Exercise 12, § 4

Let $H : \mathbf{N} \to \mathbf{N}$ be a mapping and $k$ a field. We propose to demonstrate the equivalence of the following properties: (i) $H$ is a Macaulay function (p. 90, exerc. 7). (ii) There exists an escalier (p. 92, exerc. 8) of finite type $\Delta \subset \mathscr{M}$ such that for every $l \in \mathbf{N}$, one has $H(l) = \mathrm{Card}(\Delta_l)$. (iii) There exists a graded $k$-algebra $A$, of type $\mathbf{N}$, generated by a finite number of its elements of degree 1, such that for every $l$, one has $H(l) = \dim_k A_l$. a) To prove (ii) $\Leftrightarrow$ (iii), we shall use exerc. 8, c). b) Let us prove (i) $\Rightarrow$ (ii). Let $H$ be a Macaulay function. For every $l$, let $\Delta_l \subset \mathscr{M}_l$ be the initial subset such that $\mathrm{Card}(\Delta_l) = H(l)$ and put $\Delta = \bigcup_l \Delta_l$. To show that $\Delta$ is an escalier, we shall use exerc. 9, c). c) Let us prove (ii) $\Rightarrow$ (i). Let $\Delta$ be an escalier. To show that $H : l \mapsto \mathrm{Card}(\Delta_l)$ is a Macaulay function, we shall use Macaulay's theorem (MC1) (exerc. 11) and exerc. 9, c) $^1$.