Different Hilbert module structures over the Segal-Bargmann space \(F^2\) of Gaussian square-integrable entire functions are defined. These depend on classes of diagonal operators acting on \(F^2\) with respect to decompositions into homogeneous subspaces. We first consider graded principal submodules generated by a single homogeneous polynomial. Generalizing results due to K. Guo and K. Wang, and under suitable conditions on the eigenvalue sequence defining the module structure, we prove p-essential normality for specific values of p. Starting from a specific commuting tuple of Toeplitz operators with homogeneous symbols in \(F^2\) , we assign a decreasing scale of quotient modules to dilation-invariant subsets \(\Omega \subset \mathbb {C}^n\) . Naturally, the question of essential or p-essential normality of such quotient modules arises.
