We establish the existence and uniqueness of finite free resolutions and their attendant Betti numbers for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example,...

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers OL is free as an OKG-module. Let Cp denote the cyclic group of prime order p. We show that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Sp...

In type A, the q, t-Fuß–Catalan numbers can be defined as the bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with nonnegative integer coefficients. We prove the conjectures for...

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers OL is free as an OKG-module. Let Cp denote the cyclic group of prime order p. We show that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilber...

We determine the class of Hilbert series H so that if M is a finitely generated zero-dimensional R-graded module with the strong Lefschetz property, then M ⊗k k[y]/(y ) has the strong Lefschetz property for y an indeterminate and all positive integers m if and only if the Hilbert series of M is in H. Given two finite graded R-modules M and N with the strong Lefschetz property, we determine suff...

We provide a new characterization of the notion of sheaf on an étale groupoid G, in terms of a particular kind of Hilbert module on the quantale O(G) of the groupoid. All the theory is developed in the context of the more general class of quantales known as stable quantal frames, of which examples are easy to construct because their category is algebraic. The homomorphisms of our Hilbert module...

The classical identification of the predual of B(H) (the algebra of all bounded operators on a Hilbert space H) with the projective operator space tensor product H⊗̂H is extended to the context of Hilbert modules over commutative von Neumann algebras. Each bounded module homomorphism b between Hilbert modules over a general C∗-algebra is shown to be completely bounded with ‖b‖cb = ‖b‖. The so ca...

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K, the ring of integers OL is free as an OKG-module. If OL is free over the associated order AL/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-S...

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to soq(3), it acts on the q-Euclidean space that becomes a soq(3)-module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on C∞ functions on R. On...

The problem of explicitly finding a free resolution, minimal in some suitable sense, of a module over a polynomial ring is solved in principle by the algorithm of Hilbert [H]. However, this algorithm is of enormous computational difficulty. If the module happens to be finite dimensional over the ground field, and if the module structure is given by specifying the commuting linear transformation...