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Exercise 111.28.4. Local rings of embedding dimension $1$. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring of embedding dimension $1$, i.e.,

\[ \dim _ k {\mathfrak m}/{\mathfrak m}^2 = 1. \]

Show that the function $f(n) = \dim _ k {\mathfrak m}^ n/{\mathfrak m}^{n + 1}$ is either constant with value $1$, or its values are

\[ 1, 1, \ldots , 1, 0, 0, 0, 0, 0, \ldots \]

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