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Let $ [n] = \lbrace 1,2,\dots,n \rbrace $, and $ F \subset [n] $. We denote by $ P_F \subset K[X_1,\dots,X_n] $ the monomial ideal generated by the variables $ X_i $ with $ i \in F $. Given an integer $ d \in [n] $, let $$I = \bigcap_{F, \vert F \vert =d} P_F,$$ where $P_F =(x_i, i \in F)$. Describe the set $G(I)$ of the generators of $I$.

I know that $ \lbrace \mbox{lcm}(u,v) : u \in G(I), v \in G(J) \rbrace $ is a set of generators of $ I \cap J $. But this case I don't understand.

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I'm a little confused, what is $J$? – froggie Nov 30 '12 at 15:23
up vote 3 down vote accepted

$I$ is generated by the monomials that are products of $n-d+1$ (distinct) indeterminates.

Take $\Delta_m=\{G\subset[1,n]:|G|\le m\}$. This is a simplicial complex on $[1,n]$ whose facets are the subsets $F$ of $[1,n]$ with $|F|=m$. Define $I_{\Delta_m}$ as being the ideal of $K[X_1,\dots,X_n]$ generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $\{i_1,\dots,i_s\}\notin\Delta_m$. From Bruns and Herzog, Cohen-Macaulay Rings, Theorem 5.1.4, one knows that $$I_{\Delta_m}=\bigcap P_F,\;\;\;\; (*)$$ where the intersection is taken over all facets $F$ of $\Delta_m$, and $P_F$ denotes the ideal generated by all $X_i$ with $i\notin F$. In our case $\{i_1,\dots,i_s\}\notin\Delta_m$ means that $s\ge m+1$ and then $I_{\Delta_m}$ is generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $s\ge m+1$. (In fact one can say that $I_{\Delta_m}$ is generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $s=m+1$.) On the other side, $P_F$ is generated by all $X_i$ with $i\notin F$, that is, $P_F$ is generated by all $X_i$ with $i\in [1,n]-F$. Looking back to $(*)$ we can deduce that the right hand side coincides with $\bigcap P_G$ where $G$ runs over all the subsets of $[1,n]$ with $|G|=n-m$. Now take $m=n-d$ and get the conclusion.

Remark. This is a simpler answer which I've found later: one knows that a finite intersection of monomial ideals is a monomial ideal, therefore $I=\bigcap_{|F|=d} P_F$ is a monomial ideal. A monomial $X_{i_1}^{a_{i_1}}\cdots X_{i_s}^{a_{i_s}}$, denoted by $X_G^a$, where $G=\{i_1,\dots, i_s\}$ and $a=(a_{i_1},\dots,a_{i_s})$, belongs to $P_F$ iff $G\cap F\neq\emptyset$, so $I$ contains all the monomials $X_G^a$ with $G\cap F\neq\emptyset$ for all $F\subset[1,n]$ with $|F|=d$. This shows that $|G|\ge n-d+1$ and get the (same) conclusion.

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