Exercise 8, § 4

Let E be a set. Put $\mathscr{M}(E) = \mathbf{N}^{(E)}$ and denote the composition in $\mathscr{M}(E)$ multiplicatively. An ideal of $\mathscr{M}(E)$ is a subset I of $\mathscr{M}(E)$ such that $I \cdot \mathscr{M}(E) \subset I$. A staircase of $\mathscr{M}(E)$ is the complement of an ideal. We denote by $d : \mathscr{M}(E) \to \mathbf{N}$ the unique homomorphism of monoids such that $d(x) = 1$, for all $x \in E$. Let $A \subset \mathscr{M}(E)$. For all $l \in \mathbf{N}$, we denote by $A_l$ the set $A \cap d^{-1}(l)$. We denote moreover $\delta A$ the set of $m \in \mathscr{M}(E)$ such that, for every element $x$ of $E$ which divides $m$, we have $m/x \in A$ and $dA$ the set of $m$ such that there exists an $x \in E$ with $mx \in A$. a) Prove that for every subset $A$ of $\mathscr{M}(E)$, one has $d\delta A \subset A$ and $A \subset \delta dA$; if $A$ and $B$ are two subsets of $\mathscr{M}(E)$, the relations $dA \subset B$ and $A \subset \delta B$ are equivalent. Prove that the following three properties of a subset $\Delta$ of $\mathscr{M}(E)$ are equivalent : (i) $\Delta$ is a staircase ; (ii) $\Delta \supset d\Delta$; (iii) $\delta \Delta \supset \Delta$. b) Let F be a subset of E and identify $\mathscr{M}(F)$ with its canonical image in $\mathscr{M}(E)$. Prove that a subset of $\mathscr{M}(F)$ is a staircase of $\mathscr{M}(F)$ if and only if it is a staircase of $\mathscr{M}(E)$. Prove that, for a staircase $\Delta$ of $\mathscr{M}(E)$, the following conditions are equivalent : (i) $\Delta_l$ is finite for every $l \in \mathbf{N}$; (ii) $\Delta_1$ is finite ; (iii) there exists a finite subset F of E such that $\Delta \subset \mathscr{M}(F)$. Such a staircase is said to be of finite type. c) Suppose from now on that $E = \mathbf{N}$ and put $\mathscr{M}(E) = \mathscr{M}$. Let k be a field. We denote by R the graded algebra $K^{(\mathscr{M})} = K[(X_i)_{i \in \mathbf{N}}]$. We identify $\mathscr{M}$ with the set of monomials in $K[(X_i)_{i \in \mathbf{N}}]$ and, for every subset A of $\mathscr{M}$, we denote by $k(A)$ the vector space subspace of R generated by A. Let I be an ideal of $\mathscr{M}$ and $\Delta$ the complementary staircase. Then $k(I)$ is an ideal of R and $k(\Delta)$ is a supplementary subspace of $k(I)$. Let $J \subset R$ be a graded ideal of R. Prove that there exists a staircase $\Delta \subset \mathscr{M}$ such that $k(\Delta)$ is a supplementary subspace of J. (Put on $\mathscr{M}$ the inverse lexicographic order for which one has $X_1^{a_1} ... X_n^{a_n} < X_1^{b_1} ... X_n^{b_n}$ if there exists $0 \leq p \leq n$ such that $a_j = b_j$ for $j > p$, and $a_p < b_p$. Define by recurrence $m_1, ..., m_p, ...$ in $\mathscr{M}$ in the following way : for every $p$, $m_p$ is the smallest monomial of $\mathscr{M}$ linearly independent of $m_1, ..., m_{p-1}$ modulo J. Let $\Delta$ be the set of the $m_i$. Prove that $\Delta$ is a staircase.) Prove that such a staircase is of finite type if and only if the algebra $R/J$ is of finite type over k.