial element ep ∈ K1(Z(p)),only depending on p. Furthermore, if I = pL1, J = q L2, and p and q have thesame zero set, then ep = eq if and only if p = c q, where c is a nonzero constant.

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928 K. Guo, K. Wang

5.2 In dimension d = 3

Let I be a homogeneous ideal in three variables, and let M = [I ] be a graded sub-module of 3-shift Hilbert module H2

3 . Theorem 3.2 says that each graded submoduleis essentially normal, and hence each graded quotient submodule is. This subsectionwill shows that each graded quotient submodule determines a nontrivial extension

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0, (5.4)

and hence it gives rise to an odd nontrivial K -homology element eI for the zero varietyZ(I ). This invariant contains important information on both the submodule and zerovariety.

We begin with a well known fact. From the Universal Coefficient Theorem in [11],there exists a natural short exact sequence

0 → ExtZ(K0(X),Z) → Ext (X)

γ∞→ Hom(K (X),Z) → 0.

Therefore, for a C∗-extension

0 → K ↪→ A π→ C(X) → 0, (5.5)

if the extension is trivial (or split), then the image of the corresponding K -homologyelement by γ∞ is also trivial. This implies the following result.

Lemma 5.5 If the extension (5.5) is trivial, then for each natural number n and eachFredholm operator A ∈ A ⊗ Mn, we have Ind A = 0, where Mn is the algebra of allcomplex n × n matrices.

Firstly we show the K -homology element is nontrivial in the case of the principalsubmodule.

Proposition 5.6 Let p be a homogeneous polynomial in three variables. Then theextension

0 → K ↪→ C∗〈p〉 π→ C(Z(p)) → 0,

is nontrivial, where C∗〈p〉 denotes the C∗-algebra C∗{Id, S1, S2, S3} on [p]⊥.

Proof It is easy to prove that there exists a pair of co-prime polynomials q1 = z1 +α1z2 + β1z3, q2 = z1 + α2z2 + β2z3, such that Z(p) ∩ Z(q1) ∩ Z(q2) = {0}. ByTheorem 5.1, for the tuple {S1, S2, S3} acting on [p]⊥, it holds that

σe(S1, S2, S3) = Z(p).

Since q1, q2 have no common zero point on σe(S1, S2, S3), the operator

A =[

Sq1 Sq2

−S∗q2

S∗q1

]

123


is Fredholm. Clearly, (1, 0)T ∈ ker A∗ and hence ker A∗ �= {0},where (1, 0)T denotesthe transpose of the vector (1, 0). We claim Ind A �= 0. For the claim, it is enough toprove ker A = 0. If A(ξ1, ξ2)

T = 0, then we have

(I) q1ξ1 + q2ξ2 ∈ [p],(II) S∗

q1ξ2 = S∗

q2ξ1.

Without loss of generality, we may assume that both ξ1 and ξ2 are homogeneous, andhave the same degree. From the above (I), there exists a homogeneous polynomial φsuch that q1ξ1 +q2ξ2 = pφ. Since the ideal (q1, q2) generated by q1, q2 is prime, andZ(p) ∩ Z(q1) ∩ Z(q2) = {0}, this implies that there exist homogeneous polynomialsh1, h2 such that φ = q1h1 + q2h2, and hence q1ξ1 + q2ξ2 = pq1h1 + pq2h2. Thisgives that

q1(ξ1 − ph1) = −q2(ξ2 − ph2).

Since GC D(q1, q2) = 1, there exists a polynomial h such that

ξ1 = ph1 + q2h, ξ2 = ph2 − q1h.

Writing h = g1 + g2 with g1 ∈ [p] and g2 ∈ [p]⊥, then

ξ1 = (ph1 + q2g1)+ q2g2, ξ2 = (ph2 − q1g1)− q1g2.

This means that

ξ1 = Sq2 g2, ξ2 = −Sq1 g2.

Combining this equality with the above (II) yields

S∗q1

Sq1 g2 + S∗q2

Sq2 g2 = 0,

and hence ξ1 = Sq2 g2 = 0, ξ2 = −Sq1 g2 = 0. Therefore, ker A = 0 and Ind A �= 0.The desired conclusion follows from Lemma 5.5. ��

Combining Proposition 5.6 and Remark following Proposition 5.2, we have

Corollary 5.7 For a homogeneous ideal I of C[z1, z2, z3], let I = p L be its Beurlingform. If p is not a constant and Z(p) ∩ Z(L) = {0}, then the extension

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0

is nontrivial.

Let P be a prime ideal of C[z1, . . . , zd ]. The height of P , denoted by height(P),is defined as the maximal length l of any properly increasing chain of prime ideals

0 = P0 ⊂ P1 · · · ⊂ Pl = P.

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930 K. Guo, K. Wang

Since the polynomial ring C[z1, . . . , zd ] is Noetherian, every prime ideal has finiteheight and the height of an arbitrary ideal is defined as the minimum of the heights ofits associated prime ideals. For an ideal I , one has

dimC Z(I ) = n − l,

where l = height(I ) is the height of I (cf. [22,38]).

Lemma 5.8 Let q be homogeneous polynomial in C[z1, z2, z3], and let P be a homo-geneous prime ideal of C[z1, z2, z3] with height(P) = 2. If q /∈ P, then Z(P) ∩Z(q) = {0}.Proof By the Lasker-Noether decomposition theorem [38, Vol. (I), p. 208], decompose

P + q C[z1, z2, z3] =m⋂

j=1

I j (5.6)

as its irredundant primary decomposition, where each I j is Pj -primary for some primeideal Pj . This implies P ⊆ Pj for each j . If P = Pj for some j , then the inclusion

P + q C[z1, z2, z3] ⊆ I j ⊆ Pj = P

implies q ∈ P . This contradiction shows that P is strictly contained in Pj for eachj . This means that each Pj has height 3, and hence each Pj is finite codimensional.Since each Pj is homogeneous, it follows that

Z(I j ) = Z(Pj ) = {0}

for j = 1, . . . ,m. Therefore, by (5.6)

Z(P) ∩ Z(q) =m⋃

j=1

Z(I j ) = {0},

completing the proof. ��Proposition 5.9 For a homogeneous ideal I of C[z1, z2, z3], if GCD(I ) = 1, thenthe extension

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0

is nontrival.

Proof From the Lasker-Noether decomposition theorem for polynomial ring [38,Vol.(I), p. 208], I can be decomposed as

I = I0 ∩ I1 ∩ · · · ∩ Im,

123


where I0 is a finite codimensional ideal, and each I j is Pj -primary for some primeideal Pj having height 2. Writing J = I1 ∩ · · · ∩ Im , then

[I ]⊥ = [J ]⊥ + [I0]⊥.

Since [I0]⊥ is finite dimensional, without loss of generality, we may assume that Ihas decomposition I = I1 ∩ · · · ∩ Im such that each I j is Pj -primary for some primeideal Pj having height 2. Then one can choose a homogeneous polynomial p withdegree 1 such that Z(p)∩ Z(I ) = {0}. The choice is as follows: Let Vj = Pj ∩H1 forj = 1, . . . ,m, where H1 is the space of all homogeneous polynomials with degree 1.Since height(Pj ) = 2, then Vj is a proper subspace of H1 for all j . By a simple linearalgebra, ∪m

j=1Vj � H1. Taking a nonzero p ∈ H1 \∪mj=1Vj , then by Lemma 5.8 such

a p satisfies

Z(p) ∩ Z(I ) = ∪mj=1 Z(p) ∩ Z(I j ) = ∪m

j=1 Z(p) ∩ Z(Pj ) = {0}.

For this p, we see that Sp is Fredholm. We claim that IndSp �= 0. In fact, 1 ∈ ker S∗p.

Therefore it suffices to prove that ker Sp = 0. Let h ∈ [I ]⊥ such that Sph = 0, thenph ∈ [I ]. Since I is homogeneous, without loss of generality, we assume that h ishomogeneous, and hence ph ∈ I by [22]. This means that ph ∈ I j for j = 1, . . . ,m.Using the characteristic space theory [12, Chap. 2], it is not difficult to show that h ∈ I j

for each j , and hence h ∈ I . This says that h = 0, and therefore, ker Sp = 0. Theclaim follows. The desired result comes from the claim and Lemma 5.5, completingthe proof. ��Proposition 5.10 Let I be a homogeneous ideal of C[z1, z2, z3], and let I = p L beits Beurling form. If p is not a constant, then the extension

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0

is nontrivial.

Proof From the Lasker-Noether decomposition theorem, the ideal L can be decom-posed as

L = L0 ∩ L1 ∩ · · · ∩ Lm,

where L0 is a finite codimensional ideal, and each L j is Pj -primary for some primeideal Pj having height 2. Writing J = L1 ∩ · · · ∩ Lm , then

dim p J/pL = dim J/L0 ∩ J = dim (J + L0)/L0 < ∞,

and hence p J = pL + R for some finite dimensional space R. Hence, we may assumethat L = L1 ∩ · · · ∩ Lm .

Decompose p as the product of its prime factors, p = pt11 · · · ptl

l , and set

Vj = Pj ∩ H1, j = 1, . . . ,m, Wi = pi C ∩ H1, i = 1, . . . , l.

123

932 K. Guo, K. Wang

Then both Vj and Wi are proper subspaces of H1. By a simple linear algebra, theset V = ∪m

j=1Vj⋃ ∪l

i=1Wi is strictly contained in H1. Taking a nonzero polyno-mial φ ∈ H1 \ V , Lemma 5.8 applies here to show that Z(φ) ∩ Z(L) = {0}. SinceGCD(φ, p) = 1, the ideal (φ, p) generated byφ and p, has height 2 (see [12, Corollary3.1.12]). Hence, as done in the proof of Proposition 5.9, one can choose a homogeneouspolynomial ψ with degree 1 such that

Z(ψ) ∩ Z(φ) ∩ Z(p) = {0}.

From the above discussion we have

Z(ψ) ∩ Z(φ) ∩ Z(I ) = {0}, Z(ψ) ∩ Z(φ) ∩ Z(p) = {0}. (5.7)

Noticing that

[I ]⊥ = [p]⊥ ⊕ ([p] � [I ])

and for a polynomial q, putting

S(1)q = P[I ]⊥ Mq |[I ]⊥ , S(2)q = P[p]⊥ Mq |[p]⊥ , S(3)q = P[p]�[I ]Mq |[p]�[I ],

then we have

S(1)q =[

S(2)q 0Kq S(3)q

]

. (5.8)

By Lemma 2.1 and Theorem 3.2, Kq is compact, and the operators S(1)q , S(2)q , S(3)qare all essentially normal. For each polynomial Q ∈ L , since I = p L , we haveQ(S(3)z1 , S(3)z2 , S(3)z3 ) = S(3)Q = 0 and hence

σe

(S(3)z1

, S(3)z2, S(3)z3

)⊆ Z(L). (5.9)

By (5.7), both the tuples (S(1)φ , S(1)ψ ) and (S(2)φ , S(2)ψ ) are Fredholm. By (5.9) and Z(φ)∩Z(L) = {0}, we see that S(3)φ is Fredholm, and the tuple (S(3)φ , S(3)ψ ) is Fredholm. Fur-thermore, applying [15, Proposition 11.1] gives that

Ind(

S(3)φ , S(3)ψ

)= 0.

From (5.8), it holds that

Ind(

S(1)φ , S(1)ψ

)= Ind

(S(2)φ , S(2)ψ

)+ Ind

(S(3)φ , S(3)ψ

)= Ind

(S(2)φ , S(2)ψ

).

123


By [14], and using the same reasoning as in the proof of Proposition 5.6, we have

Ind(

S(2)φ , S(2)ψ

)= Ind

[S(2)φ S(2)ψ

−S(2)∗ψ S(2)∗φ

]

�= 0,

and hence

Ind(

S(1)φ , S(1)ψ

)=

[S(1)φ S(1)ψ

−S(1)∗ψ S(1)∗φ

]

�= 0.

The desired result follows from Lemma 5.5, completing the proof. ��Combining Propositions 5.9 and 5.10, we reach at the main result in this subsection.

Theorem 5.11 Let I be a homogeneous ideal of C[z1, z2, z3] with infinite codimen-sion, then the extension

0 → K ↪→ C∗〈I 〉 π→ C(Z(I )) → 0

is nontrivial.

Acknowledgements We are grateful to Professors W. Arveson and R. Douglas for many valuable sug-gestions. The authors thank the referee for helpful suggestions which make this paper more readable. Thiswork is partially supported by NSFC, SRFDP and NKBRPC (2006CB805905).

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Essentially normal Hilbert modules and K-homology