Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.     

Star-generating vectors of Rudin's quotient modules

The purpose of this paper is to study a class of quotient modules of the Hardy module H²(Dⁿ)$H^2({\mathbb{D}}^n)$. Along with the two variables quotient modules introduced by W. Rudin, we introduce and study a large class of quotient modules, namely Rudin's quotient modules of H²(Dⁿ)$H^2({\mathbb{D}}^n)$. By exploiting the structure of minimal representations we obtain an explicit co-rank formula for Rudin's quotient modules.     

Algebraic modules and the Auslander-Reiten quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander-Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander-Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.     

Algebraic subgroup lattices of topological groups

It is shown that the lattice of all closed subgroups of a compact topological group and the lattice of all connected closed subgroups of a pro-Lie group are algebraic, even arithmetic, if they are equipped with the order opposite to the natural one. The compact elements form ideals in these lattices and are explicitly determined. In the course of the proof the question is treated whether forming lattices of closed subgroups of topological groups commutes with projective limits.Presented by Laszlo Fuchs.     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk.     

Stanley depth of multigraded modules

The Stanley's Conjecture on Cohen–Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture holds in 5 variables.     

Beurling type quotient modules over the bidisk and boundary representations

This paper mainly concerns Beurling type quotient modules of H²(D²) over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most (1,1). Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of C^∗(Sz,Sw) is a boundary representation of B(Sz,Sw) in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of C^∗-algebras, Acta Math. 123 (1969) 141-224; W. Arveson, Subalgebras of C^∗-algebras II, Acta Math. 128 (1972) 271-308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk.     

Essential normality of quasi-homogeneous quotient modules over the polydisc

In this paper, we characterize the essential normality of quasi-homogeneous quotient Hardy modules over the polydisc by giving a complete criterion in terms of partially maximal ideals. Then we generalize this result to the weighted Bergman modules.     

On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.     

Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks

In this paper,we give a complete characterization for the essential normality of quasi-homogenous quotient modules of the Hardy modules H2 (D2).Also,we show that if d 3,then all the principle homogenous quotient modules of H 2 (Dd) are not essentially normal.     

A new depth and the annihilation of local cohomology modules

Let J I be two proper ideals of a commutative Noetherian ring and M a finitely generated module. Strong relative depth is defined and characterized. It is proved that this depth is just the maximum integer n such that can be annihilated by some power of J for all i ≤ n. It turns out that the local-global principle for the annihilation of local cohomology modules can be formulated as a natural property of this new depth. Received: 28 July 2006     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk. This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program