An index theorem for quotients of Bergman spaces on egg domains
aa r X i v : . [ m a t h . OA ] S e p AN INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACESON EGG DOMAINS
MOHAMMAD JABBARI AND XIANG TANG
Dedicated to the memory of Ronald G. Douglas
Abstract.
In this paper we prove a K -homology index theorem for the Toeplitzoperators obtained from the multishifts of the Bergman space on several classes ofegg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball[21]. Introduction
Around a decade ago a multivariate operator theory approach to algebraic geometrywas suggested by Arveson and Douglas in the following way [7, 19]. Suppose that I ⊆ A := C [ z , . . . , z m ] is an ideal of the ring of polynomials in m variables. Tounderstand the geometry of the zero variety V ( I ) := { p ∈ C m : f ( p ) = 0 , ∀ f ∈ I } defined by I , algebraic geometers study the coordinate ring A/I . To find an operatortheory model for
A/I , one can replace A by the Bergman space L a (Ω) of square-integrable analytic functions on some bounded strongly pseudoconvex domain Ω ⊆ C m with smooth boundary, and mod it out by the closure I of I inside L a (Ω) . The quotientHilbert space Q I := L a (Ω) /I has a natural Hilbert A -module structure given by p · ( f + I ) = pf + I , p ∈ A , f ∈ L a (Ω) . Transporting this action to the orthogonalcomplement L a (Ω) ⊖ I = I ⊥ ∼ = Q I , makes I ⊥ a Hilbert A -module. Alternatively, the module structure of I ⊥ is given bythe compression of multiplication operators: T p := P I ⊥ M p | I ⊥ , p ∈ A, where M p : L a (Ω) → L a (Ω) is the multiplication by p , and P I ⊥ is the orthogonalprojection in L a (Ω) onto I ⊥ . Let T I be the unital C*-algebra generated by { T p : p ∈ A } ∪ K , where K is the ideal of compact operators on I ⊥ . Arveson, based on his work There is a one-to-one correspondence between commuting m -tuples of operators T := ( T , . . . , T m ) acting on a Hilbert space H and Hilbert A -module structures on H [7]. The correspondence is givenby representing each polynomial p ( z , . . . , z m ) ∈ A by the operator p ( T , . . . , T m ) . Conversely, T isidentified with the m -tuple ( M z , . . . , M z m ) of multiplication operators by coordinate functions, andis called the fundamental tuple of Toeplitz operators on the Hilbert A -module H . Based on thiscorrespondence, the properties of T are attributed to H and vice versa. For example, H is calledessentially normal (respectively p -essentially normal) if all [ T j , T ∗ k ] are compact (respectively Schatten p -summable). Also, σ e ( H ) denotes the essential Taylor spectrum associated to the fundamental tupleof Toeplitz operators of H [45, 41]. on the model theory of spherical contractions in multivariate dilation theory [3],[1,Chapters 40–41], conjectured that [5, 6]: Conjecture 1 (Arveson) . I ⊥ is essentially normal. In other words, all commutators [ T z j , T ∗ z k ] , j, k = 1 , . . . , m are compact. Suppose momentarily that this conjecture holds. Also, assume that I is homo-geneous. Then the maximal ideal space of T I / K is homeomorphic via the mapping ϕ (cid:0) ϕ ( T z ) , . . . , ϕ ( T z m ) (cid:1) to the essential Taylor spectrum of ( T z , . . . , T z m ) , whichis itself contained in X I := V ( I ) ∩ ∂ Ω [19, Theorem 4.1]. (Also see [14, Corollary3.10],[32],[33, Theorem 5.1].) The Gelfand-Naimark duality then gives the followingshort exact sequence of C*-algebras: → K ֒ → T I → C ( X I ) → . Let τ I := [ T I ] be the equivalence class represented by this exact sequence in the odd K -homologygroup K ( X I ) of Brown-Douglas-Fillmore [11, 12]. Douglas [19] (see also [8, Section25]) asked for an explicit computation of this element in other topological or geometricrealizations of K -homology: Problem 2 (Douglas) . Assume that I is homogeneous and I ⊥ is essentially normal.Identify τ I ∈ K ( X I ) . More specifically, in the same paper he conjectured that:
Conjecture 3 (Douglas) . Let I be the vanishing ideal of an algebraic variety V ⊆ C m which intersects ∂ Ω transversally. Then I ⊥ is essentially normal, and its inducedextension class τ I is identified with the fundamental class of X I , namely the extensionclass induced by the Spin c Dirac operator associated to the natural Cauchy-Riemannstructure of X I . By analogy with the Atiyah-Singer index theorem, one expects that this conjecturewould lead to new connections between geometry and operator theory. To see whatbrought Arveson and Douglas to their conjecture/problem, we refer the reader to theiroriginal papers [6, 4, 19] as well as [1, Chapter 41],[36, Sections 1.2–3]. Specially,Conjecture 3 generalizes some aspects of the Boutet de Monvel index theorem forToeplitz operators on strongly pseudoconvex domains to possibly singular algebraicvarieties [10, 9].Let us review some results about these conjectures and problem. (See also [1,Chapter 41],[34].) When Ω is the unit open ball, Conjecture 1 has been proved for thefollowing cases:(1) I is monomial [6, 18, 21].(2) I is homogeneous and m ≤ [33]. N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 3 (3) I is homogeneous and dim V ( I ) ≤ [33].(4) I is principal [33, 24, 28, 29, 20, 30]. (The last two references allow for stronglypseudoconvex domains Ω .)(5) I has a stable generating set { p , . . . , p k } of homogeneous polynomials in thesense that there exists C > such that every q ∈ I can be written as q = P kj =1 r j p j with r j ∈ A and k r j p j k L (Ω) ≤ C k q k L (Ω) [43, 47].(6) I is the vanishing ideal of a homogeneous variety smooth away from the origin[27, 22, 25, 48].When Ω is the unit ball, the articles [33] and [22] answer Problem 2 respectivelywhen m ≤ and when I is the vanishing ideal of a complete intersection variety(possibly singular away from the boundary). In [21] we gave an answer to Problem 2when Ω is the unit open ball and I is monomial: Theorem 4.
Let Ω be the unit open ball B m , and I be a monomial ideal.(a) There exist a positive integer k , essentially normal Hilbert A -modules A := L a (Ω) , A , . . . , A k , and Hilbert A -module morphisms Ψ q : A q → A q +1 , q = 0 , . . . , k − such that → I ֒ → A → A → · · · Ψ k − → A k → is exact. (This implies that I ⊥ is essentially normal.)(b) For each q , let T ( A q ) be the unital C*-algebra generated by all module actionoperators as well as all compact operators on the Hilbert module A q , and let σ qe := σ qe ( A q ) be the essential Taylor spectrum associated to A q . Then the identification τ I = k X q =1 ( − q − [ T ( A q )] holds in K (cid:0) σ e ∪ · · · ∪ σ ke (cid:1) . By its explicit construction, each A q has a tractable geometry as the Hilbert spaceof square-integrable analytic sections of a Hermitian vector bundle on a disjoint unionof subsets of B m .In this paper, we generalize Theorem 4 to the case when Ω is an egg domain of theform(1) Ω := ( z , . . . , z m ) ∈ C m : m X j =1 (cid:12)(cid:12) z j (cid:12)(cid:12) p j < , p j > , or more generally of the form(2) Ω := m X j =1 (cid:12)(cid:12) z j (cid:12)(cid:12) p j a + n X k =1 | w k | q k b + · · · < ⊆ C m + n + ··· , Bounded linear maps that preserve A -module structures. MOHAMMAD JABBARI AND XIANG TANG where the finitely many parameters p j , q k , a, b, . . . are arbitrary positive reals. (Whenall p j , q k , . . . equal , Ω is called a generalized complex ellipsoid in [38].) Theorem 5.
Let Ω be a domain of the form (1) or (2), and I be a monomial ideal.(a) There exist a positive integer k , essentially normal Hilbert A -modules A := L a (Ω) , A , . . . , A k , and Hilbert A -module morphisms Ψ q : A q → A q +1 , q = 0 , . . . , k − such that (3) → I ֒ → A → A → · · · Ψ k − → A k → is exact. (This implies that I ⊥ is essentially normal.)(b) For each q , let T ( A q ) be the unital C*-algebra generated by all module actionoperators as well as all compact operators on the Hilbert module A q , and let σ qe := σ qe ( A q ) be the essential Taylor spectrum associated to A q . Then the identification τ I = k X q =1 ( − q − [ T ( A q )] holds in K (cid:0) σ e ∪ · · · ∪ σ ke (cid:1) . The explicit construction of the resolution (3) and the proof of Theorem 5 comein Sections 2 and 3, respectively. Our proof uses crucially the fact that monomialsconstitute an orthogonal basis for L a (Ω) if Ω is a domain of type (1) or (2). Each A q has a tractable geometry as the Hilbert space of square-integrable analytic sections ofa Hermitian vector bundle on a disjoint union of subsets of Ω . Remark 6.
When I is homogeneous, the C*-algebra generated by { } ∪ { T p : p ∈ A } is irreducible (it has no proper reducing closed subspace), hence contains K if I ⊥ isessentially normal [33, Page 923],[17, Theorem 5.39]. Remark 7.
One reason why we care about monomial ideals is that a comprehensiveunderstanding of the phenomena appearing in this generically nonradical case may leadto new results beyond the recently established ones about radical ideals [22, 25, 27].
Remark 8.
When all p j are ≥ and at least one of them is > , domain (1) is weakly(but not strongly) pseudoconvex and with smooth boundary [15],[39, Section 11.5].(The same is true for domain (2) if all a, b, . . . , p j , q k , . . . are ≥ and at least one of p j , q k , . . . is > .) As far as we know, putting the polydiscs aside [49], Theorem 5 isthe only result which discusses Conjecture 1 and Problem 2 on weakly pseudoconvexdomains. Remark 9.
Note that a domain of type (2) is obtained from a domain of type (1) wheneach | z j | is replaced by an expression of the form P n j k =1 | z jk | p jk , where all coordinates z jk are distinct. Applying this process on a domain of type (2) and repeating thisprocess finitely many times, give rise to more generalized egg domains. For example, N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 5 we can get the following one: (cid:16)(cid:0) | z | p + | z | p (cid:1) p + (cid:0) | z | p + | z | p + | z | p (cid:1) p + | z | p (cid:17) p +( · · · ) p + · · · < . The arguments in this paper prove Theorem 5 for all such domains.Arveson’s statement of his essential normality conjecture was more refined than Con-jecture 1 in the sense that it addressed the Schatten class membership of commutators[6, 18]. In this paper, however, we merely focused on the membership of commutatorsin the ideal of compacts. The reason is that our proof of Theorem 5.(b) relies cruciallyon the usage of the Fuglede-Putnam theorem in the proof of Proposition 13.(b,c,d).Since the Schatten class version of the Fuglede-Putnam theorem is missing [18, 44], ourresult does not determine the Schatten class membership of the commutators for thequotients of Bergman spaces by monomial ideals. Nevertheless, it is worth pointingout that our computations (not included in this paper) show that the whole Bergmanspace L a (Ω ) , associated with domain (1), is p -essentially normal exactly when p isstrictly larger than max (cid:8) m, p j ( m −
1) : j = 1 , . . . , m (cid:9) . (See also Remark 15.) This suggests that the Schatten class property of the commu-tators may be related to the convexity and geometry of the domain [13, 23, 40]. Weplan to discuss this relation in the future [37]. Acknowledgments.
We would like to thank Ronald Douglas, Guoliang Yu and YiWang for inspiring discussions. Both authors are partially supported by NSF grants.2.
The construction of the resolution in Theorem 5
From now on, Ω and Ω are domains of type (1) and (2), respectively. We developthe details for Ω , and Ω can be treated similarly, with the only difference beingProposition 11 and the proof of Lemma 14. We always use the multi-index notation[39, Page 3], especially | α | to stand for the sum of the components of the multi-index α . N denotes the set of nonnegative integers.2.1. The monomial orthonormal basis for the Bergman space.
Monomial func-tions z α , α ∈ N m are orthogonal in L a (Ω ) , as the integration in polar coordinates ineach variable shows. On the other hand, since Ω is a complete Reinhardt domain,polynomials are dense in L a (Ω ) with respect to the topology of uniform convergenceon compacts [42, Page 47]. Then a standard shrinking argument [50, Page 43],[26,Page 11] shows that the normalized monomials z α p ω ( α ) , ω ( α ) := k z α k L a (Ω ) MOHAMMAD JABBARI AND XIANG TANG constitute an orthonormal basis for the Hilbert space L a (Ω ) . Next, we are going tofind an explicit formula for ω ( α ) as well as ω ( α, β, . . . ) := (cid:13)(cid:13) z α w β · · · (cid:13)(cid:13) L a (Ω ) . In whatfollows, for a variable x = ( x , . . . , x m ) ranging on some part of R m , dx := dx · · · dx m denotes the Riemannian density of the Euclidean space R m . The set of positive realsis denoted by R + . Lemma 10.
Given α := ( α , . . . , α m ) ∈ R m + , we have Z x ∈ R m + , P x j < x α dx = B (cid:0) α +12 (cid:1) m (cid:12)(cid:12) α +12 (cid:12)(cid:12) , Z x ∈ R m + , P x j =1 x α dσ m ( x ) = B (cid:0) α +12 (cid:1) m − , where α + 1 = ( α + 1 , . . . , α m + 1) , B (( α + 1) /
2) = Q Γ(( α j + 1) / / Γ( P ( α j + 1) / isthe multi-variable Beta function, and dσ m is the Riemannian density that dx induceson the unit sphere S m − ⊆ R m .Proof. These are standard facts [2, Section 1.8],[50, Page 13],[31, Page 80]. (cid:4)
Proposition 11. (a) Given multi-index α ∈ N m , we have ω ( α ) = k z α k L a (Ω ) = π m Q p j B (cid:16) α +1 p (cid:17)(cid:12)(cid:12)(cid:12) α +1 p (cid:12)(cid:12)(cid:12) , where α +1 p := (cid:16) α +1 p , . . . , α m +1 p m (cid:17) .(b) Given multi-indices α ∈ N m , β ∈ N n , . . . , we have ω ( α, β, . . . ) = (cid:13)(cid:13)(cid:13) z α w β · · · (cid:13)(cid:13)(cid:13) L a (Ω ) = π m + n + ··· Q p j Q q k · · · ab · · · B (cid:18) α + 1 p (cid:19) B (cid:18) β + 1 q (cid:19) · · · B (cid:18)(cid:12)(cid:12)(cid:12) α +1 ap (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) β +1 bq (cid:12)(cid:12)(cid:12) , . . . (cid:19)(cid:12)(cid:12)(cid:12) α +1 ap (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) β +1 bq (cid:12)(cid:12)(cid:12) + · · · . Proof. (a) Using polar coordinates z j = x j e √− θ j for each coordinate z j of z = ( z , . . . , z m ) ,we have ω ( α ) = Z z ∈ Ω x α Y x j dx j dθ j = (2 π ) m Z x ∈ R m + , P x pjj < x α +1 dx. After the change of variables X j := x p j j , we have ω ( α ) = (2 π ) m Q p j Z X ∈ R m + , P X j < X α +2 p − dX. We are done by Lemma 10.
N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 7 (b) Using polar coordinates z j = x j e √− θ j , w k = y k e √− ϕ k , . . . , we have ω ( α, β, . . . ) =(2 π ) m + n + ··· Z x ∈ R m + ,y ∈ R n + ,..., (cid:18)P x pjj (cid:19) a + (cid:16)P y qkk (cid:17) b + ··· < x α +1 y β +1 · · · dxdy · · · . After the change of variables X j := x pj , Y k := y qk , . . . , we have ω ( α, β, . . . ) =(2 π ) m + n + ··· Q p j Q q k · · · Z X ∈ R m + ,Y ∈ R n + ,..., ( P X j ) a + ( P Y k ) b + ··· < X α +2 p − Y β +2 q − · · · dXdY · · · . Now comes the trick we learned from [16]. Changing to the spherical coordinates X = rξ, Y = sη, . . . , where r, s, . . . are positive reals and ξ, η, . . . live on unit spheres S m − , S n − , . . . , respectively, we have ω ( α, β, . . . ) = (2 π ) m + n + ··· Q p j Q q k · · · Z r,s,... ∈ R + ,r a + s b + ··· < r (cid:12)(cid:12)(cid:12) α +2 p (cid:12)(cid:12)(cid:12) − s (cid:12)(cid:12)(cid:12) β +2 q (cid:12)(cid:12)(cid:12) − · · · drds · · · × Z ξ ∈ S m − ,η ∈ S n − ,... ξ α +2 p − η β +2 q − · · · dσ m ( ξ ) dσ n ( η ) · · · , where S m − denotes S m − ∩ R m + , and similarly for others. The first integral is given bythe first formula in Lemma 10 after the change of variables R := r a , S := s b , . . . , andthe second integral is given by the second formula in Lemma 10. (cid:4) For later uses, we do the same computations in the more general context of weightedBergman spaces. Given a domain Ω ⊆ C m with smooth boundary, L a,s (Ω) , s > − denotes the weighted Bergman space consisting of all holomorphic functions f on Ω such that R Ω | f ( z ) | ρ ( z ) s dV ( z ) < ∞ , where ρ ( z ) is a positively signed smooth definingfunction for Ω and dV is the Lebesgue measure. For Ω and Ω , we use the definingfunctions − X | z j | p j and − (cid:16)X (cid:12)(cid:12) z j (cid:12)(cid:12) p j (cid:17) a − (cid:16)X | w k | q k (cid:17) b − · · · , respectively. Proposition 12. (a) Given multi-index α ∈ N m , we have ω ,s ( α ) := k z α k L a,s (Ω ) = π m Q p j B (cid:18) α + 1 p (cid:19) Γ (cid:18)(cid:12)(cid:12)(cid:12) α +1 p (cid:12)(cid:12)(cid:12)(cid:19) s !Γ (cid:18)(cid:12)(cid:12)(cid:12) α +1 p (cid:12)(cid:12)(cid:12) + s + 1 (cid:19) . MOHAMMAD JABBARI AND XIANG TANG (b) Given multi-indices α ∈ N m , β ∈ N n , . . . , we have ω ,s ( α, β, . . . ) := (cid:13)(cid:13)(cid:13) z α w β · · · (cid:13)(cid:13)(cid:13) L a,s (Ω ) = π m + n + ··· Q p j Q q k · · · ab · · · B (cid:18) α + 1 p (cid:19) B (cid:18) β + 1 q (cid:19) · · · B (cid:12)(cid:12)(cid:12)(cid:12) α + 1 ap (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) β + 1 bq (cid:12)(cid:12)(cid:12)(cid:12) , . . . ! × s !Γ (cid:18)(cid:12)(cid:12)(cid:12) α +1 ap (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) β +1 bq (cid:12)(cid:12)(cid:12) + · · · (cid:19) Γ (cid:18) s + 1 + (cid:12)(cid:12)(cid:12) α +1 ap (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) β +1 bq (cid:12)(cid:12)(cid:12) + · · · (cid:19) . Proof. (a) Similar to the proof of Proposition 11.(a), we have ω ,s ( α ) = (2 π ) m Q p j Z X ∈ R m + , P X j < X α +2 p − (cid:16) − X X j (cid:17) s dX. Changing to the spherical coordinates X = rξ , r > , ξ ∈ S m − , we have ω ,s ( α ) = (2 π ) m Q p j Z ξ ∈ S m − ξ α +2 p − dσ m ( ξ ) × Z r (cid:12)(cid:12)(cid:12) α +2 p (cid:12)(cid:12)(cid:12) − (cid:0) − r (cid:1) s dr. The first integral is given by the second formula in Lemma 10, and the second integralis given by the formula R t a − (1 − t ) b − dt = Γ( a )Γ( b ) / Γ( a + b ) after the change ofvariable r = t .(b) Similar to the proof of Proposition 11.(b), we have ω ,s ( α, β, . . . ) = (2 π ) m + n + ··· Q p j Q q k · · · Z ξ ∈ S m − ,η ∈ S n − ,... ξ α +2 p − η β +2 q − · · · dσ m ( ξ ) dσ n ( η ) · · · × Z r,s,... ∈ R + ,r a + s b + ··· < r (cid:12)(cid:12)(cid:12) α +2 p (cid:12)(cid:12)(cid:12) − s (cid:12)(cid:12)(cid:12) β +2 q (cid:12)(cid:12)(cid:12) − · · · drds · · · (cid:16) − r a − s b − · · · (cid:17) s . The first integral is given by the second formula in Lemma 10. The second integralafter the change of coordinates R := r a , S := s b , . . . becomes ab · · · Z R,S,... ∈ R + ,R + S + ··· < R (cid:12)(cid:12)(cid:12) α +2 ap (cid:12)(cid:12)(cid:12) − S (cid:12)(cid:12)(cid:12) β +2 bq (cid:12)(cid:12)(cid:12) − · · · (cid:0) − R − S − · · · (cid:1) s dRdS · · · . Changing to the spherical coordinates ( R, S, . . . ) = rξ , r > , ξ in the unit sphere,this latter integral equals an integral in the second formula in Lemma 10 multipliedby some integral of the form R t u − (1 − t ) v − dt = Γ( u )Γ( v ) / Γ( u + v ) . (cid:4) Some notations.
From now on, we are going to use the notation(4) z n := z n . . . z n m m p ω ( n ) , n = ( n , . . . , n m ) ∈ N m for the elements of the orthonormal basis of L a (Ω ) derived in Section 2.1. N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 9
Given a positive integer q , let S q ( m ) denote the set of all q -shuffles of the set { , . . . , m } , namely S q ( m ) := (cid:8) j := ( j , . . . , j q ) ∈ Z q : 1 ≤ j < j < · · · < j q ≤ m (cid:9) . Whenever necessary, we identify shuffles in S q ( m ) with subsets of { , . . . , m } of size q . This enables us to talk about the union, intersection, etc. of shuffles of { , . . . , m } with themselves and with other subsets of { , . . . , m } .2.3. Boxes and their associated Hilbert modules.
To each j = ( j , . . . , j q ) ∈ S q ( m ) and b = ( b , . . . , b q ) ∈ N q we associate the box B bj := n ( n , . . . , n m ) ∈ N m : n j i ≤ b i for i = 1 , . . . , q o , and to each box B bj we associate the Hilbert space H bj := L a (Ω ) ⊖ D z b +1 j , . . . , z b q +1 j q E consisting of all functions X = P n ∈ N m X n z n ∈ L a (Ω ) such that X n = 0 for every n ∈ N m \ B bj . An element X ∈ H bj has the Taylor expansion X = P X n ··· n m z n withsummation over n j ≤ b , . . . , n j q ≤ b q . The general construction in Section 1 aboutthe orthogonal complements of polynomial ideals makes H bj a Hilbert A -module. ( A denotes the ring of polynomials in m variables.) More explicitly, its fundamental tupleof Toeplitz operators is given by T j , b z i ( z n ) := ( z i z n , if ( n , . . . , n i − , n i + 1 , n i +1 , . . . , n m ) ∈ B bj , , otherwise , , i = 1 , . . . , m. In the next proposition we gather several facts about essential normality which willbe used later.
Proposition 13 (Arveson-Douglas) . (a) Let Ω be an open subset of C m , I ⊆ A be ahomogeneous ideal, and P, Q := 1 − P be the orthogonal projections in L (Ω) onto I and I ⊥ , respectively. Suppose that L a (Ω) is essentially normal. Then I is essentiallynormal (module actions are given by restrictions of multiplications in L a (Ω) ) if andonly if I ⊥ is essentially normal, if and only if all [ M z α , P ] , α = 1 , . . . , m are compact,if and only if all P M z α Q are compact, if and only if all [ M z α , Q ] are compact, if andonly if all QM ∗ z α P are compact.(b) Let M and N be isomorphic Hilbert A -modules. Then M is essentially normalif and only of N is; if so, then they represent the same odd K -homology class.(c) Let M be an essentially normal Hilbert A -module, and N ⊆ M be a submodule.Then N is essentially normal if and only if the quotient module M / N is.(d) Let Ψ : A → A be a closed-range Hilbert A -module map between essentiallynormal Hilbert modules. Then the kernel and range of Ψ are essentially normal.Proof. (a) Our reference is [6, Theorem 4.3]. Recall that an operator T is compact ifand only if T ∗ is compact, if and only if T T ∗ is compact. Let the module action of p ∈ A on L (Ω) , I and I ⊥ be denoted by operators M p , R p and T p , respectively. Forbrevity, set M α := M z α , R α := R z α and T α := T z α . The last four statements are easilyseen to be equivalent. Here are the reasons. Since I is invariant under M α , we have P M α P = M α P . Then [ M α , P ] = M α P − P M α = P M α P − P M α = − P M α Q. The equality P + Q = 1 gives [ M α , P ] = − [ M α , Q ] . Also note that ( P M α Q ) ∗ = QM ∗ α P .For the rest, we need the assumption that L (Ω) is essentially normal. With anabuse of language, one says that, as mappings from L (Ω) to I , R α P and R ∗ β P equal P M α P = M α P and P M ∗ β P , respectively. Then [ R α , R ∗ β ] P = M α P M ∗ β P − P M ∗ β M α P ∼ M α P M ∗ β P − P M α M ∗ β P = [ M α , P ] M ∗ β P = − P M α QM ∗ β P = − ( P M α Q )( QM ∗ β P ) = − ( P M α Q )( P M β Q ) ∗ = − [ M α , P ][ M β , P ] ∗ , where ∼ denotes equality modulo compacts. This identity shows that all [ R α , R ∗ β ] arecompact if and only if all [ M α , P ] are. The rest of the proof is dual. As mappings from L (Ω) to I ⊥ , T α Q and T ∗ β Q equal QM α Q and QM ∗ β Q = M ∗ β Q , respectively. We alsohave the identity [ T α , T ∗ β ] Q ∼ [ M β , Q ] ∗ [ M α , Q ] , which proves that all [ T α , T ∗ β ] are compact if and only if all [ M α , Q ] are.(b, c, d) Refer to [22, Proposition 4.4], [18, Theorem 2.1] and [19, Theorem 2.2],respectively. (cid:4) Lemma 14.
Each H bj is essentially normal.Proof. We first show that L a (Ω ) is essentially normal. Let M z i ∈ B ( L a (Ω )) , i =1 , . . . , m be the multiplication by the coordinate function z i . Since these operatorscommutate with each other, according to the Fuglede-Putnam theorem, it suffices toverify that each M z i is essentially normal. Clearly (cid:2) M z i , M ∗ z i (cid:3) ( z n ) = λz n , ∀ n = ( n , . . . , n m ) ∈ N m , where λ = λ ′ − λ ′′ , λ ′ = ω ( n · · · n m ) ω ( n · · · n i − · · · n m ) , λ ′′ = ω ( n · · · n i + 1 · · · n m ) ω ( n · · · n m ) . We need to check that λ → when the norm of n (say the l norm) tends to infinity.By Proposition 11.(a), we have λ ′ = Γ (cid:16) n i +1 p i (cid:17) Γ (cid:16) n i p i (cid:17) Γ (cid:16) N + n i p i (cid:17) Γ (cid:16) N + n i +1 p i (cid:17) N + n i p i N + n i +1 p i , where N := X l = i n l + 1 p l . N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 11
Note that by Stirling’s formula (or more strongly [46]), Γ( x + a ) / Γ( x ) ≈ x a as the realvariable x grows large. Therefore, when n i is bounded and N → ∞ , λ ′ is dominatedby N − /p i , so λ → . On the other hand, when n i → ∞ , λ ′ asymptotically behaveslike (cid:18) n i p i (cid:19) pi (cid:18) N + n i p i (cid:19) − pi ≈ − N n i p i + N ! pi ≈ − Nn i + p i N .
This shows that λ = λ ′ − λ ′′ → when n i → ∞ . We have shown that L a (Ω ) isessentially normal.Let P be the orthogonal projection in L a (Ω ) onto H bj . To prove our lemma, ac-cording to Proposition 13.(a), it suffices to check that each [ M z i , P ] is compact. Foreach n ∈ B bj we have P M z i ( z n ) = q ω ( n ··· n i +1 ··· n m ) ω ( n ··· n m ) z n ··· n i +1 ··· n m , if ( n · · · n i + 1 · · · n m ) ∈ B bj , , otherwise ,M z i P ( z n ) = q ω ( n ··· n i +1 ··· n m ) ω ( n ··· n m ) z n ··· n i +1 ··· n m , if ( n · · · n i · · · n m ) ∈ B bj , , otherwise . Note that the coefficients √· · · appear because of the normalization in definition (4).Therefore (cid:2) M z i , P (cid:3) ( z n ) = q ω ( n ··· b l +1 ··· n m ) ω ( n ··· b l ··· n m ) z n ··· n i +1 ··· n m , if ( n · · · n i · · · n m ) ∈ B bj and ∃ l such that i = j l , n i = b l , , otherwise . We need to check that the ratio ρ := ω ( n · · · b l + 1 · · · n m ) ω ( n · · · b l · · · n m ) , with l and b l fixed, approaches zero when the norm of ( n , . . . , b l , . . . , n m ) tends toinfinity. This was verified during the proof of the essential normality of L a (Ω ) . Thisfinishes the proof of our lemma for domains of type (1). The proof for domains of type(2) is completely similar, having the explicit formula for ω ( α, β, . . . ) (Proposition11.(b)) at hand. (cid:4) Remark 15.
With arguments similar to the ones in the proof of Lemma 14, one canshow that L a (Ω ) is p -essentially normal if and only if p is strictly larger than max (cid:8) m, p j ( m −
1) : j = 1 , . . . , m (cid:9) . The computations will be included in our forthcoming paper [37]. It is worth pointingout that it is a new phenomenon that the p -essential normality of the Bergman moduledepends not only on the dimension of the domain but also on its geometry. Thisphenomenon will also be explored in [37]. The geometry of the Hilbert modules associated to boxes.
Consider theHilbert module H bj associated to the box B bj . Set Ω , j := (cid:8) ( z , . . . , z m ) ∈ Ω : z j = · · · = z j q = 0 (cid:9) . Observe that Ω , j is an egg domain of type (1) inside C m − q . Consider the Hilbert space e H bj := M i =( i ,...,i q ) ∈ N q i ≤ b ,...,i q ≤ b q L a, P ql =1 il +1 pjl (Ω , j ) , and the map R bj : H bj → e H bj given by sending X ∈ H bj to Y = X Y i , Y i = π q i ! Q ql =1 p j l Q ql =1 Γ (cid:18) i l +1 p jl (cid:19)(cid:18)P ql =1 i l +1 p jl (cid:19) ! ∂ | i | X∂z i j · · · ∂z i q j q (cid:12)(cid:12)(cid:12)(cid:12) Ω , j ∈ L a, P ql =1 il +1 pjl (Ω , j ) . A straightforward computation with the orthonormal bases (Propositions 11 and 12) showsthat: R bj is an isometric isomorphism of Hilbert spaces. Now consider the trivial vector bundle E bj := C ( b +1) ··· ( b q +1) × Ω , j over Ω , j , together withthe standard frame e i , i = ( i , . . . , i q ) ∈ N q , i ≤ b , . . . , i q ≤ b q , and equip it with theHermitian structure h e i , e i ′ i ( z ) = − q X l =1 | z l | p jl P ql =1 il +1 pjl δ i , i ′ , z ∈ Ω , j , where δ is the Kronecker tensor. This way, e H bj can be identified with the Bergman space ofthe L -holomorphic sections of E bj . Under the isomorphism R bj , one can identify the Toeplitzalgebra generated by T j , b z i ∈ B ( H bj ) , i = 1 , . . . , m with the algebra generated by matrix-valuedToeplitz operators on the latter Bergman space of L -holomorphic sections of E bj .2.5. The construction of the resolution.
This section constructs the resolution in The-orem 5. Let the ideal I ⊆ A be generated by distinct monomials z α i , α i := ( α i , . . . , α mi ) ∈ N m , i = 1 , . . . , l. Let the complementary space C ( I ) ⊆ N m be the set of the exponents of those monomialswhich do not belong to I . Note that the set of monomials belonging to I is a basis of I asa complex vector space [35, Theorem 1.1.2]. Also note that a monomial u belongs to I ifand only if there is a monomial v such that u = vz α i for some i = 1 , . . . , l [35, Proposition1.1.5]. In other words, z n · · · z n m m ∈ C ( I ) if and only if for every i = 1 , . . . , l there exists s i ∈ { , . . . , m } such that n s i < α s i i . Consider the finite collection S ( α , . . . , α l ) := { , . . . , m } l of l -tuples s = ( s , . . . , s l ) of integers such that ≤ s i ≤ m for every i . Given s , let j s be theshuffle associated to the set { s , . . . , s l } . For each j ∈ j s , let b j be the minimum of all α s i i − , i = 1 , . . . , l , such that s i = j . Set b s := ( b j ) j ∈ j s . The following symbolic logic computation N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 13 shows that: C ( I ) is the union of boxes B b s j s , s ∈ S ( α , . . . , α l ) . z n · · · z n m m ∈ C ( I ) ↔ (cid:16) n < α ∨ · · · ∨ n m < α m (cid:17) ∧ · · · ∧ (cid:16) n < α l ∨ · · · ∨ n m < α ml (cid:17) ↔ _ ( s ,...,s l ) ∈{ ,...,m } l (cid:16) n s < α s ∧ · · · ∧ n s l < α s l l (cid:17) . The construction of modules A q . From now on, fix a finite collection of boxes(5) B b i j i , i = 1 , . . . , k such that their union equals C ( I ) . Given I ⊆ { , . . . , k } (note that we are using the symbol I for two purposes), let B b I j I := \ i ∈ I B b i j i denote the intersection of boxes B b i j i , i ∈ I . (Note that the intersections of boxes are againboxes.) Each box B b I j I has a corresponding Hilbert module H b I j I as introduced in Section 2.3.For each q = 1 , . . . , k , set A q := M I ∈ S q ( k ) H b I j I , A := L a (Ω ) . Note that each Hilbert space A q is equipped with a Hilbert A -module structure coming fromthe A -module structures on its direct summands. It is immediate from Lemma 14 that Proposition 16.
Each A q is essentially normal. The construction of maps Ψ q . Thinking of the elements of S q +1 ( k ) as the subsets I q +1 ⊆ { , . . . , k } of size q + 1 , define the maps f iq +1 : S q +1 ( k ) → S q ( k ) , i = 1 , . . . , q + 1 by setting f iq +1 ( I q +1 ) to be the subset of { , . . . , k } obtained by dropping the i -th smallestelement in I q +1 . The map Ψ q : A q → A q +1 is defined by sending X = P I q ∈ S q ( k ) X I q ∈ A q , X I q ∈ H b Iq j Iq to Y = P I q +1 ∈ S q +1 ( k ) Y I q +1 ∈ A q +1 , Y I q +1 ∈ H b Iq +1 j Iq +1 , given by (cid:16) Y I q +1 (cid:17) n = P q +1 i =1 ( − i − (cid:16) X f iq +1 ( I q +1 ) (cid:17) n , n ∈ B b Iq +1 j Iq +1 , , otherwise . Remark 17.
Similar to the explanation in Section 2.4, each Hilbert module A q , q = 1 , . . . , k can be identified with the Bergman space of the L -holomorphic sections of a Hermitianvector bundle on a disjoint union of subsets of Ω . Under this identification, the modulemorphisms Ψ q , q = 0 , . . . , k − can be realized as the restriction maps of jets of holomorphicsections to the subsets. Although this geometric picture is not used heavily in what follows,we believe that such an intuition will play a crucial role in the study of nonradical idealsbeyond monomials. 3. The Proof of Theorem 5
In this section we prove Theorem 5. Again, we develop the details for a domain Ω of type(1), and domains of type (2) can be treated similarly. The proof of Theorem 5.(a).
In this section we prove that the construction of Section2.5 is a resolution of Hilbert modules asserted in Theorem 5.(b). This is an adjustment ofthe proof of Theorem 4, first appeared in [21, Theorem 1.1].
Proposition 18.
Each Ψ q is a morphism of Hilbert A -modules.Proof. We first verify boundedness. For each X = P I q ∈ S q ( k ) X I q ∈ A q , X I q ∈ H b Iq j Iq wedefined Ψ q ( X ) = X I ′ q +1 Y I ′ q +1 , Y I ′ q +1 ∈ H b I ′ q +1 j I ′ q +1 , Y I ′ q +1 n = P q +1 i =1 ( − i − X f iq +1 ( I ′ q +1 ) n , n ∈ B b I ′ q +1 j I ′ q +1 , , otherwise . Therefore (cid:13)(cid:13) Ψ q ( X ) (cid:13)(cid:13) = X I ′ q +1 X n ∈ B b I ′ q +1 j I ′ q +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q +1 X i =1 ( − i − X f iq +1 (cid:16) I ′ q +1 (cid:17) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X I ′ q +1 X n ∈ B b I ′ q +1 j I ′ q +1 ( q + 1) (cid:12)(cid:12)(cid:12)(cid:12) X f iq +1 ( I ′ q +1 ) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ X I ′ q +1 X n ∈ B b Iq j Iq ( q + 1) (cid:12)(cid:12)(cid:12) X I q n (cid:12)(cid:12)(cid:12) , since B b I ′ q +1 j I ′ q +1 ⊆ B b Iq j Iq ≤ ( k − q )( q + 1) X I ∈ S q ( k ) X n ∈ B b Iq j Iq (cid:12)(cid:12)(cid:12) X I q n (cid:12)(cid:12)(cid:12) = ( k − q )( q + 1) k X k . The last inequality is because every I q is contained in at most k − q number of I ′ q +1 .Next, we prove that Ψ q commutes with the module actions. For each I ∈ S q ( k ) and X I ∈ H b I j I , we defined Ψ q ( X I ) = X ≤ s ≤ k, s / ∈ I ( − sign ( I,s ) Y I ∪{ s } , Y I ∪{ s } ∈ H b I ∪{ s } j I ∪{ s } , Y I ∪{ s } n = X I n , n ∈ B b I ∪{ s } j I ∪{ s } , , otherwise , where s is the α -th smallest number in I ∪ { s } , and sign ( I, s ) = α − .Each z p action on H b I j I is implemented by T j I , b I z p (cid:16) X I (cid:17) n ··· n p +1 ··· n m = q ω ( n ··· n p +1 ··· n m ) ω ( n ··· n m ) X In ··· n p ··· n m , p / ∈ j I , q ω ( n ··· n p +1 ··· n m ) ω ( n ··· n m ) X In ··· n p ··· n m , p = j s ∈ j I , n p + 1 ≤ b s , , otherwise . This shows that T j I , b I z p preserves the component H b I j I . Similarly, the z p action on H b I ∪{ s } j I ∪{ s } isrealized by T j I ∪{ s } , b I ∪{ s } z p (cid:16) Y I ∪{ s } (cid:17) n ··· n p +1 ··· n m = q ω ( n ··· n p +1 ··· n m ) ω ( n ··· n m ) Y I ∪{ s } n ··· n p ··· n m , p / ∈ j I , p = s, q ω ( n ··· n p +1 ··· n m ) ω ( n ··· n m ) Y I ∪{ s } n ··· n p ··· n m , p = j t ∈ j I ∪{ s } , n p + 1 ≤ b t , , otherwise . N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 15
It is straightforward to directly check that on each component H b I ∪{ s } j I ∪{ s } we have Ψ q (cid:18) T j I , b I z p (cid:16) X I (cid:17)(cid:19)! I ∪{ s } = T j I ∪{ s } , b I ∪{ s } z p (cid:18) Ψ q (cid:16) X I (cid:17) I ∪{ s } (cid:19) , and we are done. (cid:4) Proposition 19. I = ker(Ψ ) .Proof. If f ∈ I , then f has no nonzero component in any of the boxes B b s j s , s ∈ S ( α , . . . , α l ) ,hence f ∈ ker(Ψ ) . This shows that I ⊆ ker(Ψ ) . For the other direction, assume f = P n ∈ N m f n z n ∈ ker(Ψ ) . Since Ψ ( f ) = 0 , it follows that f n = 0 for every i = 1 , . . . , k and n ∈ B b i j i . Let f M , M = 1 , , . . . be the truncation of the Taylor expansion of f by requiring n , . . . , n m < M . f M has no component in the boxes B b j , . . . , B b k j k , hence f M ∈ I . Therefore, f = lim f M ∈ I . (cid:4) Proposition 20.
Im(Ψ q − ) ⊆ ker(Ψ q ) for every q = 1 , . . . , k .Proof. For each I ∈ S q − ( k ) and X I ∈ H b I j I , the image of X I under Ψ q − is of the form X ≤ s ≤ k, s / ∈ I ( − sign ( I,s ) Y I ∪{ s } , where Y I ∪{ s } ∈ H b I ∪{ s } j I ∪{ s } , s is the α -th smallest number in I ∪ { s } , sign ( I, s ) = α − , and thefunction Y I ∪{ s } is given by Y I ∪{ s } n = X I n , n ∈ B b I ∪{ s } j I ∪{ s } , , otherwise . Similarly, the image of Y I ∪{ s } under Ψ q is of the form X ≤ t ≤ k, t/ ∈ I ∪{ s } ( − sign ( I ∪{ s } ,t ) Z I ∪{ s,t } , where Z I ∪{ s,t } ∈ H b I ∪{ s,t } j I ∪{ s,t } , t is the β -th smallest number in I ∪ { s, t } , sign ( I ∪ { s } , t ) = β − ,and the function Z I ∪{ s,t } is given by Z I ∪{ s,t } n = Y I ∪{ s } n , n ∈ B b I ∪{ s,t } j I ∪{ s,t } , , otherwise . Therefore Ψ q (cid:18) Ψ q − (cid:16) X I (cid:17)(cid:19) = X ≤ s = t ≤ k, s,t/ ∈ I ( − sign ( I,s )+ sign ( I ∪{ s } ,t ) Z I ∪{ s,t } = X ≤ s Im(Ψ ) ⊇ ker(Ψ ) .(b) Im(Ψ q − ) ⊇ ker(Ψ q ) for every q = 1 , . . . , k .Proof. (a) Assume X := ( X , . . . , X p ) ∈ ker(Ψ ) . Consider ξ ∈ A given by ξ n := X s n , there is s such that n ∈ B b s j s , , otherwise . This is well-defined because Ψ ( ξ ) = 0 . Note that ξ ∈ A because k ξ k = k X k + · · · + k X p k .Clearly, Ψ ( ξ ) = X .(b) We apply induction on k . When k = 1 , the map Ψ : A → A is surjective becausecomputing with the orthonormal basis shows that A can be identified with a closed subspaceof A = L a (Ω ) , with Ψ being the corresponding orthogonal projection. Assuming Im(Ψ q − ) ⊇ ker(Ψ q ) , q = 1 , . . . , k, ≤ k < p, we prove the statement for k = p . The case q = 1 is proved in (a), so from now on we assume ≤ q ≤ k .Consider the following two collections of p − boxes: • The first p − boxes: B b j , . . . , B b p − j p − .Applying the construction in Section 2.5 to these boxes, we get the Hilbert modules A s together with the Hilbert module maps Ψ s : A s → A s +1 , s = 1 , . . . , p − . Set A p := { } and Ψ p − := 0 . • The intersection of the first p − boxes with the last one: B b p j p , . . . , B b p − p j p − p .Applying the construction in Section 2.5 to these boxes, we get the Hilbert modules A s together with the Hilbert module maps Ψ s : A s → A s +1 , s = 1 , . . . , p − . Set A p := { } and Ψ p − := 0 .By the induction assumption we have Im (cid:16) Ψ q − (cid:17) ⊇ ker (cid:16) Ψ q (cid:17) , Im (cid:16) Ψ q − (cid:17) ⊇ ker (cid:16) Ψ q (cid:17) , q = 1 , . . . , p − . Define a map Φ s : A s → A s by Φ s ( X I ) = Y I ∪{ p } , I ∈ S s ( p − , where Y I ∪{ p } denotes the component corresponding to the intersection of the boxes B b i p j i p , . . . , B b isp j isp , given by Y I ∪{ p } n := ( − t X I n , n ∈ B b I ∪{ p } j I ∪{ p } , , otherwise . Similar to the proof of Proposition 18, Φ s is an A -module map. Furthermore, we can easilycheck that • A q = A q ⊕ A q − for q = 2 , . . . , p . • Ψ q = Ψ q q Ψ q − ! for q = 2 , . . . , p − .These identifications are used below to prove that Im (Ψ q − ) ⊇ ker(Ψ q ) . We split the proofinto three cases.(1) q = 2 . N INDEX THEOREM FOR QUOTIENTS OF BERGMAN SPACES ON EGG DOMAINS 17 Suppose ( X , X ) ∈ A ⊕ A = A is in ker(Ψ ) . By the identification above for Ψ q , wehave Ψ ( X ) = 0 , Φ ( X ) + Ψ ( X ) = 0 . By the induction assumption, we have ker(Ψ ) ⊆ Im (Ψ ) , so there exists Y ∈ A suchthat Ψ ( Y ) = X . By Proposition 20, for the morphism Ψ • , we have (0 , 0) = Ψ (cid:0) Ψ ( Y , (cid:1) = Ψ (cid:0) Ψ ( Y ) , Φ ( Y ) (cid:1) = (cid:16) Ψ (cid:0) Ψ ( Y ) (cid:1) , Φ (cid:0) Ψ ( Y ) (cid:1) + Ψ (cid:0) Φ ( Y ) (cid:1)(cid:17) , Ψ ( Y ) = X , Ψ (cid:0) Ψ ( Y ) (cid:1) = 0= (cid:16) , Φ ( X ) + Ψ (cid:0) Φ ( Y ) (cid:1)(cid:17) . Therefore, Φ ( X ) + Ψ (cid:0) Φ ( Y ) (cid:1) = 0 . Setting X ′ := X − Φ ( Y ) , we have Ψ ( X ′ ) = Ψ ( X ) − Ψ (Φ ( Y )) = Ψ ( X ) + Φ ( X ) = 0 , because ( X , X ) = (Ψ ( X ) , Φ ( X ) + Ψ ( X )) . Since Ψ ( X ′ ) = 0 , it followsthat the following assignment is well-defined: ( Y ) n := ( X ′ ip ) n , n ∈ B b ip j ip for some i = 1 , ..., p − , , otherwise . Arguments similar to the proof of Proposition 21 show that this assignment gives Y ∈ H b p j p such that Ψ ( Y ) = X ′ . In summary, we have found ( Y , Y ) ∈ A = A ⊕ H b p j p whichsatisfies Ψ ( Y , Y ) = (cid:16) Ψ ( Y ) , Φ ( Y ) + Ψ ( Y ) (cid:17) = (cid:0) X , Φ ( Y ) + X ′ (cid:1) = ( X , X ) . (2) q = 3 , . . . , p − .Suppose ( X , X ) ∈ A q ⊕ A q − = A q is in ker(Ψ q ) . By the identification above for Ψ q ,we have Ψ q ( X ) = 0 , Φ q ( X ) + Ψ q − ( X ) = 0 . Since Im(Ψ q − ) ⊇ ker(Ψ q ) , there exists Y ∈ A q − such that X = Ψ q − ( Y ) . Since Ψ q (Ψ q − ( Y , , it follows that Φ q ( X ) + Ψ q − (Φ q − ( Y )) = 0 . Therefore Ψ q − (cid:0) X − Φ q − ( Y ) (cid:1) = 0 . Since Im (cid:16) Ψ q − (cid:17) ⊇ ker (cid:16) Ψ q − (cid:17) , there exists Y ∈ A q − such that Ψ q − ( Y ) = X − Φ q − ( Y ) . In summary, we have found ( Y , Y ) ∈ A q which satisfies Ψ q − ( Y , Y ) = (cid:16) Ψ q − ( Y ) , Φ q − ( Y ) + Ψ q − ( Y ) (cid:17) = ( X , X ) . (3) q = p .Since Ψ p − : A p − → A p − is surjective, it follows that Ψ p − : (cid:16) A p − = A p − ⊕ A p − (cid:17) → (cid:16) A p = A p − (cid:17) is also surjective.All cases are exhausted. (cid:4) The proof of Theorem 5.(b). To deduce the index formula in Theorem 5.(b) fromthe resolution in Theorem 5.(a), we need the following proposition. Proposition 22. Let → M → M → M → be a short exact sequence of essentiallynormal Hilbert A -modules and Hilbert A -module maps between them. Suppose that the es-sential spectra of M i , i = 1 , , is contained in Ω , and let α i : C (Ω ) → Q ( M i ) be the ∗ -representation of C (Ω ) on the Calkin algebra Q ( M i ) = B ( M i ) / K ( M i ) induced by theessential normality of M i .(a) There are co-isometries U : M → M and V : M → M such that U V ∗ = 0 = V U ∗ , U ∗ U + V ∗ V = 1 , and they commute with A -module structures up to compact operators in thesense that [ U ] α [ U ] ∗ = α and [ V ] α [ V ] ∗ = α , where α i ( p ) = [ T ip ] ∈ Q ( M i ) , p ∈ A is theequivalence class of the multiplication operator T ip ∈ B ( M i ) .(b) We have [ α ] = [ α ] + [ α ] in K (cid:0) σ e (cid:1) , where [ α ] and [ α ] are identified as classes in K ( σ e ) by the co-isometries U and V .Proof. (a) [21, Proposition 3.8].(b) Set σ ie := σ e ( M i ) . The representation α i factors through ∗ -monomorphism C ( σ ie ) → Q ( M i ) . We have α = [ U ] α [ U ] ∗ by (a). The composition of [ U ] α [ U ] ∗ with α − is a ∗ -homomorphism C ( σ e ) → C ( σ e ) , and this induces a natural map σ e → σ e . Similarly, we havea natural map σ e → σ e . Therefore, α and α induce classes [ α ] and [ α ] in K (cid:0) σ e (cid:1) by thefunctoriality of K . Putting all equations U U ∗ = 1 = V V ∗ , U V ∗ = 0 = V U ∗ , U ∗ U + V ∗ V = 1 , [ U ] α [ U ] ∗ = α , [ V ] α [ V ] ∗ = α , together, we deduce that [ α ] = [ α ] + [ α ] . (cid:4) The proof of Theorem 5.(b). The idea is to decompose the resolution of I in Theorem 5.(a)into short exact sequences and then apply Proposition 22.(b). The details follow. Consider A − q := Im(Ψ q − ) = ker(Ψ q ) as a closed subspace of A q . Note that A − k = A k because Ψ k − is surjective. The morphism Ψ q : A q → A q +1 of Hilbert modules induces the short exactsequence(6) → A − q ֒ → A q Ψ q −−→ A − q +1 → , q = 1 , . . . , k − , which, according to Propositions 13 and 16, implies that A − q is essentially normal. Set σ qe := σ e ( A q ) , and let α q (respectively α − q ) be the ∗ -monomorphism C ( σ qe ) → Q ( A q ) (respectively C ( σ q − e ) → Q ( A − q ) ) induced by essential normality. Note that the essential spectra of all termsin exact sequence (6) is contained in Ω . By Proposition 22.(b), we have [ α q ] = [ α − q ] + [ α − q +1 ] in K (cid:0) σ qe (cid:1) for every q = 1 , . . . , k − . These formulas for q = k − and q = k − give [ α k − ] = [ α − k − ] + [ α k ] ∈ K (cid:16) σ k − e (cid:17) , [ α k − ] = [ α − k − ] + [ α − k − ] ∈ K (cid:16) σ k − e (cid:17) . Pushing forward these equations into K (cid:16) σ k − e ∪ σ k − e (cid:17) by inclusion maps σ k − e , σ k − e ֒ → σ k − e ∪ σ k − e , gives [ α k − ] + [ α − k − ] = [ α k ] + [ α k − ] . Continuing this argument, we have(7) [ α − ] = [ α ] − [ α ] + . . . + ( − k − [ α k ] in K ( σ e ∪ · · · ∪ σ ke ) . 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