关于伯格曼空间作为哈代代数上的模的平坦性
On the flatness of the Bergman space as a module over the Hardy algebra
AI总结：
研究复平面开单位圆盘上伯格曼空间作为哈代代数模的平坦性问题，通过特定运算证明其非平坦，且存在极大理想使\(\mathfrak{m}A^2 = A^2\)。
AI中文摘要：
设\(A^2\)为复平面上开单位圆盘\(\mathbb{D}\)的伯格曼空间，由关于圆盘面积测度平方可积的全纯函数组成，\(H^\infty\)为\(\mathbb{D}\)上有界全纯函数的哈代代数。通过逐点运算，\(A^2\)是一个\(H^\infty\) - 模。证明了\(H^\infty\) - 模\(A^2\)不是平坦的，还证明了在\(H^\infty\)中存在一个极大理想\(\mathfrak{m}\)使得\(\mathfrak{m}A^2 = A^2\)。
英文摘要：
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 $\mathfrak{m}A^2=A^2$.
