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Exercise 111.28.7. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring of dimension $1$. Assume also that the embedding dimension of $A$ is $2$, i.e., assume that

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

Notation: $f(n) = \dim _ k {\mathfrak m}^ n/{\mathfrak m}^{n + 1}$. Pick generators $x, y \in {\mathfrak m}$ and write $Gr_{\mathfrak m}(A) = k[\bar x, \bar y]/I$ for some homogeneous ideal $I$.

  1. Show that there exists a homogeneous element $F\in k[\bar x, \bar y]$ such that $I \subset (F)$ with equality in all sufficiently high degrees.

  2. Show that $f(n) \leq n + 1$.

  3. Show that if $f(n) < n + 1$ then $n \geq \deg (F)$.

  4. Show that if $f(n) < n + 1$, then $f(n + 1) \leq f(n)$.

  5. Show that $f(n) = \deg (F)$ for all $n >> 0$.

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