On the flatness of the Bergman space as a module over the Hardy algebra

Abstract

Let $A^2$ be the Bergman space of the open unit disc $\mathbb{D}$ in the complex plane consisting of holomorphic functions that are square integrable with respect to the area measure in the disc, and $H^{\infty }$ be the Hardy algebra of bounded and holomorphic functions on $\mathbb{D}$. With pointwise operations, $A^2$ is an $H^{\infty }$-module. It is shown that the $H^{\infty }$-module $A^2$ is not flat. It is also shown that there exists a maximal ideal $\mathfrak{m}$ in $H^{\infty }$ such that $\mathfrak{m}{A}^2=A^2$.

20 pages. Added a remark (Remark 2.11) contrasting the flatness versus nonflatness results for $H^2$ and $A^2$, respectively